EMS Monographs in Mathematics - 213.230.96.51:8090

EMS Monographs in Mathematics

Edited byIvar Ekeland (Pacific Institute, Vancouver, Canada)Gerard van der Geer (University of Amsterdam, The Netherlands)Helmut Hofer (Institute for Advanced Study, Princeton, USA)Thomas Kappeler (University of Zürich, Switzerland)

EMS Monographs in Mathematics is a book series aimed at mathematicians and scientists. It publishes research monographs and graduate level textbooks from all fields of mathematics. The individual volumes are intended to give a reasonably comprehensive and selfcontained account of their particular subject. They present mathematical results that are new or have not been accessible previously in the literature.

Previously published in this series:

Richard Arratia, A.D. Barbour and Simon Tavaré, Logarithmic Combinatorial Structures: A Probabilistic ApproachDemetrios Christodoulou, The Formation of Shocks in 3-Dimensional FluidsSergei Buyalo and Viktor Schroeder, Elements of Asymptotic GeometryDemetrios Christodoulou, The Formation of Black Holes in General RelativityJoachim Krieger and Wilhelm Schlag, Concentration Compactness for Critical Wave Maps

Jean-Pierre BourguignonOussama HijaziJean-Louis MilhoratAndrei MoroianuSergiu Moroianu

A Spinorial Approach to Riemannian and Conformal Geometry

Oussama HijaziInstitut Élie Cartan de Lorraine (IÉCL)Université de Lorraine, NancyB. P. 23954506 Vandoeuvre-Lès-Nancy Cedex, France

[email protected]

Andrei MoroianuUniversité de Versailles-St QuentinLaboratoire de MathématiquesUMR 8100 du CNRS45 avenue des États-Unis78035 Versailles, France

[email protected]

Authors:

Jean-Pierre BourguignonIHÉSLe Bois-Marie35 route de Chartres91440 Bures-sur-Yvette, France

[email protected]

Jean-Louis MilhoratDépartement de Mathématiques,Université de Nantes2, rue de la Houssinière, B.P. 9220844322 Nantes, France

[email protected]

Sergiu MoroianuInstitutul de Matematica al Academiei RomâneCalea Grivit‚ei 21010702 Bucures‚ti, Romania

[email protected]

2010 Mathematics Subject Classification: Primary: 53C27, 53A30, Secondary: 53C26, 53C55, 53C80, 17B10, 34L40, 35S05

Key words: Dirac operator, Penrose operator, Spin geometry, Spinc geometry, conformal geometry, Kähler manifolds, Quaternion-Kähler manifolds, Weyl geometry, representation theory, Killing spinors, eigenvalues

ISBN 978-3-03719-136-1

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2015 European Mathematical Society

Contact address:

European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich, Switzerland

Phone: +41 (0)44 632 34 36 | Email: [email protected] | Homepage: www.ems-ph.org

Typeset using the authors’ TEX files: M. Zunino, Stuttgart, GermanyPrinting and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany∞ Printed on acid free paper9 8 7 6 5 4 3 2 1

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I Basic spinorial material . . . . . . . . . . . . . . . . . . . . . . . . 9

1 Algebraic aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 De�nitions . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.2 Classi�cation of Clifford algebras . . . . . . . . . . . . . . 18

1.2 Spin groups and their representations . . . . . . . . . . . . . . . . 20

1.2.1 Spin groups . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.2 Representations of spin groups . . . . . . . . . . . . . . . 28

1.2.3 Real and quaternionic structures . . . . . . . . . . . . . . 36

2 Geometrical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 Spinorial structures . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1 Spin structures and spinorial metrics . . . . . . . . . . . . . 39

2.1.2 Spinorial connections and curvatures . . . . . . . . . . . . 42

2.2 Spinc and conformal structures . . . . . . . . . . . . . . . . . . . . 45

2.2.1 Spinc structures . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.2 Weyl structures . . . . . . . . . . . . . . . . . . . . . . . 49

2.2.3 Spin and Spinc conformal manifolds . . . . . . . . . . . . . 52

2.3 Natural operators on spinors . . . . . . . . . . . . . . . . . . . . . 56

2.3.1 General algebraic setting . . . . . . . . . . . . . . . . . . 57

2.3.2 First-order differential operators . . . . . . . . . . . . . . . 59

2.3.3 Basic differential operators on spinor �elds . . . . . . . . . 61

2.3.4 The Dirac operator: basic properties and examples . . . . . 63

2.3.5 Conformal covariance of the Dirac and Penrose operators . . 69

2.3.6 Conformally covariant powers of the Dirac operator . . . . . 72

vi Contents

2.4 Spinors in classical geometrical contexts . . . . . . . . . . . . . . . 73

2.4.1 Restrictions of spinors to hypersurfaces . . . . . . . . . . . 73

2.4.2 Spinors on warped products . . . . . . . . . . . . . . . . . 75

2.4.3 Spinors on Riemannian submersions . . . . . . . . . . . . 76

2.5 The Schrödinger–Lichnerowicz formula . . . . . . . . . . . . . . . 81

3 Topological aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.1 Topological aspects of spin structures . . . . . . . . . . . . . . . . 85

3.1.1 Cech cohomology and principal bundles . . . . . . . . . . . 86

3.1.2 Lifting principal bundles via central extensions . . . . . . . 88

3.1.3 Stiefel–Whitney classes . . . . . . . . . . . . . . . . . . . 90

3.2 Topological classi�cation of Spinc structures . . . . . . . . . . . . . 91

3.3 Spin structures in low dimensions . . . . . . . . . . . . . . . . . . 93

3.3.1 Dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.3.2 Dimension 2 . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3.3 Dimensions 3 and 4 . . . . . . . . . . . . . . . . . . . . . 95

3.4 Examples of obstructed manifolds . . . . . . . . . . . . . . . . . . 96

4 Analytical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Pseudo-differential calculus . . . . . . . . . . . . . . . . . . . . . 101

4.2.1 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.2 Asymptotic summation . . . . . . . . . . . . . . . . . . . 102

4.3 Pseudo-differential operators . . . . . . . . . . . . . . . . . . . . 104

4.4 Composition of pseudo-differential operators . . . . . . . . . . . . 107

4.5 Action of diffeomorphisms on pseudo-differential operators . . . . . 108

4.6 Pseudo-differential operators on vector bundles . . . . . . . . . . . 109

4.7 Elliptic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.8 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.9 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.10 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.11 Eigenvalues of self-adjoint elliptic operators . . . . . . . . . . . . . 121

Part II Lowest eigenvalues of the Dirac operator on closed spin manifolds . . 125

5 Lower eigenvalue bounds on Riemannian closed spin manifolds . . . . . . 127

5.1 The Lichnerowicz theorem . . . . . . . . . . . . . . . . . . . . . . 127

5.2 The Friedrich inequality . . . . . . . . . . . . . . . . . . . . . . . 129

5.3 Special spinor �elds . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 The Hijazi inequality . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.5 The action of harmonic forms on Killing spinors . . . . . . . . . . . 140

Contents vii

5.6 Other estimates of the Dirac spectrum . . . . . . . . . . . . . . . . 142

5.6.1 Moroianu–Ornea’s estimate . . . . . . . . . . . . . . . . . 142

5.7 Further developments . . . . . . . . . . . . . . . . . . . . . . . . 146

5.7.1 The energy–momentum tensor . . . . . . . . . . . . . . . 147

5.7.2 Witten’s proof of the positive mass theorem and

applications . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.7.3 Further applications . . . . . . . . . . . . . . . . . . . . . 158

6 Lower eigenvalue bounds on Kähler manifolds . . . . . . . . . . . . . . . 161

6.1 Kählerian spinor bundle decomposition . . . . . . . . . . . . . . . 161

6.2 The canonical line bundle . . . . . . . . . . . . . . . . . . . . . . 164

6.3 Kählerian twistor operators . . . . . . . . . . . . . . . . . . . . . 166

6.4 Proof of Kirchberg’s inequalities . . . . . . . . . . . . . . . . . . . 170

6.5 The limiting case . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7 Lower eigenvalue bounds on quaternion-Kähler manifolds . . . . . . . . . 175

7.1 The geometry of quaternion-Kähler manifolds . . . . . . . . . . . . 176

7.2 Quaternion-Kähler spinor bundle decomposition . . . . . . . . . . 179

7.3 The main estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.4 The limiting case . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.5 A systematic approach . . . . . . . . . . . . . . . . . . . . . . . . 202

7.5.1 General Weitzenböck formulas . . . . . . . . . . . . . . . 202

7.5.2 Application to quaternion-Kähler manifolds . . . . . . . . . 206

7.5.3 Proof of the estimate . . . . . . . . . . . . . . . . . . . . 210

Part III Special spinor �eld and geometries . . . . . . . . . . . . . . . . . . 213

8 Special spinors on Riemannian manifolds . . . . . . . . . . . . . . . . . 215

8.1 Parallel spinors on spin and Spinc manifolds . . . . . . . . . . . . . 215

8.1.1 Parallel spinors on spin manifolds . . . . . . . . . . . . . . 215

8.1.2 Parallel spinors on Spinc manifolds . . . . . . . . . . . . . 219

8.2 Special holonomies and relations to warped products . . . . . . . . 223

8.2.1 Sasakian structures . . . . . . . . . . . . . . . . . . . . . 223

8.2.2 3-Sasakian structures . . . . . . . . . . . . . . . . . . . . 225

8.2.3 The exceptional group G2 . . . . . . . . . . . . . . . . . . 227

8.2.4 Nearly Kähler manifolds . . . . . . . . . . . . . . . . . . . 229

8.2.5 The group Spin7 . . . . . . . . . . . . . . . . . . . . . . . 233

8.3 Classi�cation of manifolds admitting real Killing spinors . . . . . . . 236

8.4 Detecting model spaces by Killing spinors . . . . . . . . . . . . . . 239

8.5 Generalized Killing spinors . . . . . . . . . . . . . . . . . . . . . 241

8.6 The Cauchy problem for Einstein metrics . . . . . . . . . . . . . . 243

viii Contents

9 Special spinors on conformal manifolds . . . . . . . . . . . . . . . . . . 251

9.1 The conformal Schrödinger–Lichnerowicz formula . . . . . . . . . . 251

9.2 Parallel spinors with respect to Weyl structures . . . . . . . . . . . . 254

9.2.1 Parallel conformal spinors on Riemann surfaces . . . . . . . 254

9.2.2 The non-compact case . . . . . . . . . . . . . . . . . . . . 256

9.2.3 The compact case . . . . . . . . . . . . . . . . . . . . . . 262

9.3 A conformal proof of the Hijazi inequality . . . . . . . . . . . . . . 263

10 Special spinors on Kähler manifolds . . . . . . . . . . . . . . . . . . . . 265

10.1 An introduction to the twistor correspondence . . . . . . . . . . . . 265

10.1.1 Quaternion-Kähler manifolds . . . . . . . . . . . . . . . . 265

10.1.2 3-Sasakian structures . . . . . . . . . . . . . . . . . . . . 267

10.1.3 The twistor space of a quaternion-Kähler manifold . . . . . 268

10.2 Kählerian Killing spinors . . . . . . . . . . . . . . . . . . . . . . . 269

10.3 Complex contact structures on positive Kähler–Einstein manifolds . . 272

10.4 The limiting case of Kirchberg’s inequalities . . . . . . . . . . . . . 275

11 Special spinors on quaternion-Kähler manifolds . . . . . . . . . . . . . . 279

11.1 The canonical 3-Sasakian SO3-principal bundle . . . . . . . . . . . 280

11.2 The Dirac operator acting on projectable spinors . . . . . . . . . . . 283

11.3 Characterization of the limiting case . . . . . . . . . . . . . . . . . 288

11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Part IV Dirac spectra of model spaces . . . . . . . . . . . . . . . . . . . . 301

12 A brief survey on representation theory of compact groups . . . . . . . . . 305

12.1 Reduction of the problem to the study of irreducible representations . 305

12.2 Reduction of the problem to the study of irreducible representations

of a maximal torus . . . . . . . . . . . . . . . . . . . . . . . . . . 315

12.3 Characterization of irreducible representations by means of dominant

weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

12.3.1 Restriction to semi-simple simply connected groups . . . . . 321

12.3.2 Roots and their properties . . . . . . . . . . . . . . . . . . 322

12.3.3 Irreducible representations of the group SU2 . . . . . . . . 325

12.3.4 Proof of the fundamental properties of roots . . . . . . . . . 327

12.3.5 Dominant weights . . . . . . . . . . . . . . . . . . . . . . 334

12.3.6 The Weyl formulas . . . . . . . . . . . . . . . . . . . . . 339

Contents ix

12.4 Application: irreducible representations of the classical groups SUn,

Spinn, and Spn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

12.4.1 Irreducible representations of the groups SUn

and Un, n � 3 . . . . . . . . . . . . . . . . . . . . . . . . 343

12.4.2 Irreducible representations of the groups Spinn

and SOn, n � 3 . . . . . . . . . . . . . . . . . . . . . . . 348

12.4.3 Irreducible representations of the group Spn . . . . . . . . . 361

13 Symmetric space structure of model spaces . . . . . . . . . . . . . . . . . 369

13.1 Symmetric space structure of spheres . . . . . . . . . . . . . . . . . 372

13.2 Symmetric space structure of the complex projective space . . . . . . 374

13.3 Symmetric space structure of the quaternionic projective space . . . . 375

14 Riemannian geometry of model spaces . . . . . . . . . . . . . . . . . . . 379

14.1 The Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . 381

14.2 Spin structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

14.3 Spinor bundles on symmetric spaces . . . . . . . . . . . . . . . . . 390

14.4 The Dirac operator on symmetric spaces . . . . . . . . . . . . . . . 392

15 Explicit computations of the Dirac spectrum . . . . . . . . . . . . . . . . 395

15.1 The general procedure . . . . . . . . . . . . . . . . . . . . . . . . 395

15.2 Spectrum of the Dirac operator on spheres . . . . . . . . . . . . . . 407

15.3 Spectrum of the Dirac operator on the complex projective space . . . 409

15.4 Spectrum of the Dirac operator on the quaternionic projective space . 412

15.5 Other examples of spectra . . . . . . . . . . . . . . . . . . . . . . 414

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

Introduction

Spin Geometry is the hidden facet of Riemannian Geometry. It arises from therepresentation theory of the special orthogonal group SOn, more precisely, fromthe spinor representation, a certain representation of the Lie algebra son which isnot a representation of SOn. Spinors can always be constructed locally on a givenRiemannian manifold, but globally there are topological obstructions for their exis-tence.

Spin Geometry lies therefore at the cross-road of several sub�elds of modernMathematics. Algebra, Geometry, Topology, and Analysis are subtly interwoven inthe theory of spinors, both in their de�nition and in their applications. Spinors havealso greatly in�uenced Theoretical Physics, which is nowadays one of themain driv-ing forces fostering their formidable development. The Noncommutative Geometryof Alain Connes has at its core the Dirac operator on spinors. The same Diracoperator is at the heart of the Atiyah–Singer index formula for elliptic operators oncompact manifolds, linking in a spectacular way the topology of a manifold to thespace of solutions to elliptic equations. Signi�cantly, the classical Riemann–Rochformula and its generalization by Hirzebruch; the Gauß–Bonnet formula and itsextension byChern; and �nally Hirzebruch’s topological signature theorem, providemost of the examples in the index formula, but it was the Dirac operator actingon spinors which turned out to be the keystone of the index formula, both in itsformulation and in its subsequent developments. The Dirac operator appears to bethe primordial example of an elliptic operator, while spinors, although younger thandifferential forms or tensors, illustrate once again that aux âmes bien nées, la valeurn’attend point le nombre des années.

Our book aims to provide a comprehensive introduction to the theory of spinorson oriented Riemannian manifolds, including some (but by no means all) recentdevelopments illustrating their effectiveness. Our primordial interest comes fromRiemannian Geometry, and we adopt the point of view that spinors are some sort of“forgotten” tensors, which we study in depth.

Several textbooks related to this subject have been published in the last two orthree decades. Without trying to list them all, we mention Lawson and Michelson’sfoundational Spin geometry [LM89], the monograph by Berline, Getzler and Vergnecentered on the heat kernel proof of the Atiyah–Singer index theorem [BGV92],and more recently the books by Friedrich [Fri00] and Ginoux [Gin09] devoted to

2 Introduction

the spectral aspects of the Dirac operator. One basic issue developed in our bookwhich is not present in the aforementioned works is the interplay between spinorsand special geometric structures on Riemannian manifolds. This aspect becomesparticularly evident in small dimensions n � 8, where the spin group acts transi-tively on the unit sphere of the real (half-) spin representation, i.e., all spinors arepure. In this way, a non-vanishing (half-) spinor is equivalent to a SU2-structure forn D 5, a SU3-structure for n D 6, a G2-structure for n D 7, and a Spin7-structurefor n D 8. Many recent contributions in low-dimensional geometry (e.g., concern-ing hypo, half-�at, or co-calibrated G2 structures) are actually avatars of the verysame phenomena which have more natural interpretation in spinorial terms.

One further novelty of the present book is the simultaneous treatment of the spin,Spinc, conformal spin, and conformal Spinc geometries. We explain in detail therelationship between almost Hermitian and Spinc structures, which is an essentialaspect of the Seiberg–Witten theory, and in the conformal setting, we develop thetheory of weighted spinors, as introduced by N. Hitchin and P. Gauduchon, and de-rive several fundamental identities, e.g., the conformal Schrödinger–Lichnerowiczformula.

The Clifford algebra

We introduce spinors via the standard construction of the Clifford algebra Cln ofa Euclidean vector space R

n with its standard positive-de�nite scalar product. Thede�nition of the Clifford algebra in every dimension is simple to grasp: it is theunital algebra generated by formal products of vectors, with relations implying thatvectors anti-commute up to their scalar product:

u � v C v � u D �2hu; vi:

In low dimensions n D 1 and n D 2, the Clifford algebra is just C, respectively H,the quaternion algebra. Already for n D 1, we encounter the extravagant idea of“imaginary numbers”, which complete the real numbers and which were acceptedonly as late as the eighteenth century. Quaternions took another century to bedevised, while their generalization to higher-dimensional Clifford algebras is ratherevident. The same construction, but with the bilinear form replaced by 0, gives riseto the exterior algebra, while the universal enveloping algebra of a Lie algebra isclosely related. The Clifford algebra acts transitively on the exterior algebra, andthus in particular it is nonzero. This algebraic construction has unexpected appli-cations in Topology, as it is directly related to the famous vector �eld problem onspheres, namely �nding the maximal number of everywhere linearly independentvector �elds on a sphere.

Introduction 3

The spin group

Inside the group of invertible elements of Clnwe distinguish the subgroup formed byproducts of an even number of unit vectors, called the spin group Spinn. It is a com-pact Lie group, simply connected for n � 3, endowed with a canonical orthogonalaction on R

n de�ning a non-trivial 2 W 1 cover Spinn ! SOn. In other words, Spinnis the universal cover of SOn for n � 3. Every representation of SOn is of coursealso a representation space for Spinn, but there exists a fundamental representa-tion of Spinn which does not come from SOn, described below. It is this “shadoworthogonal representation” which gives rise to spinor �elds.

The complex Clifford algebra Cln is canonically isomorphic to the matrix alge-bra C.2n=2/ for n even, respectively to the direct sum of two copies of C.2.n�1/=2/for n odd, and thus has a standard irreducible representation on C

2Œn=2�for n even

and two non-equivalent representations on C2Œn=2�

for n odd. In the �rst case, therestriction to Spinn of this representation splits as a direct sum of two inequiva-lent representations of the same dimension (the so-called half-spin representations),whereas for n odd the restrictions to Spinn of the two representations of Cln areequivalent and any of them is referred to as the spin representation.

The classi�cation of real Clifford algebras is slightly more involved, and is basedon the algebra isomorphisms ClnC8 D R.16/ ˝ Cln : Note that the periodicity ofreal and complex Clifford algebras is intimately related to the Bott periodicity ofreal, respectively complex K-theory, which provides a very convenient algebraicsetting for the index theorem.

Spinors

The geometric idea of spinors on an n-dimensional Riemannian manifold .M n; g/

is to consider the spin group as the structure group in a principal �bration extend-ing, in a natural sense, the orthonormal frame bundle ofM . The existence of sucha �bration, called a spin structure, is not always guaranteed, since it is equiva-lent to a topological condition, the vanishing of the second Stiefel–Whitney classw2 2 H 2.M;Z=2Z/. Typical examples of manifolds which do not admit spinstructures are the complex projective spaces of even dimensions. When it is non-empty, the set of spin structures is an af�ne space modeled onH 1.M;Z=2Z/. Forinstance, on a compact Riemann surface of genus g, there exist 22g inequivalentspin structures. In this particular case, a spin structure amounts to a holomorphicsquare root of the holomorphic tangent bundle T1;0M .

Once we �x a spin structure on M , spinors are simply sections of the vectorbundle associated to the principal Spinn-bundle via the fundamental spin represen-tation. In general, we may think of spinors as square roots of exterior forms. Moreto the point, on Kähler manifolds in every dimension, spin structures correspond toholomorphic square roots of the canonical line bundle.

4 Introduction

The Dirac operator

In the same way as the Laplacian � D d�d is naturally associated to every Rie-mannian metric, on spin manifolds there exists a prominent �rst-order elliptic dif-ferential operator, the Dirac operator. The �rst instance of this operator appearedindeed in Dirac’s work as a differential square root of the Laplacian on Minkowskispace-time R

4, by allowing the coef�cients to be matrices, more precisely the Paulimatrices, which satisfy the Clifford anti-commutation relations. The generaliza-tion of Dirac’s operator to arbitrary spin manifolds was given by Atiyah–Singer andLichnerowicz. The spin bundle inherits the Levi-Civita connection from the framebundle, and hence the spinor bundle is endowed with a natural covariant derivativer. Moreover, vector �elds act on spinors by Clifford multiplication, denoted by .The Dirac operator is then de�ned as the Levi-Civita connection composed with theClifford multiplication:

D D ı rW�.†M/ �! �.†M/; D‰ DnX

iD1ei � rei

‰;

where fe1; : : : ; eng is a local orthonormal frame. The Dirac operator satis�es thefundamental Schrödinger–Lichnerowicz formula:

D2 D r�r C 1

4Scal; (1)

where Scal is the scalar curvature function. A spectacular and elementary applica-tion of this formula is the Lichnerowicz theorem which says that if the manifoldis closed and the scalar curvature is positive, then there is no harmonic spinor, i.e.,KerD D 0. On the other hand, Atiyah and Singer had computed the index of theDirac operator on compact even-dimensional spin manifolds. They noted that thespinor bundle on such a manifold is naturally Z2-graded, and the Dirac operator isodd with respect to this grading. The operator DC is de�ned as the restriction ofDto positive spinors, DCW�.†CM/ ! �.†�M/, and its index is the Fredholmindex, namely the difference of the dimensions of the spaces of solutions for DC

and its adjoint. The Atiyah–Singer formula gives this index in terms of a character-istic class, the yA genus. It follows thatM does not admit metrics of positive scalarcurvature if yA.M/ ¤ 0. It also follows that yA.M/ is an integer class whenM is aspin manifold. The Atiyah–Singer formula for twisted Dirac operators was also acrucial ingredient in the recent classi�cation of inner symmetric spaces and positivequaternion-Kähler manifolds with weakly complex tangent bundle [GMS11].

Another unexpected application of the Dirac operator is Witten’s proof of thepositive mass theorem for spin manifolds. In this book we present a variant ofWitten’s approach by Ammann and Humbert, which works on locally conformally�at compact spin manifolds. The idea is to express the mass as the logarithmic term

Introduction 5

in the diagonal expansion of the Green kernel for the conformal Laplacian, then usea scalar-�at conformal metric to construct a harmonic spinor on the complementof a �xed point, and �nally to use the Schrödinger–Lichnerowicz formula (1) andintegration by parts to deduce that the mass is non-negative.

Elliptic theory and representation theory

A main focus in this text is on the eigenvalues of the Dirac operator on compactspin manifolds. It appeared desirable to include a self-contained treatment of several(by now, classical) facts about the spectrum of an elliptic differential operator. Oncethese are established, as a bonuswe apply the corresponding results for the Laplacianon functions to prove the Peter–Weyl theorem. Then we develop the representationtheory of semisimple compact Lie groups, in order to compute explicitly the spectraof the Dirac operator on certain compact symmetric spaces.

The lowest eigenvalues of the Dirac operator and special spinors

As already mentioned, the Atiyah–Singer index theorem illustrates how the spec-trum of the Dirac operator encodes subtle information on the topology and thegeometry of the underlying manifold. Another seminal work on the subject is dueto N. Hitchin and concerns harmonic spinors; see [Hit74]. He �rst discovered that,in contrast with the Laplacian on exterior forms, the dimension of the space ofharmonic spinors is a conformal invariant which can (dramatically) change withthe metric.

A signi�cant part of the book is devoted to a detailed study of the relationshipbetween the spectrum of the Dirac operator and the geometry of closed spin mani-folds with positive scalar curvature.

In that context, the Schrödinger–Lichnerowicz formula not only states that on aclosed spin manifold with positive scalar curvature there is no harmonic spinor, butalso that there is a gap in the spectrum of the square of the Dirac operator. The �rstimportant achievement is Friedrich’s inequality, which says that the �rst eigenvalueof the Dirac operator is bounded from below by that of the sphere, the model space ofsuch a family of manifolds. The original proof of Th. Friedrich involved the notionof “modi�ed connection.” The inequality may also be proved by using elementaryarguments in representation theory, but the simplest proof relies on the Schrödinger–Lichnerowicz formula together with the spinorial Cauchy–Schwarz inequality:

jr‰j2 � 1

njD‰j2; for any spinor �eld ‰.

6 Introduction

Another remarkable consequence of this point of view is that how far this inequalityis from being an equality is precisely measured by jP‰j2, where P is the Penroseoperator (also called the Twistor operator).

Closedmanifolds for which the �rst eigenvalue of the Dirac operator (in absolutevalue) satis�es the limiting case of Friedrich’s inequality are called limiting mani-folds. They are characterized by the existence of a spinor �eld in the kernel of thePenrose operator (called twistor-spinor) which is also an eigenspinor of D. Thesespecial spinors are called real Killing spinors (this terminology is due to the factthat the associated vector �eld is Killing).

It has been observed by Hijazi and Lichnerowicz that manifolds having realKilling spinors cannot carry non-trivial parallel forms, hence there is no real Killingspinor on a manifold with “special” holonomy. With this in mind, Friedrich’s in-equality could be improved in different directions.

First, by relaxing the assumption on the positivity of the scalar curvature. Forinstance, based on the conformal covariance of the Dirac operator, a property sharedwith the Yamabe operator (the conformal scalar Laplacian), it is surprising to notethat if one considers the Schrödinger–Lichnerowicz formula over a closed spin man-ifold for a conformal class of metrics, then for a speci�c choice of the conformalfactor, basically a �rst eigenfunction of the Yamabe operator, it follows that thesquare of the �rst eigenvalue of the Dirac operator is, up to a constant, at least the�rst eigenvalue of the Yamabe operator (this is known as the Hijazi inequality).Again the limiting case is characterized by the existence of a real Killing spinor.Another approach is to consider a deformation (which generalizes that introduced byFriedrich) of the spinorial covariant derivative byCliffordmultiplication by the sym-metric endomorphism of the tangent bundle associated with the energy–momentumtensor corresponding to the eigenspinor. This approach is of special interest in thesetup of extrinsic spin geometry.

Secondly, a natural question is to improve Friedrich’s inequality for manifoldswith “special” holonomy. By the Berger–Simons classi�cation, one knows thatamong all compact spin manifolds with positive scalar curvature, we haveKähler manifolds and quaternion-Kähler manifolds. They carry a parallel 2-form(the Kähler form) and a parallel 4-form (the Kraines form), respectively. Modelspaces of such manifolds are respectively the complex projective space (note thatcomplex projective spaces of even complex dimension are not spin, but are stan-dard examples of Spinc manifolds) and the quaternionic projective space. It isnatural to expect that for these manifolds the �rst eigenvalue of the Dirac opera-tor is at least equal to that of the corresponding model space. This is actually thecase (with the restriction that for Kähler manifolds of even complex dimension,the lower bound turns out to be given by the �rst eigenvalue of the product of thecomplex projective space with the 2-dimensional real torus). These lower boundsare due to K.-D. Kirchberg in the Kähler case and to W. Kramer, U. Semmelmann,

Introduction 7

and G. Weingart in the quaternion-Kähler setup. There are different proofs of theseinequalities, but it is now clear that representation theory of the holonomy groupplays a central role in the approach. Roughly speaking, the proofs presented hereare based on the use of Penrose-type operators given by the decomposition of thespinor bundle under the action, via Clifford multiplication, of the geometric parallelforms characterizing the holonomy.

Thirdly, a natural task is to classify closed spin manifolds M of positive scalarcurvature admitting real Killing spinors and characterize limiting manifolds ofKähler or quaternion-Kähler type. For the �rst family of manifolds, the classi�-cation (obtained by C. Bär) is based on the cone construction, i.e., the manifold xMde�ned as a warped product ofM with the interval .0;C1/. This warped productis de�ned in such a way that for M D S

n then xM is isometric to RnC1nf0g. The

cone construction was used by S. Gallot in order to characterize the sphere as beingthe only limiting manifold for the Laplacian on exterior forms. M. Wang charac-terized complete simply connected spin manifolds xM carrying parallel spinors bytheir possible holonomy groups. Bär has shown that Killing spinors on M are inone-to-one correspondence with parallel spinors on xM , hence he deduced a list ofpossible holonomies for xM , and consequently a list of possible geometries forM .

The classi�cation of limiting manifolds of Kähler type is due to A. Moroianu.The key idea is to interpret any limiting spinor as a Killing spinor on the unit canon-ical bundle of the manifold. It turns out that in odd complex dimensions, limitingmanifolds of dimension 4l � 1 are exactly twistor spaces associated to quaternion-Kähler manifolds of positive scalar curvature, and those of dimension 4l C 1 arethe complex projective spaces. In even complex dimensions m D 2l � 4, the uni-versal cover zM of a limiting manifold M 2m is either isometric to the Riemannianproduct CPm�1 � R

2, for l odd, or to the Riemannian product N 2m�2 � R2, for

l even, where N is a limiting manifold of odd complex dimension. A spectacularapplication of these classi�cation results is an alternative to LeBrun’s proof of thefact that every contact positive Kähler–Einstein manifold is the twistor space of apositive quaternion-Kähler manifold.

Finally, in the case of compact spin quaternion-Kähler manifolds M 4m, it wasproved by W. Kramer, U. Semmelmann, and G. Weingart, that the only limitingmanifold is the projective space HPm. We present here a more elementary proof.

Dirac spectra of model spaces

As pointed out previously, it is important to compute the Dirac spectrum for someconcrete examples, and the archetypal examples in geometry are given by sym-metric spaces. For these manifolds, computing the spectrum is a purely algebraicproblem which can be theoretically solved by classical harmonic analysis

8 Introduction

methods, involving representation theory of compact Lie groups. Those methodswere already well known in case of the Laplacian.

In the last part of the book we give a short self-contained review of the represen-tation theory of compact groups. Our aim is to provide a sort of practical guide inorder to understand the techniques involved. From this point of view, representa-tions of the standard examples (unitary groups, orthogonal and spin groups, sym-plectic groups) are described in detail. In the same spirit, we give an elementaryintroduction to symmetric spaces. This point of view is also followed to explain thegeneral procedure for an explicit computation of the spectrum of the Dirac opera-tor of compact symmetric spaces, and then the spectrum of the standard examples(spheres, complex and quaternionic projective spaces) is computed in order to illus-trate the method.

Part I

Basic spinorial material

Chapter 1

Algebraic aspects

This chapter is devoted to the basic algebraic ingredients of spin geometry: Cliffordalgebras and spin groups. Clifford algebras are naturally associated to vector spacesendowed with symmetric bilinear forms. The Clifford algebra of the n-dimensionalEuclidean space contains the spin group Spinn, which is the 2-fold cover of thespecial orthogonal group SOn for n � 3. We study the complex representationsof the spin groups induced by algebra representations of the Clifford algebras andhighlight some of their important properties.

1.1 Clifford algebras

1.1.1 De�nitions

Definition 1.1. Let V be an n-dimensional vector space over the �eld K D R or C

and q a bilinear symmetric form on V . The Clifford algebra Cl.V; q/ associatedwith .V; q/ is the associative algebra with unit, de�ned by

Cl.V; q/ ´ V ˝ıI.V; q/

where V ˝ D ˚i�0V ˝i is the tensor algebra of V and I.V; q/ the two-sided idealgenerated by all elements of the form x˝ xC q.x; x/1, for x 2 V . The product inthe Clifford algebra will be denoted by “ � ”.

Remark 1.2. There is a natural map �WV ! Cl.V; q/ obtained by consideringthe natural embedding V ,! V ˝, followed by the projection V ˝ ! Cl.V; q/.By Proposition 1.5 below, this map is an embedding. Viewing V as a subset ofCl.V; q/ in that way, the algebra Cl.V; q/ is generated by V (and the unit 1), subjectto the relations

v � v D �q.v; v/1:

The Clifford algebra can be characterized by the following universal property.

12 1. Algebraic aspects

Proposition 1.3. LetA be an associative algebra with unit and f WV ! A a linearmap such that for all v 2 V

f .v/2 D �q.v; v/1A: (1.1)

Then there exists a unique K-algebra homomorphism

Qf WCl.V; q/ �! A

satisfyingQf ı � D f:

Furthermore, if C is an associative K-algebra with unit carrying a linear map�0WV ! C satisfying �0.v/2 D �q.v; v/1C, with the property above, then C isisomorphic to Cl.V; q/.

Proof. If i WV ,! V ˝ is the natural inclusion, then by the universal property of thetensor algebra, there exists a unique algebra homomorphism F such that f D F ıi .We have to show that I.V; q/ � KerF . Indeed, for any v 2 V , we have

F.v ˝ v C q.v; v/1/ D F.v/ � F.v/C q.v; v/1A

D f .v/ � f .v/C q.v; v/1A

D 0;

hence F factors through the quotient and we have the commutative diagram

V ˝ Proj //

F❆❆

❆❆❆❆

❆

❆❆❆

❆❆❆❆

Cl.V; q/

Qf

��V

i

OO

f// A

where Proj is the natural projection. We thus have

f D Qf ı Proj ıi D Qf ı �:Since Proj is onto, the uniqueness of F implies that of Qf . Applying this to f D �

shows that the identity is the unique algebra homomorphism a of Cl.V; q/ whichsatis�es a ı � D �.

Let C be any associative K-algebra with unit, with a linear map �0WV ! C, veri-fying the same universal property as Cl.V; q/. Applying the universal properties ofC and Cl.V; q/ to the maps � and �0 yields algebra homomorphisms Q�W C ! Cl.V; q/and Q�0WCl.V; q/ ! C satisfying Q� ı �0 D � and Q�0 ı � D �0. Moreover, the compositiona ´ Q� ı Q�0 is an algebra morphism of Cl.V; q/ satisfying a ı � D �, so by the aboveremark a is the identity. Similarly, Q�0 ı Q� is the identity of C. �

1.1. Clifford algebras 13

From now on we will only consider Clifford algebras of real Euclidean spaces(i.e., with a positive de�nite bilinear form q) or C-vector spaces endowed with anon-degenerate quadratic form.

Remarks 1.4. (1) The Clifford algebra Cl.V; q/ can be abstractly de�ned as thealgebra generated by nC 1 elements fv0; v1; : : : ; vng, subject to the relations

v0 � vi D vi � v0 D vi ; v20 D v0; v2i D �v0 .i � 1/;

vi � vj D �vj � vi .1 � i ¤ j � 1/:

Of course, v0 corresponds to 1 and fv1; : : : ; vng to a q-orthonormal basis of V .

(2) If fe1; : : : ; eng is a q-orthonormal basis of V , then the system

f1; ei1 � � � eik I 1 � i1 < � � � < ik � n; 1 � k � ng (1.2)

spans Cl.V; q/ as vector space, thus dimCl.V; q/ � 2n. It will be shown in the nextproposition that this system is actually a basis.

(3) Note that Cl.V; 0/ is nothing else than ƒ�V , the exterior algebra of V .

It turns out that the exterior algebra and the Clifford algebra can be naturallyidenti�ed as vector spaces (see Chevalley [Che62] and Kähler [Käh62]).

Proposition 1.5. There is a canonical isomorphism of vector spaces between theexterior algebra and the Clifford algebra of .V; q/, which, in terms of an orthonor-mal basis fe1; : : : ; eng of V , is given by

ƒ�V �! Cl.V; q/; ei1 ^ � � � ^ eik 7�! ei1 � � � eik :Proof. For any vector v 2 V , let v^ and v³ be the exterior multiplication, respec-tively the interior multiplication by v relative to q, and consider the map

f WV �! End.ƒ�V /; v 7�! f .v/.!/ D v ^ ! � v ³ !:Then one can easily check that the homomorphism f satis�es condition (1.1), withA D End.ƒ�V /, hence by the universal property, f uniquely extends to a homo-morphism

Qf WCl.V; q/ �! End.ƒ�V /:

Let us de�ne the linear map

F WCl.V; q/ �! ƒ�V ; ' 7�! . Qf .'//.1ƒ�V /:

If fe1; : : : ; eng is a q-orthonormal basis of V , then for every i1 < � � � < ik , weobtain

F.ei1 � � � eik / D ei1 ^ � � � ^ eik ;so F is onto. On the other hand, Remark 1.4 (2) shows that dimCl.V; q/ � 2n Ddimƒ�V , thus F is an isomorphism. �


Remark 1.6. The Clifford algebra and the universal enveloping algebra U.g/ of aLie algebra g are both de�ned as quotients of the tensor algebra. A classical resultin Lie algebra theory is the Poincaré–Birkhoff–Witt theorem stating that U.g/ hasa basis formed by ordered monomials in the elements of a well-ordered basis of g.In this light, Proposition 1.5 or even the embedding of V in Cl.V / are perhaps lessobvious than they might seem at �rst sight. We note that there is a gap in the proofof this result in [LM89], Proposition 1.2, p. 10.

Clifford algebras are endowed with the following fundamental automorphisms.

(1) Using the universal property, the injective morphism V ,! Cl.V; q/ inducedby the map �IdW v 7! �v, gives rise to the automorphism

˛WCl.V; q/ �! Cl.V; q/; ei1 � � � eik 7�! .�1/kei1 � � � eik :

As ˛2 D Id, we get the decomposition

Cl.V; q/ D Cl0.V; q/˚ Cl1.V; q/;

whereCli .V; q/ ´ f' 2 Cl.V; q/I ˛.'/ D .�1/i'g: (1.3)

Clearly, for i , j 2 Z=2Z, we have

Cli .V; q/ � Clj .V; q/ � CliCj .V; q/:

Thus the Clifford algebra Cl.V; q/ is a Z=2Z-graded algebra, i.e., a superal-gebra. The subspace Cl0.V; q/ (resp. Cl1.V; q/) is called the even (resp. odd)part of Cl.V; q/.

(2) Consider the K-algebra anti-automorphism de�ned by

tWV ˝ �! V ˝; xi1 ˝ � � � ˝ xik 7�! xik ˝ � � � ˝ xi1 :

Since t.I.V; q// � I.V; q/, there is an induced automorphism

tWCl.V; q/ �! Cl.V; q/; xi1 � � � xik 7�! xik � � � xi1 :

An immediate application of the universal property shows that if there exists aK-isomorphism f between two K-vector spaces V and V 0, endowed respectivelywith two bilinear symmetric forms q and q0, such that f �q0 D q, then f uniquelyextends to a K-algebra isomorphism between Cl.V; q/ and Cl.V 0; q0/.

Lemma 1.7. Let f W .V; q/ ! .V 0; q0/ be an isometry. Then f uniquely extends toa K-algebra isomorphism

Qf WCl.V; q/ �! Cl.V 0; q0/:


By the above remark, we may restrict to the following canonical examples.

Definition 1.8. The Clifford algebra Cln ´ Cl.Rn; qR/ .resp.Cln ´ Cl.Cn; qC//

associated with the canonical Euclidean scalar product qR .resp. qC/ de�ned byqR.x; y/ D

Pi xiyi .resp. q

C.z; w/ DPi ziwi / is called the n-dimensional real

.resp. complex/ Clifford algebra.

Examples 1.9. If fe1; : : : ; eng denotes the canonical orthonormal basis of Rn, then

the relationsei � ej C ej � ei D �2ıij : (1.4)

hold in Cln. Hence, we have the following fact.

(1) A basis of Cl1 is given by f1; e1g. Since e21 D �1, one has Cl1 Š C.

(2) A basis of Cl2 is given by f1; e1; e2; e1 �e2g. Since the three vectors e1, e2, ande1 � e2 verify the same multiplication rules as the standard basis of imaginaryquaternions, one has Cl2 Š H.

(3) The volume element ! ´ e1 � e2 � e3 2 Cl3 is central and ! � ! D 1. De�ne�˙ ´ 1

2.1˙ !/. Since

�C C �� D 1;

.�˙/2 D �˙;

�� C D �C � �� D 0;

one has the algebra decomposition Cl3 D ClC3 ˚Cl�3 where Cl˙3 D �˙ � Cl3D Cl3 ��˙. A basis of Cl˙3 is given by f�˙; �˙ � e1; �˙ � e2; �˙ � e1 � e2g,whence Cl˙3 Š H and Cl3 Š H ˚ H.

The following important results can be derived from the universal property.

Proposition 1.10. The complex Clifford algebra is isomorphic to the complexi�ca-tion of the real Clifford algebra, i.e.,

Cln Š Cln ˝RC:

Proof. By the universal property, the inclusion Rn ,! C

n � Cln can be extendedto a morphism {WCln ! Cln which induces the morphism

{CWCln ˝RC �! Cln; a˝ z 7�! {.a/z:

On the other hand, the C-linear map

|CW Cn �! Cln ˝RC; v 7�! Re.v/˝ 1C Im.v/˝ i


satis�es condition (1.1), i.e.,

|C.v/2 D �qC.v; v/1;

hence it can be extended to a morphism Cln ! Cln˝RC, which is the inverseof {C. �

Proposition 1.11. The n-dimensional real (resp. complex) Clifford algebra is iso-morphic to the even part of the .n C 1/-dimensional real (resp. complex) Cliffordalgebra, i.e.,

Cln Š Cl0nC1 and Cln Š Cl0nC1 : (1.5)

Proof. Denote by fe1; : : : ; eng and fe1; : : : ; enC1g the canonical bases of Kn and

KnC1, K D R (resp. C). This suggests to identify K

n with the image of its canonicalinjection in K

nC1 as the subspace generated by the �rst n vectors. De�ne the linearmap

f W Kn �! Cl0nC1 (resp.Cl0nC1/; ei 7�! ei � enC1:

By the de�nition of f , we have f .ei /2 D �1, thus f extends to

Qf WCln �! Cl0nC1 (resp. Cln �! Cl0nC1).

Clearly, Qf is an injective linear map between vector spaces of the same dimension,thus the map Qf is an algebra isomorphism. �

Proposition 1.12. For any couple of positive integers .m; n/, the .mC n/-dimen-sional real (resp. complex) Clifford algebra is isomorphic to the graded tensor prod-uct of the m-dimensional real (resp. complex) Clifford algebra with the n-dimen-sional real (resp. complex) Clifford algebra:

ClmCn ' Clm y Cln and ClmCn ' Clm y Cln;

the “graded tensor product” being the usual tensor product endowed with the mul-tiplication

.a ˝ b/ � .a0 ˝ b0/ D .�1/deg.b/ deg.a0/a � a0 ˝ b � b0;

where degWCl.V; q/ ! f0; 1g is de�ned by the grading (1.3).

Proof. Let .ei/iD1;:::;mCn be the standard basis ofKmCn,K D R orK D C. Identify

Km (resp. K

n) with the subspace of KmCn spanned by the ei , i D 1; : : : ; m (resp.

the ei , i D mC 1; : : : ; n). De�ne the linear map

f W KmCn �! Cl.Km; qK/ y Cl.Kn; qK/


by

f .ei / D

8<:ei ˝ 1 if i D 1; : : : ; m;

1˝ ei if i D mC 1; : : : ; n.

Since f .ei / � f .ej / C f .ej / � f .ei / D �2ıij , the map f extends to a K-algebrahomomorphism

Cl.KmCn; qK/ �! Cl.Km; qK/ y Cl.Kn; qK/:

Considering generators of Cl.KmCn; qK/, it is clear that this linear map is onto be-tween vector spaces of the same dimension, hence an isomorphism. �

By Proposition 1.5, there exists a canonical vector space isomorphism betweenCln andƒ�

Rn (resp. Cln andƒ�

Cn). The following proposition gives the relation

between the product in the Clifford algebra in terms of the exterior and interiorproducts in the exterior algebra.

Proposition 1.13. For all v 2 Kn, and all

' 2 ƒpKn � ƒ�

Kn Š

8<:Cln if K D R;

Cln if K D C;

we have

v � ' Š v ^ ' � v ³ ' and ' � v Š .�1/p.v ^ ' C v ³ '/;where ^ denotes the exterior product and ³ the interior product.

Proof. Let v D ej and ' D ei1 � � � eip with i1 < � � � < ip . If there exists ik suchthat j D ik , then v ^ ' D 0 and

v ³ ' D .�1/k�1ei1 ^ � � � ^ eik�1^ eikC1

^ � � � ^ eipŠ .�1/k�1ei1 � � � eik�1

� eikC1� � � eip

D �v � 'D .�1/p' � v:

If j … fi1; : : : ; ipg, then v ³ ' D 0 and

v ^ ' D ej ^ ei1 ^ � � � ^ eipŠ ej � ei1 � � � eipD v � 'D .�1/p' � v:

We then conclude by bilinearity. �


Wewill also need the following interesting feature of the Clifford product viewedon ƒ�

Kn.

Proposition 1.14. For any

' 2 ƒpKn � ƒ�

Kn Š

8<:Cln if K D R;

Cln if K D C;

and for any orthonormal basis fe1; : : : ; eng we havenX

jD1ej � ' � ej D .�1/pC1.n � 2p/':

Proof. It suf�ces to consider ' D ei1 : : : eip . Then, by equation (1.4) and for a�xed j 2 f1; : : : ; ng, it is straightforward to see that

ej � ' � ej D

8<:.�1/p' if j 2 ¹i1; : : : ; ipº;

.�1/pC1' if j … ¹i1; : : : ; ipº;

hence

nX

jD1ej � ' � ej D Œ.�1/pp C .�1/pC1.n � p/�'

D .�1/pC1.n� 2p/': �

1.1.2 Classi�cation of Clifford algebras

Since in the following we mainly deal with complex Clifford algebras, we now givesome basic facts which lead to their classi�cation. We also mention the tools whichlead to the classi�cation of real Clifford algebras. Henceforth, if K is a �eld, K.n/

denotes the algebra of n � n matrices with coef�cients in K.

Theorem 1.15 (Classi�cation of complex Clifford algebras). Let n 2 N.

.1/ There exists a canonical algebra isomorphism ClnC2 Š Cln ˝ Cl2.

.2/We have the following algebra isomorphisms:

Cl2nC1 Š C.2n/˚ C.2n/;

and

Cl2nC2 Š C.2nC1/:


Proof. (1) Denote by fe1; : : : ; enC2g the canonical basis of CnC2 and consider the

identi�cations Cn D spanfe1; : : : ; eng, C

2 D spanfenC1; enC2g. One can easilycheck that the linear map

f W CnC2 �! Cln ˝ Cl2; ek 7�!

8<:ek ˝ ienC1 � enC2; for k � n,

1˝ ek; for k > n,

satis�es (1.1). By Proposition 1.3, f can be extended to a homomorphism

ClnC2 7�! Cln ˝ Cl2 :

With the help of the standard bases (1.2) of the Clifford algebras, one checks that thishomomorphism is onto, and hence that it is an isomorphism, since dimC ClnC2 DdimC Cln˝ Cl2.

(2) By (1), using the isomorphism C.k/˝ C.l/ Š C.kl/, induced by the iden-ti�cation C

k ˝ Cl Š C

kl , it is suf�cient to prove the assertion for n D 0. Wealready noted that Cl1 Š C, hence Cl1 Š Cl1 ˝RC Š C ˝R C Š C ˚ C, the lastisomorphism being given on decomposable elements by z1 ˝ z2 7! .z1z2; Nz1z2/.

We also noted that Cl2 Š H. It is easy to verify that the identi�cation H Š C2

given by the R-isomorphism

a1 C bi C cj C dk D .a1 C bi /C j .c � d i / 7�! .� D a C ib; � D c � id/;

induces the injective R-homomorphism of algebras

H �! C.2/; q D �C j� 7�!�� N�� N�

�: (1.6)

Extending this injective homomorphism to an injective C-homomorphism

H ˝R C �! C.2/;

we conclude by a dimensional argument that H ˝R C Š C.2/. �

The classi�cation of real Clifford algebras is obtained by considering the Clif-ford algebra

Cl0n ´ Cl.Rn;�qRn

/:

Using the universal property, we can prove the following proposition.

Proposition 1.16. For all n 2 N,

ClnC2 Š Cln ˝Cl02 and Cl0nC2 Š Cl0n ˝Cl2 :


Using this result and the explicit description of Cl02, one deduces the �rst row ofthe following table.

Table 1

n 1 2 3 4

Cln C H H ˚ H H.2/

Cln C ˚ C C.2/ C.2/˚ C.2/ C.4/

n 5 6 7 8

Cln C.4/ R.8/ R.8/˚ R.8/ R.16/

Cln C.4/˚ C.4/ C.8/ C.8/˚ C.8/ C.16/

Finally, using again Proposition 1.16 together with the results of the above table,one deduces the following proposition.

Proposition 1.17. For all n 2 N,

ClnC8 Š Cln ˝R Cl8 :

This gives the complete classi�cation of real Clifford algebras.

1.2 Spin groups and their representations

In this section we de�ne certain subgroups of the multiplicative group of units inthe real or the complex Clifford algebra. We are mainly interested in the real andcomplex spin groups and the conformal spin groups. We then establish explicitrealizations of the corresponding Lie algebras. Finally, we study the irreduciblerepresentations of these Lie groups.

1.2.1 Spin groups

We begin by the following remark. Let v be any non-zero vector of an Euclideanspace .V; q/. The symmetry �v with respect to the hyperplane orthogonal to v isde�ned by

�v.x/ D x � 2q.x; v/q.v; v/

v:

Viewing V as a subset of Cl.V; q/, we can write

�v.x/ D x � .x � v C v � x/ � v�2 � v D �v � x � v�1 D �Ad.v/.x/: (1.7)

1.2. Spin groups and their representations 21

Since any element of the group SO.V; q/ is the product of an even number of sym-metries of that type, SO.V; q/ is the image under the map Ad of a certain subgroupof invertible elements in Cl.V; q/. We are thus led to having a closer look at thegroup of units in Cl.V; q/.

We consider the map

N WCl.V; q/ �! Cl.V; q/; a 7�! ˛. ta/ � a:

Note that N.v/ D jvj2, for all v 2 V . Denote by Cl�.V; q/ the group of invertibleelements of Cl.V; q/. It can be easily checked that N.˛. t.a�1/// � N.a/ D 1.This implies N.Cl�.V; q// � Cl�.V; q/. Since v � v D �jvj2, it is clear that anynonzero vector v 2 V � Cl.V; q/ lies in Cl�.V; q/, and v�1 D �v=jvj2. For eacha 2 Cl�.V; q/, we consider the inner automorphism

AdaWCl.V; q/ �! Cl.V; q/; b 7�! a � b � a�1;

and the map

fAdaWCl.V; q/ �! Cl.V; q/; b 7�! ˛.a/ � b � a�1:

As we pointed out in (1.7), for any nonzero vector v, the vector space V � Cl.V; q/is stable under fAdv:

fAdvjV D �v: (1.8)

For anyK-vector space V endowed with a symmetric bilinear form q, let P.V; q/be the group de�ned by

P.V; q/ D fa 2 Cl�.V; q/I fAda.V / � V g:

As before, we will restrict to the case .V; q/ D .Kn; qK/. (For a non-degeneratesymmetric bilinear form q, this choice is not restrictive in case K D C, whereas incase K D R, it means that we will not deal with non-degenerate bilinear forms withnonzero signature).

Lemma 1.18. (i) The kernel of the map

fAdW P.V; q/ �! GLn.K/

is K�.

(ii) N.P.V; q// � K�.

(iii) The mapN jP.V;q/ W P.V; q/ 7�! K

�

is a group homomorphism.


Proof. (i) Consider a 2 Cl.V; q/ such that

˛.a/ � v D v � a; v 2 Kn (1.9)

and, as above, denote by fe1; : : : ; eng the canonical basis of Kn. By Proposition 1.5,

for each �xed i; 1 � i � n, any element a 2 P.V; q/ can be uniquely written asa D bC ei � c, where b and c are linear combinations of elements in the basis (1.2)that do not involve ei . Condition (1.9), written for v D ei , is equivalent to

˛.b/ � ei D ei � b and ˛.c/ � ei D �ei � c:

The last relation implies c D 0. Since the argument holds for each ei , it followsthat a is a scalar, and since a 2 Cl�.V; q/, it should be a nonzero scalar.

(ii) For all v 2 Kn and a 2 P.V; q/ we have fAda.v/ 2 K

n, hence

fAda.v/ D t.fAda.v//:

This implies fAdN.a/.v/ D v, hence from (i) we conclude that N.a/ 2 K�.

(iii) By (ii),

N.a � b/ D ˛. tb � ta/ � a � b D ˛. tb/ �N.a/b D N.a/N.b/: �

Definition 1.19. (i) The spin group Spinn is the subgroup of P.Rn; qR/ generatedby elements of the form v1 � � �v2k , with k � 1 and kvik D 1, for 1 � i � 2k.

(ii) The conformal spin group CSpinn is the group Spinn � R�C.

Proposition 1.20. For n � 2, the homomorphism

� ´ fAdjSpinn

is a nontrivial double covering of the special orthogonal group SOn. In particularfor n � 3, the group Spinn is the universal cover of SOn.

Proof. By (1.8) we know that the image of a nonzero vector by the map

fAdW Rn n f0g � Cl�n �! GLn

is the symmetry with respect to the hyperplane orthogonal to this vector. The imageof fAdjSpinn

is then the group of even products of such symmetries, which by theCartan–Dieudonné theorem, is exactly the group SOn. Using (iii) of Lemma 1.18,any element a of Spinn satis�es N.a/ D 1. By (i) of Lemma 1.18, we conclude

that the kernel of fAdW Spinn ! SOn is f˙1g:


To show that the covering is nontrivial it is suf�cient to check that 1 and �1belong to the same connected component of Spinn. To see this, choose unit orthog-onal vectors v; w 2 R

n (n � 2) and note that the curve

cW Œ0; �=2� �! Spinn;

t 7�! c.t/ D .v sin t C w cos t / � .v sin t �w cos t /;

satis�es c.0/ D 1, c.�=2/ D �1. Finally, the last assertion follows from the factthat, for n � 3, we have �1.SOn/ D Z=2Z. �

Remark 1.21. Note that since P.Rn; qR/ is a closed subgroup of the Lie group Cl�n,it is actually a Lie group. Hence Spinn D fAd�1.SOn/, being a closed subgroup ofP.Rn; qR/, is also a Lie group. Furthermore, the condition that Spinn is generatedby elements of the form v1 � � �v2k , with kvik D 1, for 1 � i � 2k, implies that itis a compact Lie group.

Examples 1.22. With obvious notations, we consider

(1) the unitary group

Un ´ fA 2 C.n/I tA xA D IdngI

(2) the special unitary group

SUn ´ fA 2 UnI detA D 1gI

(3) the symplectic group1

Spn ´ fA 2 HomH.Hn;Hn/I

hA � x; A � yi D hx; yi; .x; y/ 2 Hn � H

ng;

where

hx; yi ńX

iD1xi Nyi ;

and conjugation is de�ned, for any element v D v01 C v1i C v2j C v3k 2H Š R

4, byNv ´ v01 � v1i � v2j � v3kI

(4) the unit sphereSn ´ fx 2 R

nC1I hx; xi D 1g:

1In order to represent elements of Spn by matrices with the usual convention for products, Hn

is viewed as a right vector space over H.


Then it is not dif�cult to describe the spin groups in low dimensions; see [LM89].

(1) Spin1 � Cl01 Š Cl0 Š R, thus Spin1 Š f˙1g D Z=2Z and

�W Spin1 Š Z=2Z �! SO1 Š f1gis the map t 7! t2.

(2) One has Spin2 � Cl02 Š Cl1 Š C. It is easy to verify that S1 Š U1 Š Spin2via the map

ei�Š7�!

�cos

�

2e1 C sin

�

2e2

��

� cos�

2e1 C sin

�

2e2

�

D cos � C sin � e1 � e2:The map

�WU1 Š Spin2 �! SO2

is then given by

ei� 7�!�cos.2�/ � sin.2�/

sin.2�/ cos.2�/

�;

hence, under the identi�cation SO2 Š U1, � is the “square” map

U1 �! U1; z 7�! z2:

The algebra Cl03 is isomorphic to H by the isomorphism

ˆW

8ˆˆ<ˆˆ:

1 7�! 1;

e1 � e2 7�! i ;

e2 � e3 7�! j ;

e3 � e1 7�! k:

By this isomorphism, Spin3 � Cl03 is identi�ed with a subgroup of H�. If a 2

Cl03 and q D ˆ.a/, then N.a/ D 1 H) q Nq D 1, so ˆ.Spin3/ � Sp1, andsince these two groups have the same dimension, we have ˆ.Spin3/ D Sp1.Under those identi�cations, the covering map � is then given by

Spin3 Š Sp1��! SO3; q 7�! .x 2 Im.H/ 7�! qxq�1 D qx Nq/:

Furthermore, it is easy to verify that the injective homomorphism of algebrasH ! C.2/ given in (1.6), induces an isomorphism between the groups Sp1and SU2. So we have the isomorphisms

Spin3 Š Sp1.Š S3/ Š SU2 :


(3) To have a description of Spin4, we consider the map

�W Sp1 � Sp1 �! GL.H/; .q; q0/ 7�! .x 7! qxq0�1 D qxSq0/:

It is easy to check that � is a group homomorphism with values in O4 and evenSO4, since Sp1 � Sp1 is connected. It is also straightforward that its kernel isthe group f˙.1; 1/g Š Z=2Z. Finally �.Sp1 � Sp1/ D SO4, since the twogroups have the same dimension. So � is a two-fold covering of SO4, hence

Spin4 Š Sp1 � Sp1 Š SU2 � SU2 :

Note that, identifying H with a subspace of C.2/ by the injective homomor-phism given in (1.6), the covering

�W SU2 � SU2 7�! SO4

is given by the map

.A; B/ 7�!�X D

�x � Nyy Nx

�2 H 7�! AX t xB

�:

We now introduce the conformal spin group CSpinn ´ Spinn � R�C. Recall

that the conformal group COCn is identi�ed with SOn �R

�C via the canonical iso-

morphism�W SOn �R

�C �! COC

n ;

.A; t/ 7�! tA:

Hence, by Proposition 1.20 it is clear that, for n � 3, the morphism

�W CSpinn �! COCn ;

.a; t / 7�! t�.a/;

is the universal cover of the conformal group. So we have the fundamental exactsequences

f1g �! Z=2Z �! Spinn��! SOn �! f1g

and

f1g �! Z=2Z �! CSpinn��!COC

n �! f1g:Consider next the group homomorphism

| W Spinn � U1 �! Cl�n;

.a; z/ 7�! {.a/z;


where {WCln ! Cln Š Cln ˝RC is the standard inclusion of Cln in Cln. Forsimplicity, we shall identify Cln with its image by {. Since the kernel of | isf.1; 1/; .�1;�1/g Š Z=2Z, its image is isomorphic to .Spinn � U1/=.Z=2Z/,which will be denoted by Spinn �Z=2Z U1.

Definition 1.23. (i) The complex spin group is the group

Spincn ´ Spinn �Z=2Z U1;

which, via | , is identi�ed with a subgroup of Cl�n.

(ii) The conformal complex spin group is the group

CSpincn ´ .Spinn �Z=2Z U1/ � R�C;

which via | � Id, is identi�ed with a subgroup of Cl�n �R�C.

Example 1.24 (The group Spinc4). As we have seen, the spin group Spin4 is iso-morphic to SU2 � SU2, hence it can be realized as

Spin4 D˚� a 00 b

�I a; b 2 SU2

:

The complex spin group can thus be realized as

Spinc4 D˚� za 0

0 zb

�I a; b 2 SU2; z 2 U1

:

An element of the complex (resp. conformal complex) spin group can be consid-ered as an equivalence class Œa; z�, a 2 Spinn (resp. a 2 CSpinn), z 2 U1, moduloŒa; z� D Œ�a;�z�. The homomorphisms � and � induce double coverings

�cW Spincn �! SOn�U1; Œa; z� 7�! .�.a/; z2/

and

�cWCSpincn �! COCn � U1; Œb; z� 7�! .�.b/; z2/:

We have the following commutative diagram of exact sequences relating � to �c:

Z=2Z //

o

��

Spinn � U1��Id //

|

��

SOn �U1

##●●●

●●●●

●●●●

●●

Id�sq

��

f1g

<<③③③③③③③③③③③③

""❉❉❉

❉❉❉❉

❉❉❉❉

❉f1g

Z=2Z // Spincn �c// SOn �U1

;;✇✇✇✇✇✇✇✇✇✇✇✇✇


where sq is the square map

sqWU1 �! U1; z 7�! z2:

There is also a similar commutative diagram corresponding to the conformal setuprelating � to �c. We shall use these exact sequences to get the explicit in�nitesimalaction of the morphisms �, �c, � and �c. Since the spin group sits in the Cliffordalgebra Cln, its Lie algebra, spinn, is a Lie subalgebra of Cln, seen as a Lie algebraby setting

Œa; b� ´ a � b � b � a:

Theorem1.25. The Lie algebras of the Lie groups Spinn, Spincn,CSpinn andCSpin

cn

and the differentials of the corresponding representations �, �c, � and �c satisfy thefollowing relations:

spinn Š son; ��.ei � ej / D 2ei ^ ej ;

spincn Š son ˚ iR; �c� D �� ˚ 2IdiR;

cspinn Š son ˚ R; �� D �� ˚ IdR;

cspincn Š son ˚ R ˚ iR; �c� D �� ˚ IdR ˚ 2IdiR;

where ei ^ ej 2 son � gln is de�ned by

.ei ^ ej /.ek/ D ıikej � ıjkei :

Proof. Recall that ƒ2Rn is identi�ed with a vector subspace of Cln by

ei ^ ej 7�! ei � ej ; 1 � i < j � n:

This subspace lies in spinn, since ei � ej is the tangent vector at t D 0 to the curvein Spinn given by

c.t/ D�ei sin

t

2� ej cos

t

2

��ei sin

t

2C ej cos

t

2

�

D cos t C ei � ej sin t:

Since dimR spinn D dimR son D dimRƒ2Rn, we get ƒ2Rn Š spinn. The other

assertions follow directly.


Finally, for all i; j; k, 1 � i < j � n and 1 � k � n we have �.c.t//.ek/ Dc.t/ � ek � c.t/�1, hence

��.ei � ej /.ek/ D d

dt

ˇˇtD0

�.c.t//.ek/

D d

dt

ˇˇtD0

.cos t C ei � ej sin t / � ek � .cos t � ei � ej sin t /

D ei � ej � ek � ek � ei � ej D 2ıikej � 2ıjkeiD 2.ei ^ ej /.ek/:

The other relations are immediate consequences of the above exact sequences. �

1.2.2 Representations of spin groups

In order to introduce the spinor representation of the spin group, which plays a cen-tral role in the sequel, we give a short description of the irreducible representationsof Spinn (details may be found in Chapter 12).

We only consider complex �nite-dimensional representations of Spinn, i.e., (con-tinuous) Lie groups homomorphisms

�W Spinn �! GL.V /;

where V is a �nite-dimensional C-vector space. Recall that any such representationis actually real analytic (cf. for instance [God82]). Note also that any representationof the group SOn is obtained as a quotient of a representation � of the covering groupSpinn verifying the condition �.˙1/ D Id.

The group Spinn being compact, any of its representations is totally reducible,cf. Theorem 12.1, hence we may restrict to irreducible representations.

Furthermore, the group Spinn being simply connected and compact, the gen-eral theory (see the comments following Theorem 12.61) asserts that all irreduciblerepresentations of Spinn may be described by considering a �nite family of “basic”irreducible representations called “fundamental.”2

Now the fundamental irreducible representations of the group Spinn are givenby considering the standard irreducible representations of the group SOn, togetherwith the spinor representations, induced by irreducible representations of the cor-responding Clifford algebras.

2In the sense that each irreducible representation is a summand in some tensor product of fun-damental representations.


1.2.2.1 The standard irreducible representations of the Group SOn, n � 3.

Let us consider the standard complex representation � of the group SOn in the spaceCn induced by the injective homomorphism SOn ,! SOn;C.

Theorem 1.26. For any positive integer k such that k � m � 1, if n D 2m, andk � m, if n D 2mC1, the representationƒk� in the spaceƒkC

n is an irreduciblerepresentation of the group SOn.

Proof. Let fe1; : : : ; eng be the canonical basis of Rn, identi�ed with the canonical

basis of Cn. The canonical basis of ƒkC

n is then given by the vectors

eI D ei1 ^ � � � ^ eik ;

where I D fi1 < � � � < ikg runs through the set of k-element subsets of f1; : : : ; ng.Let V be a SOn-invariant subspace ofƒkC

n and v a non-zero vector in V . Onehas v D

PI cIeI , where cI 2 C. We will prove by induction on the number

of non-zero coef�cients cI that V D ƒkCn. If there is only one such non-zero

coef�cient, say cI , then eI 2 V . But, up to a sign, all the eJ are obtained from eIby a convenient permutation of the vectors e1; : : : ; en. Substituting �e1 for e1 inthe case of an odd permutation, every eJ is then the image of eI under the action ofan element of SOn, so all the eJ are in V , and hence V D ƒkC

n.Suppose now that v has at least two non-zero coef�cients cI and cJ . Under the

hypothesis made on k, and since I ¤ J , it is always possible to �nd a couple .i; j /such that i 2 I , i … J , j … I [ J . Consider the transformation g 2 SOn such that

g � ei D �ei ; g � ej D �ej ; g � ek D ek; k ¤ i; j:

Then

g � eI D �eI ; g � eJ D eJ ; g � eK D ˙eK ; K ¤ I; J:

Hence v C g � v 2 V has fewer non-zero coef�cients than v, but still at least oneand the proof follows by induction. �

Remark 1.27. Let k be a positive integer such that k � m � 1, if n D 2m, andk � m, if n D 2mC1. Then theC-linear extension of the Hodge �-operator de�nesan isomorphism between ƒn�k� and ƒk�.

If n D 2m and k D m, then the representation ƒm� is reducible: since therestriction of the �-operator to ƒm� squares to .�1/m times the identity, there is adecomposition of ƒmC2m into eigenspaces corresponding to the eigenvalues ˙1for m even and ˙i for m odd. By the SOn-invariance of the �-operator, theseeigenspaces are invariant under the action of SOn.


1.2.2.2 The spinor representations. In this section we will show that by consid-ering the standard complex representations of the Clifford algebras, the spin groupSpinn inherits two irreducible complex representations if n is even and one if n isodd. As those representations do not descend to the group SOn, they are calledspinor representations. These representations are of fundamental importance since,together with the standard irreducible representations of the group SOn, they consti-tute the “fundamental” (in the sense mentioned before) irreducible representationsof the spin group.

We �rst consider complex representations of the Clifford algebras. Since everycomplex representation of Cln induces a complex representation of Cln and con-versely, we shall only consider complex Clifford algebras.

By a theorem of Burnside, up to equivalence, the only complex irreducible repre-sentation of the matrix algebra C.N / is the canonical representation on C

N , henceby the classi�cation of the complex Clifford algebras (Theorem 1.15) we have atonce the following theorem.

Theorem 1.28. The complex Clifford algebra Cln has a unique irreducible repre-sentation

�nW Cln �! End.†n/;

for n even, and two inequivalent irreducible representations

�˙n W Cln �! End.†n/;

for n odd, where †n is a complex vector space of complex dimension N ´ 2Œn2 �.

(Here Œ � stands for the integer part).

For a better understanding of the structure of the Clifford algebras we introducethe complex volume element

!C ´ i ŒnC1

2 �e1 � � � en 2 Cln : (1.10)

It is straightforward to check the relations

.!C/2 D 1;

and

x � !C D .�1/n�1!C � x; x 2 Rn � Cln : (1.11)

We now prove the following two propositions.

Proposition 1.29. For n odd,

Cln D ClCn ˚ Cl�n ;

where Cl˙n D �˙ � Cln D Cln ��˙ and �˙ D 12.1˙ !C/. Moreover, ˛.Cl˙n / D

Cl�n .


Proof. Since .!C/2 D 1, one has

�C C �� D 1;

.�˙/2 D �˙;

�� C D �C � �� D 0:

Since n is odd, !C and �˙ are central in Cln. It is then clear that Cl˙n D �˙ � Clnare two ideals of Cln and Cln D ClCn ˚ Cl�n .

The volume element being odd, i.e., !C 2 Cl1n, we have ˛.�˙/ D ��, hence

˛.Cl˙n / D Cl�n and the two subalgebras are isomorphic. �

Proposition 1.30. For n odd, let �n be a complex irreducible representation ofCln.Then either �n.!C/ D Id, or �n.!C/ D �Id. The two possibilities occur and theyare inequivalent.

Proof. Since .!C/2 D 1 and �.!C/2 D Id, the vector space †n can be writ-ten as the sum of the eigenspaces †˙

n associated with the ˙1-eigenvalues, i.e.,†n D †C

n ˚ †�n . The volume element !C being central, the eigenspaces †˙

n areCln-invariant. The representation being irreducible, we conclude that †n D †C

n

or †n D †�n . It is then clear that these two representations are inequivalent. By

considering the action of Cln on Cl˙n by left multiplication, we see that the twopossibilities occur. �

Definition 1.31. The map

Cln ˝†n �! †n;

� ˝ 7�! � � ´

8<:�n.�/. / if n is even,

�Cn .�/. / if n is odd,

is called Clifford multiplication of � with .

Proposition 1.32. (i) For n even, the restriction of �n to Spinn .resp. Cl0n/ splitsinto †n D †C

n ˚ †�n , where †

Cn and †�

n are complex inequivalent irreduciblerepresentations of Spinn .resp. Cl0n/.

(ii) For n odd, the restrictions of �˙n to Spinn .resp. Cl0n/ are irreducible and

equivalent.

(iii) For n even, for all x 2 R n f0g, the linear maps �n.x/W†˙n ! †�

n areisomorphisms. Moreover, under the isomorphism Cl0n Š Cln�1, the vector spaces†˙n correspond to the two inequivalent irreducible representations of Cln�1.


Proof. Note that the complex subalgebra generated by Spinn � Cln is the even partCl0n of Cln. Hence, two representations of Cl0n are irreducible or equivalent if andonly if it is the case for their restrictions to Spinn. It is therefore suf�cient to provethe assertions for Cl0n instead of Spinn.

(i) Let n D 2m. Since !C commutes with Cl0n, the restriction of �n to Cl0n ŠCln�1 splits into †n D †C

n ˚†�n , where

†˙n D �n

�1˙ !C

2

�.†n/:

By Theorem 1.28, it is suf�cient to prove that †Cn and †�

n are nontrivial vectorspaces. Since the linear map �n is an isomorphism, �n.!C/ cannot be equal to˙Id†n D �n.˙1/.

(ii) Again, we make use of the isomorphism Cl0n Š Cln�1. For n odd, we knowthat ˛.Cl˙n / D Cl�n . Note that the even part Cl0n sits diagonally in the decomposi-tion Cln D ClCn ˚ Cl�n . In fact, since !C 2 Cl1n, we have !

CW Cl0n ! Cl1n, henceCl0n \ Cl˙n D f0g. More precisely, we have

Cl0n D fu˙ C ˛.u˙/I u˙ 2 Cl˙n g:

Since the two inequivalent irreducible representations of Cln are distinguished bythe isomorphism ˛, by restriction to Cl0n, they become equivalent.

(iii) By (1.11), x � !C D �!C � x. Then,

�n.x/†˙n D �n.x/�n

�1˙ !C

2

�.†n/

D �n

�1� !C

2

��n.x/.†n/

� †�n :

The linear map �n.x/ is in fact an isomorphism since �n.x/2 D �kxk2Id.For the last statement, it is suf�cient to note that the isomorphism (1.5) maps

the volume element !Cn�1 ´ i Œ

n2 �e1 � � � en�1 of Cln�1 to the volume element

!Cn ´ i Œ

nC12 �e1 � � � en�1 � en of Cln. �

Definition 1.33. The representation

�n ´

8<:�nˇSpinn

for n even,

�Cn

ˇSpinn

for n odd,

is called the canonical complex spin representation and is denoted by .�n; †n/.


If n is odd, �n is irreducible, whereas if n is even, it splits into two irreduciblecomponents �˙

n . Note that since �n.�1/ D �Id, the canonical complex spin repre-sentation does not descend to the group SOn, and neither do its irreducible compo-nents �˙

n if n is even.

Definition 1.34. For n even, the Clifford multiplication by the complex volumeelement

!C�W †n D †Cn ˚†�

n �! †n D †Cn ˚†�

n ;

D C C � 7�! x ´ !C � D C � �;

is called the conjugation map.

The existence of a real or quaternionic structure on the canonical complex spinrepresentation will be discussed below. Note that, looking to the table of real Cliffordalgebras we see that Cln has, up to isomorphism, one irreducible representationfor n � 0; 1; 2 mod 4, and two irreducible representations for n � 3 mod 4.In this last case, the two representations become equivalent when restricted to Spinn.For all n, we thus get a representation of Spinn called the real spin representation.

Note that since the algebra generated by the groups Spinn and Spincn in Cln isexactly Cl0n, it is clear that the preceding facts are also true for the group Spincn.In fact, observe that the complex representations of Spinn are the same as that ofthe group Spincn. Hence the restriction of �n (resp. �

Cn ) to Spin

cn, also denoted by �

cn,

satis�es Proposition 1.32 too.

For each k 2 R, we consider the representation �kn (resp. �c;kn ) of CSpinn (resp.CSpincn) on †n de�ned, for all a 2 Spinn (resp. a 2 Spincn) and t 2 R

�C, by

�kn..a; t // D tk�n.a/ .resp: �c;kn ..a; t // D tk�cn.a//: (1.12)

As before, one checks that these representations are irreducible for n odd, and arethe direct sum of two inequivalent irreducible representations for n even.

Now since it is inherited from a representation of the complex Clifford algebra,the canonical complex spin representation .�n; †n/ enjoys the following importantproperty.

Proposition 1.35 (the natural Hermitian product on the spinor space). There existsa Hermitian product .� ; � / on †n such that

.�1; �2/ D .x � �1; x � �2/; for all x 2 Rn; kxk D 1; �1; �2 2 †n: (1.13)

A Hermitian product on †n with the above property is unique up to a positiveconstant.


Proof. Let �n be the multiplicative subgroup of Cl�n generated by a g-orthonormalbasis fe1; : : : ; eng of R

n. Using the relations .�1/2 D 1, e2i D �1, and ei � ej D�ej � ei , 1 � i; j � n, i ¤ j , we see that �n is �nite and j�nj D 2nC1. Now wechoose an arbitrary Hermitian product h�; �i on †n and de�ne for �1, �2 2 †n

.�1; �2/ D 1

j�njX

�2�n

h� � �1; � � �2i:

First, for ei 2 �n, one has that

.ei � �1; ei � �2/ D 1

j�njX

�2�n

h� � ei � �1; � � ei � �2i

D 1

j�njX

�2�n

h� � �1; � � �2i

D .�1; �2/:

Then, for x 2 Rn with kxk D 1, i.e., x D

PniD1 xiei , with

PniD1 x

2i D 1, we get

.x � �1; x � �2/ DX

i

x2i .ei � �1; ei � �2/CX

i 6Djxixj .ei � �1; ej � �2/

DX

i

x2i .�1; �2/CX

i<j

xixj ..ei � �1; ej � �2/C .ej � �1; ei � �2//

D .�1; �2/

since, for i < j ,

.ei � �1; ej � �2/ D .ei � ei � �1; ei � ej � �2/

D �.�1; ei � ej � �2/

D �.ej � �1; ej � ei � ej � �2/

D .ej � �1; ei � ej � ej � �2/

D �.ej � �1; ei � �2/:

Assume now that .�; �/0 is another Hermitian product on †n satisfying (1.13). Then.�; �/0 D .A.�/; �/ for some Cln-invariant automorphism A. The representation ofthe Clifford algebra on †n being irreducible, A has to be a constant multiple of theidentity by the Schur lemma. �


Corollary 1.36. For all x 2 Rn and for all �1, �2 2 †n, we have

.x � �1; �2/ D �.�1; x � �2/:

Proof. Let x be in Rn n f0g. Then

.x � �1; �2/ D�x � x

kxk � �1;x

kxk � �2�

and

.x � �1; �2/ D 1

kxk2 .x � x � �1; x � �2/ D �.�1; x � �2/: �

The following corollary is an immediate consequence of Proposition 1.35.

Corollary 1.37. The representations .�n; †n/ and .�cn; †n/ of Spinn and Spincn,respectively, are unitary for the above Hermitian product on †n.

Finally, it is often useful to have an explicit description of the spinor representa-tion. We only give it for n even, since the description for n odd may then be simplyderived.

For this, let fe1; : : : ; em; emC1; : : : ; e2mg be the canonical orthonormal basisof R

2m and fzj ; Nzj gjD1;:::;m, the corresponding Witt basis of R2m ˝R C, i.e.,

zj ´ 1

2.ej ˝ 1 � ejCm ˝ i/ and Nzj ´ 1

2.ej ˝ 1C ejCm ˝ i/:

These vectors satisfy the relations

qC.zj ; zk/ D qC. Nzj ; Nzk/ D 0;

and

qC. Nzj ; zk/ D qC.zj ; Nzk/ D 1

2ıjk;

which yield

zj � zk C zk � zj D 0; (1.14a)

Nzj � Nzk C Nzk � Nzj D 0; (1.14b)

zj � Nzk C Nzk � zj D �ıjk; (1.14c)

since x � y C y � x D �2qC.x; y/ for x; y 2 C2m � Cl2m. It is convenient to

introduce the notations zLr ´ zl1 � � � zlr , for 1 � l1 < � � � < lr � m. Then, byrelations (1.14), it is straightforward to show that, as a vector space, we have

Cl2m D spanfzJp � NzKq I 1 � j1 < � � � < jp � m; 1 � k1 < � � � < kq � m;

zJ0D NzK0

D 1; 0 � p; q � mg:


Note that, if we de�ne the element N! by

N! ´ Nz1 � � � Nzm;

then again by (1.14) one has Nzk � N! D 0 for 1 � k � m. De�ne the complex vectorspace †2m by

†2m D spanfzLr � N!I 1 � l1 < � � � < lr � m; 0 � r � mg:

Clearly the complex dimension of †2m is at most 2m. Now consider the map

�W Cl2m �! End.†2m/;

v D zJp � NzKq 7�! �.v/ D .zLr � N! 7! zJp � NzKq � zLr � N!/:

This is obviously a representation of Cl2m, which, by Theorem 1.28, is isomor-phic to the unique irreducible representation of Cl2m. Thus, the spinor complexrepresentation of Spin2m can be described by considering the restriction �jSpin2m

.

1.2.3 Real and quaternionic structures

In this section we investigate the existence of real or quaternionic structures of thecanonical complex spin representation �n (see De�nition 1.33).

Definition 1.38. Let �WG ! † be a complex representation of a Lie group G.A real, resp. quaternionic structure of � is a C-anti-linear G-equivariant automor-phism j of † satisfying j2 D 1, resp. j2 D �1.

In order to avoid possible confusions we emphasize here that the complex spinrepresentation �n (de�ned as the restriction to Spinn of an irreducible representationof Cln) is not in general the complexi�cation of the real spin representation (de�nedas the restriction to Spinn of an irreducible representation of Cln).

In fact, for dimensional reasons, using the table of real Clifford algebras (Table 1in Section 1.1.2), it is clear that the restriction to Cln of an irreducible representa-tion of Cln is again irreducible for n � 1; 2; 3; 4; 5 mod 8, and therefore in thesedimensions the real and complex spin representations coincide. In the remainingdimensions, n � 6; 7; 8 mod 8, the complex spin representation is the complexi�-cation of the real spin representation.

Since �n is irreducible for n odd and reducible for n even, we need to distinguishtwo cases.

Case I: n odd. Since Spinn is a subset of Cl0n Š Cln�1, Table 1 in Section 1.1.2

shows that the complex spin representation �n on †n has a real structure for n � 1

and n � 7 mod 8, and a quaternionic structure for n � 3 and n � 5 mod 8.We denote this structure by j.


We would like to understand the behavior of j with respect to the Clifford actionon †n. Clearly j is C-anti-linear and Cl0n-equivariant. It remains to see whetherj commutes or anti-commutes with the Clifford product by odd elements of Cln.Recall that the complex volume element !C, de�ned in an orthonormal basis by

!C D i ŒnC1

2 �e1 � � � en; (1.15)

acts on †n as the identity, so obviously commutes with j. On the other hand,!C (which has odd degree), is real for n � 3 and n � 7 mod 8, and purelyimaginary for n � 1 and n � 5 mod 8. Thus j commutes (resp. anti-commutes)with the Clifford product by real vectors for n � 3 and n � 7 mod 8 (resp. n � 1

and n � 5 mod 8). This, by the way, corroborates the fact (visible on the table ofClifford algebras) that the representation of the whole complex Clifford algebra isquaternionic for n � 3 mod 8, and real for n � 7 mod 8.

Case II: n even. The complex spin representation splits into the direct sum of spinhalf-representations †n D †C

n ˚ †�n . Looking at Table 1 in Section 1.1.2 again,

we deduce the following. For n � 2 and n � 6 mod 8, Cl0n Š Cln�1 is the algebra

of complex matrices C.2Œn�1

2 �/, acting irreducibly on E D C2

Œ n�12

�

. Taking thetensor product with C yields a representation of Cln�1 on E ˝R C Š E ˚ xE,showing that †C

n can be identi�ed with E and †�n with xE. They carry no real

or quaternionic structure separately, but their direct sum carries both a real and aquaternionic structure, canonically de�ned by

jR D�0 f �1

f 0

�

and

jH D�0 f �1

�f 0

�;

where f WE ! xE denotes the (C-anti-linear) identi�cation of E and xE. Since Clnis an algebra of quaternionic matrices for n � 2 mod 8, and of real matrices forn � 6 mod 8, we obtain that jH is Cln-equivariant for n � 2 mod 8 and jR isCln-equivariant for n � 6 mod 8.

Finally, for n � 4 mod 8 (resp. n � 8 mod 8), Cln�1 is the direct sum of twocopies of some algebra of quaternionic (resp. real) matrices, so each half-spin rep-resentation †˙

n carries a quaternionic (resp. real) structure j˙. Moreover, the directsum jC ˚ j� acting on †n is Cln-equivariant since Cln has the same quaternionic(resp. real) nature as Cln�1.

Summarizing, we have proved the following theorem.


Theorem 1.39. The complex spin representation†n has a real structure jR for n �1; 2; 6; 7; 8 mod 8, and a quaternionic structure jH for n � 2; 3; 4; 5; 6 mod 8.The Clifford product with real vectors commutes with jR for n � 6; 7; 8 mod 8,and anti-commutes with it for n � 1; 2 mod 8. The Clifford product with realvectors commutes with jH for n � 2; 3; 4 mod 8, and anti-commutes with it forn � 5; 6 mod 8.

Chapter 2

Geometrical aspects

This chapter is devoted to the general properties of the connections induced onthe complex spinor bundle in several natural geometric settings: Riemannian, con-formal, spin and Spinc. We study the corresponding Dirac operators and derivethe well-known Bochner-type formulas relating the square of the Dirac operatorand the rough Laplacian. We also study spinors in particular geometric instances:hypersurfaces, warped products, and Riemannian submersions. This turns out tohave important consequences in the study of manifolds with special spinor �elds.

2.1 Spinorial structures

In this section we introduce the basic concepts needed to deal geometrically withspinors. We review them in decreasing order of generality, starting with the puzzlingconcept of a spin structure. We refer to Chapter 3 for a complete discussion on theexistence of such structures and for their interaction with the global structure of amanifold (see also [Bou05] for a short introduction).

2.1.1 Spin structures and spinorial metrics

Let M n be an oriented n-dimensional manifold and let P be a principal bundleoverM with group G. Recall that every representation �WG ! Aut.V / de�nes anassociated vector bundle E, denoted by E D P �� V , de�ned as the quotient ofP � V by the right G-action

g � .u; X/ ´ .ug; �.g�1/.X//:

The equivalence class of .u; X/ is denoted by Œu; X� and the space of smooth sec-tions of E is denoted by �.E/.

The principal bundle of positive linear frames ofM will be denoted by

PGLCnM �! M:

40 2. Geometrical aspects

Let„W eGLC

n �! GLCn

denote the universal covering of GLCn .

Definition 2.1. A spin structure on an n-dimensional manifold M is given by a

principal eGLCn -bundle PeGLC

n

M together with a projection

‚WPeGLC

n

M �! PGLCnM;

making the diagram

eGLCn

a 7! Qua //

„

��

PeGLC

n

M

‚

��

""❊❊❊

❊❊❊❊

❊

M

GLCn A 7!‚. Qu/A

// PGLCnM

;;✇✇✇✇✇✇✇✇✇✇

commute for each Qu 2 PeGLC

n

M .

Equivalently, ‚ is �bre-preserving and satis�es ‚. Qua/ D ‚. Qu/„.a/; for allQu 2 P

eGLCn

M and a 2 eGLCn .

Consider now a Riemannian metric g on M and let PSOnM be the principal

SOn-bundle of positive g-orthonormal frames overM . We denote by { the repre-sentation of SOn on Cln obtained by extending every linear map AW R

n ! Rn of

SOn to an algebra morphism of Cln (see Corollary 1.7). We denote by Cl.M/ thevector bundle associated to PSOn

M for the representation

Cl.M/ D PSOnM �{ Cln :

Recall that Cl.M/ (which is called the Clifford bundle of .M; g/) is the set ofequivalence classes Œu; a� of pairs (u 2 PSOnM; a 2 Cln) with respect to theequivalence relation Œu; a� D ŒuA; {.A�1/.a/�, for all A 2 SOn. Each �bre ofCl.M/ has a natural structure of a complex algebra, the product being de�ned byŒu; a� � Œu; b� ´ Œu; a � b� (actually the �bre of Cl.M/ at some point x 2 M is justthe Clifford algebra Cl.TxM;gx/). Consequently, the sections of Cl.M/ form acomplex algebra over C1.M/.

The algebraic results of the �rst chapter yield directly the following proposition.

2.1. Spinorial structures 41

Proposition 2.2. The tangent bundle of M is a sub-bundle of Cl.M/. The Clif-ford bundle ofM is a complex vector bundle of rank 2n, canonically isomorphic toƒ�M ˝ C, the vector bundle of complex-valued exterior forms onM .

Note that this construction can be carried over to E-objects, i.e., to the casewhere one considers a general linear representation � of SOn on a vector space Eand the associated bundles

EgM D SOgM �� E:Definition 2.3. A spinorial metric on a Riemannian manifold .M; g/ of dimensionn is given by a principal Spinn-bundle PSpinn


� WPSpinnM �! PSOn

M;

making the diagram

Spinna 7! Qua //

�

��

PSpinnM

$$❍❍❍

❍❍❍❍

❍❍

�

��

M

SOnA 7!�. Qu/A

// PSOnM

::✈✈✈✈✈✈✈✈✈

commute for each Qu 2 PSpinnM .

The spinorial metric is said to be subordinated to a spin structure P if PSpinnM

is a reduction to Spinn of theeGLCn -principal bundle PeGLC

n

M .

Proposition 2.4. On a manifold endowed with a spin structure, each Riemannianmetric gives rise to a spinorial metric.

The proof is clear. The Riemannian metric g on M determines the subbundle

PSOnM inside PGLC

nM . Since the group Spinn is the inverse image of SOn in

eGLCn

under„, the inverse image ofPSOnM inPeGLC

n

M is a Spinn-principal bundle, which

determines the associated spinorial metric.By Proposition 2.4, it is natural to give up the not so widely used term spinorial

metric, which will be used later in the book, for the more classical term, in thecontext of Riemannian geometry, of a spin manifold.

To PSpinnM we associate a complex vector bundle †M D PSpinn

M ��n †n,called the spinor bundle. The equivariance of � shows that there exists a represen-tation of C1.M/-algebras

Cl.M/ �! End.†M/;


given by

Œu; a�.Œ Qu; �/ D Œ Qu; �n.a/. /�

for each Qu 2 PSpinnM , u D �. Qu/ 2 PSOn

M , a 2 Cln and 2 †n. This action ofCl.M/ on †M is called the Clifford product and is denoted by

.�; ‰/ 7�! � � ‰:

Local sections of PSpinnM are called local spinorial gauges and sections of †M

spinor �elds or simply spinors.

Proposition 2.5. There exists a canonical Hermitian product h�; �i on †M withrespect to which the Clifford product with every vector of unit norm is unitary.

Proof. Let .�; �/ denote the Hermitian product on†n introduced in Proposition 1.13.If ‰ D Œ Qu; � and ˆ D Œ Qu; '�, de�ne h‰;î ´ . ; '/ and use Corollary 1.37 tocheck that this does not depend on the choice of the spinorial frame Qu. �

Similarly, Theorem 1.39 of Chapter 1, yields directly the following theorem.

Theorem 2.6. IfM is a spin manifold of real dimension n, the complex spinor bun-dle†M carries a real structure jR, for n � 1; 2; 6; 7; 8 mod 8, and a quaternionicstructure jQ, for n � 2; 3; 4; 5; 6 mod 8. The Clifford product with real tangentvectors commutes with jR, for n � 6; 7; 8 mod 8, and anti-commutes with jR, forn � 1; 2 mod 8. The Clifford product with real tangent vectors commutes with jQ,for n � 2; 3; 4 mod 8, and anti-commutes with jQ, for n � 5; 6 mod 8.

2.1.2 Spinorial connections and curvatures

We now develop more advanced tools to deal with spinors.The Levi-Civita connection on PSOn

M induces on PSpinnM a canonical con-

nection as follows (for full details, see [KN63], Section 2.6). When viewed as ason-valued equivariant 1-form onPSOn

M , the connection form! of the Levi-Civitaconnection can be lifted, through the map

� WPSpinnM �! PSOnM

de�ning the spinorial metric, to a son-valued 1-form on PSpinnM . This can be made

into a Spinn-connection form by using the inverse isomorphism between the Liealgebras of Spinn and SOn.

2.1. Spinorial structures 43

One can then, in a very canonical way, de�ne a covariant derivative on the spinorbundle. We denote byr the covariant derivative on TM , as on any Riemannian bun-dle such as Cl.M/, as well as that on †M , induced by the Levi-Civita connection.These covariant derivatives satisfy the following Leibniz rule relative to the Cliffordproduct:

rX .� � �/ D .rX�/ � � C � � rX�; �; � 2 �.Cl.M//;

and

rX.� �‰/ D .rX�/ � ‰ C � � rX‰; � 2 �.Cl.M//; ‰ 2 �.†M/:

It is easy to see that the canonical Hermitian product on†M de�ned in Lemma 2.5is parallel with respect to r.

LetR be the Riemannian curvature tensor, which can be seen as aƒ2M -valued2-form onM . We then have the following theorem.

Theorem 2.7. Let‰ D Œ Qu; � be a .local/ spinor �eld onM written in the gauge Quand let �. Qu/ D .X1; : : : ; Xn/ be the local orthonormal frame onM correspondingto Qu. Then the covariant derivative of ‰ can be computed by the formula

rX‰ D Œ Qu;X. /�C 1

2

X

i<j

g.rXXi ; Xj /Xi � Xj �‰: (2.1)

Moreover, the spin curvature operator

RX;Y ´ ŒrX ;rY �� rŒX;Y �

of r on †M satis�es

RX;Y‰ D 1

2R.X; Y / �‰; ‰ 2 �.†M/:

Proof. Let ! 2 �1.PSOnM; son/ be the connection form of the Levi-Civita con-nection ofM and let Q! D ��1

� ı ��! be the connection form induced on PSpinnM .

By de�nition, if X is a vector �eld on M , uWU ! PSOnM is a local section andvWU ! R

n is a local function, then

rX Œu; v� D Œu; X.v/C !.u�X/.v/�: (2.2)

Similarly, if QuWU ! PSpinnM is a local section and WU ! †n is a local

function, thenrX Œ Qu; � D Œ Qu;X. /C Q!. Qu�X/ � �:


We �x a spinor ‰ locally written ‰ D Œ Qu; � and denote by u D .X1; : : : ; Xn/

the projection of Qu by � (Xi D Œu; ei �). We let v D ei in (2.2) and take the scalarproduct with Xj to obtain

!.u�X/ DX

i<j

eijg.rXXi ; Xj /;

where eij 2 son is de�ned by eij .ek/ D ıikej � ıjkei . We then get

Q!. Qu�X/ D ��1� .!.�� Qu�X//

D ��1� .!.u�X//

D ��1�X

i<j

eijg.rXXi ; Xj / D 1

2

X

i<j

g.rXXi ; Xj /ei ^ ej

D 1

2

X

i<j

g.rXXi ; Xj /ei � ej ;

thus proving (2.1). Let now � 2 �2.PSOnM; son/ be the curvature form of ! and

let z� D ��1� ı �� be the curvature form on PSpinn

M . With the notations abovewe have

RX;YXi D RX;Y Œu; ei � D Œu;�.u�X; u�Y /.ei/�;

so that�.u�X; u�Y / D

X

i<j

eijg.RX;YXi ; Xj /;

and �nally

RX;Y‰ D Œ Qu; z�. Qu�X; Qu�Y / � �D Œ Qu; ��1

� .�.u�X; u�Y / � �

Dh

Qu; 12

X

i<j

g.RX;YXi ; Xj /ei � ej � i

D 1

2R.X; Y / � ‰: �

Corollary 2.8. If Xi is an orthonormal frame at x 2 M and X 2 TxM , then

nX

iD1Xi � RX;Xi

‰ D �12Ric.X/ �‰: (2.3)

2.2. Spinc and conformal structures 45

Proof. According to Theorem 2.7 together with the �rst Bianchi identity and therelation in Cl.M/

Xj �Xk �Xi D Xi �Xj �Xk � 2Xj ıik C 2Xkıij (2.4)

we may write

nX

iD1Xi � RX;Xi

‰ D 1

2

nX

iD1Xi �R.X;Xi/ �‰

D 1

4

X

i;j;k

g.R.X;Xi/Xj ; Xk/Xi �Xj �Xk �‰

D 1

12

X

i;j;k

.g.R.X;Xi/Xj ; Xk/Xi �Xj �Xk �‰C g.R.X;Xj /Xk ; Xi/Xj �Xk � Xi �‰C g.R.X;Xk/Xi ; Xj /Xk �Xi �Xj �‰/

D 1

12

X

i;j;k

..�2Xj ıik C 2Xkıij /g.R.X;Xj /Xk ; Xi /

C .�2Xj ıik C 2Xiıkj /g.R.X;Xk/Xi ; Xj // �‰

D 1

12.�6Ric.X// �‰: �

2.2 Spinc and conformal structures

2.2.1 Spinc structures

Definition 2.9. Let .M; g/ be an oriented Riemannian manifold. A Spinc structureon .M; g/ is given by a principal Spincn-bundle PSpincnM , a principal U1-bundlePU1

M called the auxiliary line bundle, and a principal bundle morphism

� D .�1; �2/WPSpincnM 7�! PSOnM � PU1M;

making the diagram

Spincna 7! Qua //

�c

��

PSpincnM

&&◆◆◆◆

◆◆◆◆

◆◆◆

�

��

M

SOn �U1.A;z/7!�. Qu/.A;z/

// PSOnM � PU1

M

77♣♣♣♣♣♣♣♣♣♣♣♣

commute for every Qu 2 PSpincnM .


Remark 2.10. In this de�nition, PSOnM � PU1M denotes the Whitney product,

that is, the inverse image of the Cartesian product of these principal bundles underthe diagonal inclusion ofM intoM �M .

Given a Spinc structure on M , we introduce as before the bundle of complexspinors†M D PSpincnM ��c†n, on which Cl.M/ acts, on the left, by Clifford mul-tiplication. Suppose now that we are given a connection A on PU1

M . This induces,together with the Levi-Civita connection of PSOn

M , a connection on PSpincnM , andhence a covariant derivative, denoted by rA, on †M , which satis�es the Leibnizrule

rAX .� �‰/ D .rX�/ �‰ C � � rA

X‰; � 2 �.Cl.M//; ‰ 2 �.†M/;

and preserves the canonical Hermitian product on †M .In general, by a Spinc manifold we will understand a 5-tuple .M; g;S; L; A/,

where .M; g/ is an oriented Riemannian manifold, S is a Spinc structure, L is thecomplex line bundle associated with the auxiliary bundle of S for the canonicalrepresentation � of U1 on C, and A is a connection on L.

Consider a local spinor �eld‰ D Œ Qu; �, where Qu is a local section of PSpincnM ,and let �. Qu/ D .u; /, where is a local section ofPU1

M and u D .X1; : : : ; Xn/ isa local section of PSOnM . If i! 2 �1.PU1

M; iR/ denotes the connection form ofA and id! 2 �2.PU1

M; iR/ is its curvature form (which is projectable to a 2-formG 2 �2.M; iR/), we have the following analogue to Theorem 2.7

Theorem 2.11. The covariant derivative of the spinor �eld ‰ is given by

rAX‰ D Œ Qu;X. /�C 1

2

X

i<j

g.rXXi ; Xj /Xi �Xj �‰ C 1

2i!. �X/‰:

The curvature operator

RAX;Y ´ ŒrA

X ;rAY � � rA

ŒX;Y �

of rA has the following expression:

RAX;Y‰ D 1

2R.X; Y / � ‰ C 1

2G.X; Y / � ‰:

Note that L is isomorphic to the complex line bundle associated to PSpincnM forthe representation �2 of Spin

cn on C,

L D PU1M �� C D PSpincnM ��2

C:

The line bundle L is called the determinant line bundle of the Spinc structure, andwe denote by rL;A the covariant derivative induced by A on L.


Lemma 2.12. A Spinc structure on M with trivial determinant line bundle canbe canonically identi�ed with a spin structure. If moreover, the connection A isthe canonical �at connection, then rA corresponds to the spinorial connection rthrough the identi�cation of spinor bundles.

Proof. If L is trivial, the same holds for PU1M , so we may �nd a global section

of the latter. Then the pull-back (denoted PSpinnM ) of PSOnM � by � is a spin

structure onM , thus proving the �rst statement. Moreover, if A is the canonical �atconnection, may be chosen parallel, so the connection on PSpincnM restricts to theLevi-Civita connection on PSpinn

M . �

Consequently, all the results concerning Spinc structures obtained below willhold, in particular, for usual spin structures.

The next theorem justi�es the above de�nitions, in the sense that it shows thatspin and Spinc structures come up naturally when one deals geometrically with realand complex spinors.

Theorem 2.13. Let .M n; g/ be an oriented Riemannian manifold and let E bea Cl.M/-module .resp. a Cl.M/-module/, that is, a vector bundle over M suchthat there exists a bundle morphism Cl.M/ � E ! E .resp. Cl.M/ � E ! E/

compatible with the Clifford product of Cl.M/ .resp. Cl.M//. Furthermore, sup-pose that for each x 2 M , Ex is irreducible as a Cl.M/x-module .resp. as aCl.M/x-module/. Then E is isomorphic to the spinor bundle of a Spinc structure.resp. spin structure/ onM .

Proof. We will not treat here the real case, which is similar to the complex one, butjust algebraically more involved.

Let h�; �i be a Hermitian product on E with the property that the action of unitvector �elds on E is unitary. The existence of such a Hermitian product followsfrom Proposition 1.35, using a partition of unity. We consider �rst the case where thedimension n ofM is even. A frame u 2 .PSOnM/x can be seen as an isomorphism

uW Cln �! Cl.M/x; 7�! Œu; �;

satisfying u.Adb.a// D u�c1.b/.a/, for all b 2 Spinn; a 2 Cln, where �c1.b/denotes the projection of �c.b/ 2 SOn �U1 on SOn. For each x 2 M , we de�ne

Px D fvW†n ! Ex unitaryI there exists u 2 .PSOnM/x; v.a � / D u.a/ � v. /;for all a 2 Clng;

as well as the projection � WPx ! .PSOnM/x which maps v to u. Of course, u is

unique because the representation of SOn on Cln is faithful.


For �xed u, the elements of ��1.u/ are exactly the Cln-module isomorphisms†n ! Ex preserving the Hermitian products of these two spaces. Note that theCln-module structure on Ex is obtained from the identi�cation by u of Cl.M/xand Cln. The group Spin

cn acts freely on the right on Px by v 7! vb, where

vb. / ´ v.b � /

for every b 2 Spincn. Indeed, on the one hand, vb is unitary, by Corollary 1.37. Onthe other hand, if B D �c1.b/ 2 SOn, then for all a 2 Cln; 2 †n

v � b.a / D v.b � a � /

D u.b � a/ � v. / D u.b � a/ � u.b�1/ � v.b � /D u.Adb.a// � vb. /D uB.a/ � vb. /;

thus showing that vb 2 Px and

�.vb/ D �.v/�c1.b/: (2.5)

Theorem 1.28 implies that (for every �xedu) theCln-representations†n andExare equivalent, so there exists an isomorphism of Cln-modules †n ! Ex . Corol-lary 1.37 shows that such an isomorphism is necessarily unitary, hence Px is notempty.

Finally, we check that the action of Spincn onPx is transitive. According to (2.5),every orbit of this action intersects each �ber ��1.u/, so it is enough to showthat every two elements of the same �ber belong to the same orbit. Indeed, ifv1; v2 2 ��1.u/, then v�1

2 ı v1 is a unitary endomorphism f of †n which com-mutes with every element of Cln. This shows that f 2 U1, since the center of theendomorphism algebra of any complex vector space is C. This shows that thereexists z 2 U1 � Spincn such that v1 D v2z.

Consider now local trivializations of PSOnM and E. The above discussionshows that the union P of all sets Px is a principal Spincn-bundle overM . More-over, (2.5) implies that P is actually a Spinc structure. Indeed, one de�nes

PU1M D P ��c2 U1

and� WP 7�! PSOn

M � PU1M; �.u/ D .�.u/; Œu; 1�/;

which satisfy

�.va/ D .�.va/; Œva; 1�/D .�.v/�c1.a/; Œv; �c2.a/�/ D �.v/�c.a/ :


By construction, the spinor bundle associated with this Spinc structure is isomorphicto E as a Cl.M/-module.

For n odd, we note that End.E/ is isomorphic to ClC.M/ or to Cl�.M/,according to whether the complex volume element acts by 1 or �1 onE. ReplacingCl.M/ by Cl˙.M/, the above argument then holds verbatim. �

The results in this section were independently obtained by Th. Friedrich andA. Trautman in the framework of Lipschitz structures [FT00].

2.2.2 Weyl structures

Let M n be an oriented manifold and let PGLCnM be the principal GLC

n -bundle of

oriented frames on M . We de�ne a family of oriented real line bundles Lk on Mby

Lk D PGLCnM �detk=n R :

Obviously, we have Lk ˝ Ll ' LkCl . The bundle Lk is called the density bundleof weight k. For each k 2 R, the complement of the zero section in Lk has twoconnected components, de�ned by

Lk˙ D PGLCnM �detk=n R

�˙:

A conformal structure onM is a section c of Sym2.T�M/˝L2 such that c.X;X/ 2L2C for every non-zero vector X on M . The existence of such a structure yields areduction PCOC

nM of the positive frame bundle ofM to the conformal group COC

n .IfM is a conformal manifold, we also can write

Lk D PCOCnM �detk=n R:

Definition 2.14. A Weyl structure on an oriented conformal manifold .M n; c/ isa torsion-free connection on PCOC

nM , or equivalently, a torsion-free connection

on PGLCnM whose induced covariant derivative on sections of Sym2.T�M/ ˝ L2

preserves c.

The foundational theorem of conformal geometry, due to H. Weyl, says thatthere is a bijective correspondence between Weyl structures and linear connectionson L ´ L1. Indeed, every Weyl structure induces a connection on the principalR

�C-bundle

LC ´ PGLCnM �detk=n R

�C;


hence a linear connection on L. Conversely, if D and DL are the covariant deriva-tives induced by a Weyl structure on TM and on L, respectively, then the followinganalogue of the Koszul formula holds:

2c.DXY;Z/ D DLX.c.Y; Z//CDL

Y .c.X;Z//�DLZ.c.X; Y //

C c.Z; ŒX; Y �/ � c.Y; ŒX;Z�/� c.X; ŒY; Z�/:

This shows the existence and the uniqueness ofD, givenDL.One usually calls the above covariant derivative D on TM a Weyl structure,

too. AWeyl structureD is called closed (resp. exact) ifDL is �at (resp. if L carriesa global DL-parallel section). As Riemannian metrics of the conformal class ccorrespond to sections of LC, it follows immediately that D is closed (resp. exact)if and only of D is locally (resp. globally) the Levi-Civita connection of a metricg 2 c.

For every metric g in the conformal class c there exists a unique non-vanishingsection l of LC (trivializing L) such that

c D g ˝ l2: (2.6)

LetDL be a torsion-free connection on L and let � 2 �1.M;R/ be its connec-tion form, written in the gauge l :

DLX l D �.X/l; X 2 TM: (2.7)

An easy calculation using (2.6), together with the Koszul formula above, gives

DXY D rXY C �.X/Y C �.Y /X � g.X; Y /T; X; Y 2 TM; (2.8)

where r is the Levi-Civita covariant derivative of g and T is the vector �eld dualto � with respect to g. The 1-form � is called the Lee form of .D; g/.

Consider a conformal change of the metric Ng D e�2f g, where f is a functiononM . Then Nl D ef l , and (2.7) shows that the Lee form N� associated with .D; Ng/is related to � by

N� D � C df:

Consequently, a Weyl structure is closed (resp. exact) if and only if its Lee form isclosed (resp. exact) for one metric in the conformal class – and thus for all of them.

We now study the curvature tensor of a Weyl structure D, de�ned as usual forvector �elds X , Y , and Z by

RDX;YZ ´ ŒDX ; DY �Z �DŒX;Y �Z:

Let F denote the curvature tensor of the connectionDL, viewed as a 2-form onMand called the Faraday form of D. In contrast to the Riemannian case, the image


of a pair of vectors by RD is an endomorphism of TM which is not always skew-symmetric. Of course, the symmetric part of RDX;Y is just F.X; Y /Id, where Id

denotes the identity of TM . We denote by RDa .X; Y / the skew-symmetric partof RDX;Y , that is

RD D RDa C F ˝ Id: (2.9)

In Chapter 9 we will need the following lemma.

Lemma 2.15. We have

SX;Y;Z

c.RDa .T; X/Y;Z/ D SX;Y;Z

.F.Y; X/c.Z; T //: (2.10)

Proof. Using formula (2.9) and the �rst Bianchi identity

ST;X;Y

RDT;XY D 0 ;

we compute

SX;Y;Z

c.RDT;XY;Z/ D SX;Y;Z

.c.RDY;XT;Z/C c.RDT;YX;Z//

D SX;Y;Z

.�c.RDY;XZ; T /C 2F.Y; X/c.Z; T /

� c.RDT;YZ;X/C 2F.T; Y /c.X;Z//:

Thus

SX;Y;Z

c.RDT;XY;Z/ D SX;Y;Z

.F.Y; X/c.Z; T /C F.T; Y /c.X;Z// (2.11)

which, together with (2.9), yields (2.10). �

We de�ne the Ricci tensor of D by

RicD.X; Y / D trfV 7! RDa .V; X/Y gand the scalar curvature ScalD as the conformal trace of RicD . In particular, ScalD

is not a function, but a density of weight �2 (as c maps T�M ˝ T�M to L�2).

Lemma 2.16. The skew-symmetric part RicDa of the Ricci tensor RicD is equal to2�n2F .

Proof. Let fRicD.X; Y / D trfV 7! RDV;XY g. The Bianchi identity applied to RD

easily yieldsfRicD.X; Y / � fRicD.Y; X/ D �nF.X; Y /: (2.12)

On the other hand, (2.9) shows that

fRicD.X; Y / � RicD.X; Y / D trfV 7! F.V;X/Y g D �F.X; Y /;which, together with (2.12), proves the desired result. �


Definition 2.17. A Weyl structure is called Weyl–Einstein if the trace-free part ofthe symmetric part of the Ricci tensor RicD vanishes, or, equivalently, if

RicD D 1

nScalDc C 2� n

2F:

Weyl–Einstein structures will play an important role in Section 9.2.

2.2.3 Spin and Spinc conformal manifolds

Definition 2.18. A spin structure on an oriented conformal manifold .M; c/ is givenby a principal CSpinn-bundle PCSpinn


� WPCSpinnM �! PCOC

nM

making the diagram

CSpinna 7! Qua //

�

��

PCSpinnM

$$■■■

■■■■

■■

�

��

M

COCn A 7!�. Qu/A

// PCOCnM

;;✈✈✈✈✈✈✈✈✈✈

commute for every Qu 2 PCSpinnM .

Definition 2.19. A Spinc structure on .M; c/ is given by a principal CSpincn-bundlePCSpincnM , a principal U1-bundle PU1

M , and a projection

� D .�1; �2/WPCSpincnM �! PCOCnM � PU1

M

making the diagram

CSpincna 7! Qua //

�c

��

PCSpincnM

''◆◆◆◆

◆◆◆◆

◆◆◆

�

��

M

COCn � U1

.A;z/7!�. Qu/.A;z/// PCOC

nM � PU1

M

88qqqqqqqqqqqq

commute for every Qu 2 PCSpincnM .


Like before, every spin structure can be canonically identi�ed with a Spinc struc-ture with trivial determinant bundle.

Given a Spinc structure on .M; c/, and k 2 R, we de�ne the spinor bundles ofweight k by

†.k/M D PCSpincnM ��c;k

n†n;

where �c;kn were de�ned in (1.12). Note that †.k/M is canonically isomorphic to†.0/M ˝R Lk for every k. Since we deal with complex bundles, it is more con-venient to complexify the density bundle L by de�ning LC D L ˝R C, so that†.k/M D †.0/M ˝C Lk

C. From now on all tensor products are taken over the �eld

of complex numbers.If g is any choice of metric in the conformal class c, the usual spinor bundle

†gM may be identi�ed with any of the weighted spinor bundles †.k/M by re-stricting the frame bundle of the latter to PSpincnM . This induces a family of isomor-phisms

ˆ.k/W†gM �! † NgM ; Œs; u� 7�! Œs�; ��ku�;

where Ng D ��2g, s 2 PgSpincn

M and u is the expression of a spinor �eld in thespinor frame s. It is easily checked that

jˆ.k/ j2Ng D ��2k j j2g (2.13)

for any spinor .

Corollary 2.20. There exists a canonical Hermitian product on †.0/M , and moregenerally a Hermitian pairing

†.k/M ˝†.l/M �! LkClC

;

satisfying

hX �‰;î D �h‰;X �î 2 LkCl�1C

for X 2 TM , ‰ 2 †.k/M , and ˆ 2 †.l/M .

Similarly to the Riemannian case, ifM has a spin structure, then anyWeyl struc-ture D induces a connection on PCSpinn

M , and therefore a covariant derivative,denoted by D.k/, on †.k/M . If M has a Spinc structure, then any Weyl structureonM and any linear connection A on the determinant bundle of the Spinc structure,induce together a covariant derivative, denoted by DA;.k/, on †.k/M . The Hermi-tian pairing de�ned above is parallel with respect to this connection.


In contrast to the Riemannian case, on a conformal manifold there is no Clif-ford bundle containing the tangent bundle as a sub-bundle. Some care to de�ne theClifford product is thus required. One possibility would be to view c as a metricon TM ˝L�1 and de�ne the Clifford bundle associated with this Euclidean vectorbundle, as well as the Clifford product acting on weighted spinors. Nevertheless,we will not adopt this point of view since it does not seem well adapted to spinorialcalculus.

A better idea is the following: rather than having a Clifford product on each†.k/M by abstract objects which cannot be identi�ed with forms or vectors onM ,we de�ne a Clifford product by usual forms (or vectors), which no longer preservesthe weight of spinors.

More precisely, if ‰ D Œ Qu; � is a spinor of weight k and ! D Œu; �� a p-form(where the conformal frame u is the projection of the spin frame Qu), then we de�nethe Clifford product ! �‰ as the spinor of weight k �p given by ! �‰ D Œ Qu; � � �.It is easy to see that this de�nition does not depend on the representatives of ‰and !.

One similarly de�nes the Clifford product of a spinor by a vector, which raisesthe conformal weight by one. The Clifford product de�ned in this way respects theLeibniz rule

D.k�p/X .! �‰/ D .DX!/ �‰ C ! �D.k/

X ‰

for X 2 TM , ! 2 �pM , and ‰ 2 �.†.k/M/, and

D.kC1/X .Y �‰/ D .DXY / �‰ C Y �D.k/

X ‰;

for X 2 TM , Y 2 �.TM/, and ‰ 2 �.†.k/M/.We now �x a metric g in the conformal class. As already mentioned, for each

k 2 R there exists a canonical isomorphism

I k W†.k/M �! †M

identifying the spinor bundle of weight k of .M n; c/ with the spinor bundleof .M n; g/. This isomorphism is compatible with the Clifford product, that is,I k�p.! � ‰/ D ! � I k.‰/, for all ! 2 ƒpM; ‰ 2 †.k/M: To simplify nota-tions, we will from now on omit I k when there is no risk of confusion. LetD be aWeyl structure onM and let � be the Lee form of .D; g/.

Theorem 2.21. The covariant derivatives induced on †M by the Levi-Civita con-nection of g and on †.k/M by the Weyl structure D are related by

D.k/X ˆ D rXˆ � 1

2X � � �ˆC

�k � 1

2

��.X/ˆ; ˆ 2 �.†.k/M/: (2.14)


Proof. Let ! 2 �1.PSOnM; son/ be the connection form of the Levi-Civita con-nection of .M; g/ and let Q! D ��1

� ı ��! be the induced connection form onPSpinn

.M; g/. We denote by !D 2 �1.PCOCnM; con/ the connection form of D

and by Q!D D ��1� ı ��!D 2 �1.PCSpinn

M; con/, the induced connection formon PCSpinn

M . We will identify for the rest of the proof PSOnM and PSpinn.M; g/

with sub-bundles of PCOCnM and PCSpinn

.M; c/ respectively. For each point ofM ,there exist a neighborhood U , a spin frame QuWU ! PSpinn

.M; g/, and a function�WU ! †n, such that ˆ D Œu; �� over U . Let fe1; : : : ; eng be the standard basis ofRn, u D � ı QuWU ! PSOn

M , and denote Xi D Œu; ei �, so that u D fX1; : : : ; Xng.If �i are the components of � with respect to this basis, (2.8) yields

Œu; .!D � !/.u�Xi /ej � D DXiXj � rXi

Xj D �iXj C �jXi � T ıij

Dhu; �iej C �j ei �

nX

lD1�lelıij

i

Dhu;��i In C

nX

lD1�leli

�ej

i;

so

.!D � !/.u�Xi / D �i In CnX

lD1�leli 2 RIn ˚ son ' con:

According to Theorem 1.25 we obtain

. Q!D � Q!/. Qu�Xi / D ��1� ..!D � !/.u�Xi //

D �i C 1

2

nX

lD1�l .el � ei C ıli /;

and consequently

DXiˆ � rXi

ˆ D Œu; .�kn/�. Q!D � Q!/. Qu�Xi /��

D Œu; k�i��Chu;1

2

nX

lD1�l .el � ei C ıli/�

i

D k�.Xi /ˆC 1

2.� �Xi �ˆC �.Xi /ˆ/;

which is equivalent to (2.14). �


2.3 Natural operators on spinors

There are different approaches to de�ne �rst-order differential operators acting ona given so.n/ irreducible vector bundle. We �rst mention the classical approachbased on representation theory.

We start by introducing the general notion of Stein–Weiss operators, sometimesknown as gradients cf. [SW68], [Bra96], and [Bra97]. If .M; g/ is an orientablen-dimensional Riemannian manifold, every �nite-dimensional SOn representationon some vector space V induces a vector bundle V over M . If V and W are twosuch vector bundles, denote by Diff.V;W/ the space of smooth-coef�cient lineardifferential operators from sections of V to sections ofW. Take an SOn-irreduciblebundle U, and consider the covariant derivative

r 2 Diff.U;T�M ˝ U/:

The bundle on the right-hand side splits up into SOn-irreducible bundles:

T�M ˝ U ´ V D V0 ˚ � � � ˚ VN ;

where of course the integer N depends on U. It is well known [Feg87] that this is amultiplicity free decomposition:

Vi ŠSOnVj H) i D j:

Thus we can project onto the Vj summand and de�ne the gradients

rj ´ ProjjV ır:

The Dirac and twistor operators, and the exterior and interior derivatives d and ı areall (constant multiples of) Stein–Weiss operators. The point of view in [SW68] isto consider gradients as generalizations of the two-dimensional Cauchy–Riemannoperator.

The second approach which we shall develop in this section is due to P. Gaudu-chon [Gau95a] and uses the decomposition of a vector bundle into eigenbundlesassociated with the principal symbol morphism of the corresponding �rst-order dif-ferential operator. Of course, these two points of view are equivalent, but the secondone has the advantage of being explicit and elementary.

2.3. Natural operators on spinors 57

2.3.1 General algebraic setting

Let V andW be two �nite-dimensional K-vector spaces endowedwith an Euclideanor Hermitian scalar products depending on whether K D R or C.

Lemma 2.22. Let H be a nontrivial homomorphism

H W V �! W;

' 7�! H.'/ D �;

and H� its metric adjoint, mapping W to V . We denote by 0 < �1 < � � � < �Nthe positive eigenvalues of the endomorphism H ı H� and by Vj .resp. Wj / theassociated eigenspaces ofH� ıH .resp.H ıH�/.

Then, for all j 2 f1; : : : ; N g, the map1

p�jH WVj �! Wj

is an isometry.

Proof. First note that the endomorphisms H� ı H and H ı H� have the samepositive eigenvalues. It is clear that the restriction of H to the eigenspace Vj is anisomorphism from Vj to Wj whose inverse is 1

�jH�. Hence, the composition

Vj

1p�jH

��! Wj

1p�jH�

��! Vj

is the identity of Vj . �

LetV0 ´ KerH and W0 ´ KerH�:

It is interesting to note that we have the diagram

V D V0 ˚ V1

H

��

˚ � � � ˚ Vj

H

��

˚ � � � ˚ VN

H

��W1

��11 H�

OO

˚ � � � ˚ Wj

��1jH�

OO

˚ � � � ˚ WN

��1N H�

OO

˚W0

Now, for k 2 f0; 1; : : : ; N g, de�ne the two projections ProjkV and ProjkW , by

ProjkV W V D V0 ˚V1 ˚ � � � ˚VN �! Vk ;

' D '0 ˚ '1 ˚ � � � ˚ 'N 7�! 'k ;

andProjkW WW DW0 ˚W1 ˚ � � � ˚WN �!Wk ;

� D �0 ˚ �1 ˚ � � � ˚ �N 7�! �k:


Lemma 2.23. (1) For all ' D '0 C '1 C � � � C 'N 2 V , one has

j'j2 �NX

jD1

1

�jj ProjjW ıH.'/j2;

and equality holds if and only if '0 D 0, i.e., ' 2 .KerH/?.(2) Let� D �N be the maximum eigenvalue of the endomorphismH ıH�. One

has

j'j2 � 1

�jH'j2; ' 2 V; (2.15)

and

j'j2 D 1

�jH'j2 ”

8<:'0 D 0 and

ProjjW ıH.'/ D 0; for j 2 ¹1; : : : ; N � 1º:

In other words, equality holds in .2.15/ if and only if

H�.H'/ D �':

Proof. By Lemma 2.22, for all ' 2 V , one has

j'j2 D j'0j2 C j'1j2 C � � � C j'N j2

D j'0j2 CNX

jD1

1

�jj ProjjW ıH.'/j2:

This gives the �rst part of the lemma. For the last statement, it is suf�cient to notethat

j'j2 D j'0j2 CNX

jD1

1

�jj ProjjW ıH.'/j2

� j'0j2 C 1

�

NX

jD1j ProjjW ıH.'/j2

� 1

�jH.'/j2:

Moreover, equality in (2.15) holds if and only if ' 2 VN , i.e.,H�.H'/ D �'. �


2.3.2 First-order differential operators

In this section we apply the preceding algebraic set-up to K-vector bundles U,V, and W on an n-dimensional Riemannian manifold .M n; g/ induced by �nite-dimensional SOn representations on U , V , andW , endowed with real or Hermitianscalar products. For any K-vector bundle U of �nite rank, let V ´ T�M ˝R U,where T�M is the cotangent (real) vector bundle. In case K D C, the vector bun-dleV is considered to be a complex vector bundle by the action of complex numberson U.

LetHW�.V/ ´ �.T�M ˝R U/ �! �.W/

be a bundle morphism induced by some SOn-equivariant map

H WV ´ Rn ˝ U �! W;

and denote byH�, its metric adjoint. The positive eigenvalues of the endomorphismH ıH�, denoted by �1 < � � � < �N , are also the eigenvalues of the bundle endo-morphismHıH� of �.W/ and coincide with the positive eigenvalues of the bundleendomorphism H� ı H of �.V/. Moreover, the associated eigenspaces Wj and Vjare sub-bundles ofW and V, j 2 f1; : : : ; N g.

For any linear connection r acting on sections of U, consider the �rst-orderdifferential operator de�ned by

D ´ H ı r:Note that the homomorphism H is equal (up to some factor i , depending on the

convention used) to the principal symbol of the differential operator D. For eachj 2 f1; : : : ; N g, we de�ne the operator D

j by

Dj ´ ProjWj

ıD

and for each k 2 f0; 1; : : : ; N g, the differential operator rk by

rk ´ ProjVkır:

The relation between these operators is expressed by the diagram

�.V/H //

ProjVj

��

�.W/

ProjWj

��

�.U/

r;;✈✈✈✈✈✈✈✈✈

rj ##❍❍❍

❍❍❍❍

❍❍

�.Vj / �.Wj /:1

�jH�

oo


In fact, for all j 2 f1; : : : ; N g, we have the relations

rj D 1

�jH� ı D

j ; D DNX

jD1Dj ; r D r0 C zD;

where the operator zD, called the suspension of D, is given by

zD ŃX

jD1rj :

By de�nition, we thus have

zDW �.U/ �! .KerH/? � �.T�M ˝ U/;

7�! ' DNX

jD1rj ;

which could be identi�ed with the operator D via the isomorphism

zH D Hj.KerH/? W .KerH/? 7�! ImH;

so that

D D zH ı zD:

The operator

r0 ´ r �NX

jD1

1

�jH� ı D

j (2.16)

is called the Penrose operator. It can be characterized by

H ı r0 D 0:

For each point x 2 M , we can apply Lemma 2.23 to get

Lemma 2.24. (1) For all 2 �.U/ and at any x 2 M ,

jr j2 D jr0 j2 CNX

jD1

1

�jjDj j2:


(2) For all 2 �.U/ and at any x 2 M ,

jr j2 �NX

jD1

1

�jjDj j2;

with equality if and only if

r0 D 0:

(3) Let � D �N be the largest eigenvalue of the endomorphism H ı H�; thenfor all 2 �.U/ and at any x 2 M , we have

jr j2 � 1

�jD j2 (2.17)

and

jr j2 D 1

�jD j2 ”

8<:

r0 D 0 and

Dj D 0; for j 2 ¹1; : : : ; N � 1º:

In other words, equality holds in .2.17/ if and only if

H�.D / D �r :

2.3.3 Basic differential operators on spinor �elds

We now apply the results of Section 2.3.2 to the spinor bundle †M of ann-dimensional Riemannian spin manifold .M n; g/. This will lead to two naturaldifferential operators, the Dirac and the Penrose operators. We then examine theirbasic properties and establish their conformal covariance. More precisely, we takeU D W D †M , and r the Levi-Civita connection acting on sections of †M .The morphism H D jT�M˝†M , denoted by the same symbol , where

W Cl.M/˝†M �! †M;

� ˝‰ 7�! � �‰;

is the pointwise Clifford multiplication. In this setup, the operator D will be denotedby D. We have


Definition 2.25. The Dirac operator is the �rst-order differential operator actingon sections of the spinor bundles, given by

D ´ ı r:

Locally, on an open set U � M , we get

DW�.†M/r�! �.T�M ˝†M/

�! �.†M/;

‰ 7�!nX

iD1e�i ˝ rei

‰ 7�!nX

iD1ei � rei

‰;

where fe1; : : : ; eng 2 �U .PSOnM/ is a local orthonormal frame of the tangent bun-dle and fe�

1 ; : : : ; e�ng the dual frame.

Lemma 2.26. The metric adjoint

�W†M �! T�M ˝†M

of the Clifford multiplication is given, on any element ‰ 2 †M , by

�.‰/ D �nX

iD1ei ˝ ei �‰;

and ı � D nId†M :

Proof. Since Cliffordmultiplication is pointwise and bilinear, it is suf�cient to makeuse of Corollary 1.36, to get for all ‰;ˆ 2 †M and X 2 TM , the relations

h .X ˝ˆ/;‰i D hX �ˆ;‰iD �hˆ;X � ‰i

D �nX

iD1X i hˆ; ei �‰i

DDX ˝ˆ;�

nX

iD1ei ˝ ei �‰

E: �

From Lemma 2.26 it follows that the endomorphism ı � has n as the uniquepositive eigenvalue and hence, the suspension zD of the Dirac operator is given, forany tangent vector �eld X and any spinor �eld ‰, by

zDX‰ D � 1nX � D‰;


and the Penrose (or twistor) operator P , i.e., the complement of the Dirac operator,by

PX‰ D rX‰ C 1

nX � D‰:

From Lemma 2.24 it follows that for any spinor �eld ‰, one has

jr‰j2 D jP‰j2 C 1

njD‰j2: (2.18)

2.3.4 The Dirac operator: basic properties and examples

We saw that associated with a spin structure of a Riemannian manifold .M n; g/,there are three essential data sets.

i) The spinor bundle†M D PSpinn

M ��n †n

with the Clifford multiplication

W TM ˝†M �! †M;

X ˝ ‰ 7�! X �‰ ´ �n.X/‰;

where �n is the Clifford multiplication and �n is the spinor representationde�ned in (1.33). This multiplication extends to

W ƒp.TM/˝†M; �! †M;

˛ ˝ ‰ 7�!P

1�i1<��<ip�n˛i1:::ipei1 � � � eip � ‰;

where locally˛ D

X

1�i1<��<ip�n˛i1:::ipe

�i1

^ � � � ^ e�ip;

and e�i D g.ei ; �/ is the dual basis of ei .

ii) The natural Hermitian product h�; �i on sections of †M .

iii) The Levi-Civita connection r on †M .

As we saw, these data satisfy the compatibility conditions

hX �‰;î C h‰;X �î D 0;

Xh‰;î � hrX‰;î � h‰;rXî D 0;

and

rX .Y � ‰/ � rXY �‰ � Y � rX‰ D 0;

for all X , Y 2 �.TM/, ‰, ˆ 2 �.†M/.


Lemma 2.27. The commutator of the Dirac operator with the action, by pointwisemultiplication on the spinor bundle, of a function

f WM �! C;

is given byŒD; f �‰ ´ df � ‰; ‰ 2 �.†M/:

Proof. A local calculation shows that

ŒD; f �‰ D .Df � f D/‰

DnX

iD1ei � rei

.f ‰/ � f D‰

DnX

iD1df .ei /ei �‰ C f D‰ � f D‰

D df �‰: �

Lemma 2.28. The Dirac operator is a �rst-order differential operator, which is

(i) elliptic and

(ii) formally self-adjoint on compactly-supported spinors with respect to the L2

inner productRM h�; �i�g , where �g denotes the volume element.

Proof. (i) Let x 2 M , � 2 T�xM n f0g and f 2 C1.M;R/ such that .df /x D �.

Then the principal symbol

��.D/W†xM �! †xM

is given by

��.D/�‰.x/

�´ DŒ.f � f .x//‰�.x/D .f D‰ C df �‰ � f .x/D‰/.x/

D .df /x �‰.x/D � �‰.x/;

i.e., ��.D/ is Clifford multiplication by �. To see thatD is elliptic, we have to checkthat, for all � 2 T�M n f0g, ��.D/W†xM ! †xM is an isomorphism. Indeed,

� �‰ D 0 H) � � � �‰ D 0 H) �k�k2‰ D 0 H) ‰ D 0:


(ii) Let ˆ;‰ be smooth spinors on M , one of them (say ˆ) with compactsupport. To show that D is formally self-adjoint, we choose normal coordinatesat x 2 M , i.e.,

.reiej /.x/ D 0; 1 � i; j � n;

and compute �rst

hD‰;î DD nX

iD1ei � rei

‰;Ê

D �nX

iD1hrei

‰; ei �î

D �nX

iD1Œei h‰; ei �î � h‰;rei

.ei �ˆ/i�

D �nX

iD1ei h‰; ei �î C h‰;Dî:

Here, we note that the sum on the right-hand side is the divergence of a complexvector �eld. To see this, consider the two vector �elds X1; X2 2 �.TM/ de�nedfor all Y 2 TM by

g.X1; Y /C ig.X2; Y / D h‰; Y �î:

Clearly, X1; X2 have compact support. Therefore,

divX1 C idivX2 DnX

kD1g.rek

X1; ek/C i

nX

lD1g.rel

X2; el/

DnX

kD1.ekg.X1; ek/ � g.X1;rek

ek//

C i

nX

lD1.elg.X2; el/ � g.X2;rel

el//

DnX

kD1ek.g.X1; ek/C ig.X2; ek//

DnX

kD1ekh ; ek � 'i:


Finally we have

hD‰;î D �divX1 � idivX2 C h‰;Dî:

This equation no longer depends on the choice of the coordinates, so we canintegrate overM and use Stokes theorem for the compactly supported vector �eldsX1; X2 to obtain Z

M

hD‰;î�g DZ

M

h‰;Dî�g ;

sinceM has no boundary. �

Lemma 2.29. Let n D 2m. Then

(i) DW�.†˙M/ ! �.†�M/, i.e., the Dirac operator exchanges positive andnegative spinors and

(ii) the eigenvalues of D are symmetric with respect to the origin.

Proof. (i) Recall that

†˙n ´ 1

2.1˙ !C/ �†n;

so †Cn is the subspace on which multiplication by !C is the identity, and †�

n theone on which the multiplication by !C is minus the identity. We therefore get, for‰C 2 �.†CM/,

!C � D‰C D !C �nX

iD1ei � rei

‰C

D �X

i

ei � !C � rei‰C

D �nX

iD1ei � rei

.!C � ‰C/

D �D‰C:

(ii) Let ‰ be an eigenspinor for D, i.e., D‰ D �‰ for � 2 R, with ‰ D‰C C‰�. Then D‰C C D‰� D �‰� C �‰C, yields with i): D‰˙ D �‰�. Sothe spinor x‰ ´ ‰C � ‰� is an eigenspinor of D associated with the eigenvalue��, since

D x‰ D D.‰C � ‰�/ D �‰� � �‰C D ��.‰C �‰�/ D ��x‰: �


Examples 2.30. (i) For M D Rn, †R

n D Rn � C

N , N D 2Œn2 �, so every spinor

‰ 2 �.†Rn/ is in fact a map ‰W R

n ! CN . The Dirac operator is given by

D DnX

iD1ei � @i ;

and acts on differentiable maps from Rn to CN , where @i D rei. Then

D2 D� nX

iD1ei � @i

�� nX

jD1ej � @j

�

DX

i;j

ei � ej � @i@j

D �X

i

@2i CX

i<j

ei � ej � @i@j CX

i>j

ei � ej � @i@j

D �X

i

@2i CX

i<j

ei � ej � @i@j CX

i<j

ej � ei � @j @i

D �X

i

@2i CX

i<j

ei � ej � .@i@j � @j@i /

D �nX

iD1@2i

D

0B@�

:: :

�

1CA :

(ii) In the caseM D R2, we have Cl2 D C.2/, the complex volume element is

!C D ie1 � e2, and one can identify the spinor bundle †2 D †C2 ˚†�

2 D C ˚ C,with †C

2 D spanC.e1 C ie2/ and †�2 D spanC.1 � ie1 � e2/. Then, each spinor

�eld ‰ 2 �.†M/ is given by two complex functions f , gW R2 ! C, such that

‰ D f .e1 C ie2/C g.1 � ie1 � e2/:

The Dirac operator acting on ‰, is then

D‰ D .e1 � @1 C e2 � @2/Œ.e1 C ie2/f C .1 � ie1 � e2/g�D �.@1 C i@2/f .1 � ie1 � e2/C .@1 � i@2/g.e1 C ie2/

D 2.�@ Nzf .1 � e1 � e2/C @zg.e1 C ie2//;


where @ Nz D 12.@1 C i@2/ and @z D 1

2.@1 � i@2/. That is,

D D�

0 2@z

�2@ Nz 0

�

in the basis fe1 C ie2; 1 � ie1 � e2g of †2. Hence the Dirac operator D can beconsidered as a generalization of the Cauchy–Riemann operator.

(iii) The Clifford bundle Cl.M/. For a Riemannian manifold .M n; g/, recallthat the vector bundle Cl.M/ de�ned by

.Cl.M//x D Cl.TxM;gx/

can be seen as a vector bundle associated to PSOnM . Indeed, by the universal prop-erty (1.7), one can extend the standard representation

SOn �! Aut.Rn/ to �W SOn �! Aut.Cln/;

so that

Cl.M/ D PSOnM �� Cln :

From (1.13) we have

v �‰ ' v ^‰ � v ³‰

under the isomorphism Cl.TxM;gx/'! ƒ�.TxM/. The differential and its adjoint

can be locally written as

d DnX

iD1ei ^ rei

and ı D �nX

iD1ei ³ rei

:

If we de�ne the Dirac operator as before, we have

D ńX

iD1ei � rei

' d C ı:

This is the “square root” of the Laplacian, since on ƒ�.TM/

D2 ' .d C ı/2 D dı C ıd D �:


2.3.5 Conformal covariance of the Dirac and Penrose operators

Let .M n; g/ be a Riemannian spin manifold. For a conformal change of the metric

Ng ´ e2ug; uWM �! R;

the given spin structure on .M n; g/ induces a spin structure on .M n; g/. An iso-morphism of the two SOn-principal �bre bundles is given by

GuW SOgM �! SO NgM;

fX1; : : : ; Xng 7�! fe�uX1; : : : ; e�uXng:

Note that in this section we denote by SOgM (resp. SpingM ), the SOn-principal(resp. Spinn-principal) bundle associated with .M n; g/ instead of PSOn

M (resp.PSpinn

M ).The spin structure induced on .M n; Ng/ is de�ned up to isomorphism by the com-

mutative diagram

SpingzGu //

�g

��

Spin NgM

� Ng

��SOgM

Gu

// SO NgM:

All the arrows in the diagram are supposed to be invariant under the group action.Let �nW Spinn ! Aut.†n/ be the spinor representation. An isomorphism of theassociated spinor bundles is explicitly given by

zGuW †M D SpingM ��n †n �! † xM D Spin NgM ��n †n;

‰ D Œs; � 7�! x‰ D Œ zGu.s/; �:

This map is an isometry with respect to the Hermitian product on the spinor bun-dles. Together with the corresponding isometry of the tangent bundle, given byX 7! xX ´ e�uX , we have the following relation between the associated Cliffordmultiplications “ � ” and “ N� ”

xX N� x‰ D X �‰

for every X 2 �.TM/ and ‰ 2 �.†M/.We are ready now to prove explicitly the conformal covariance of the Dirac

operator, established by N. Hitchin [Hit74] (see also [Bau81] and [Hij86b]).


Proposition 2.31. Let .M n; g/ be an n-dimensional Riemannian spin manifold andNg D e2ug a conformal change of the metric, then one has

xD.e� n�12 u x‰/ D e� nC1

2 uD‰ (2.19)

for every ‰ 2 �.†M/.

Remark 2.32. It follows therefore, that the dimension of the space of harmonicspinors

H ´ f‰ 2 �.†M/I D‰ D 0gis conformally invariant. However, in contrast to the Laplace–Beltrami operatoron p-forms, the dimension of H for D is not topologically invariant (see [Hit74]and [Bär96]).

In order to prove Proposition 2.31, we start by relating the Levi-Civita connec-tions associated with two conformally related metrics:

Proposition 2.33. Let .M n; g/ be a Riemannian spin manifold and consider a con-formal change of the metric given by Ng D e2ug. Then

xrX x‰ D rX‰ � 1

2X � du �‰ � 1

2X.u/x‰ (2.20)

for all ‰ 2 �.†M/, X 2 �.TM/.

Proof. This is a direct consequence of Theorem 2.21 applied to the Weyl structureD D xr acting on spinors of weight 0 whose Lee form with respect to g is � D du.Nevertheless, we will also provide a direct proof for the convenience of the reader.

By the Koszul formula and the fact that r and xr are torsion free, their con-nection forms with respect to local g-orthonormal bases u ´ fX1; : : : ; Xng andNu ´ f xX1; : : : ; xXng, are

N!ij .Xk/ D !ij .Xk/CXi .u/ıjk �Xj .u/ıik; i; j; k D 1; : : : ; n:

Let Qu be a local section of SpingM such that �g. Qu/ D u. Then zGu. Qu/ is a localsection of Spin NgM such that � Ng. zGu. Qu// D Nu. Consider the local spinor ‰ ´Œ Qu; � 2 �.†M/ de�ned by some constant vector 2 †n and the correspondingspinor x‰ ´ Œ zGu. Qu/; � 2 �.† xM/. Theorem 2.11 shows that

xrXkx‰ D 1

4

X

i;j

N!ij .Xk/Xi �Xj �‰

D rXk‰ � 1

2Xk � grad.u/ �‰ � 1

2Xk.u/x‰: �


Lemma 2.34. The Dirac operator associated with the metric Ng satis�esxD.f x‰/ D f xD x‰ C e�udf �‰: (2.21)

Proof.

xD.f x‰/ DX

i

xXi N� xr xXi.f x‰/

DX

i

xXi N� Œ xXi .f /x‰ C f xr xXi

x‰�

D f xD x‰ C e�udf �‰:

Recall that xXi D e�uXi . �

Lemma 2.35. The Dirac operators D and xD are related by

xDx‰ D e�uhD‰ C n � 1

2du �‰

i: (2.22)

Proof. We have

xD x‰ DX

i

xXi N� xr xXi

x‰

D e�uX

i

xXi N� xrXix‰

D e�uX

i

xXi N�hrXi

‰ � 1

2Xi � grad.u/ � ‰ � 1

2Xi .u/x‰

i

D e�uhD‰ C n � 1

2du �‰

i�

Proof of Proposition 2.31. Let

f ´ e� n�12 u:

Then

xD.f x‰/ D f xD x‰ C e�udf �‰

D fe�uD‰ C fe�un � 12

du �‰ C e�udf �‰

D fe�uD‰;

since

df D �n � 12

fdu: �


We now show that the Penrose operator is also conformally covariant [Lic88].More precisely, we have the following proposition.

Proposition 2.36. For any spinor �eld ‰, the Penrose operators P and xP associ-ated with the conformally related metrics g and Ng D e2ug satisfy

xP.e u2 x‰/ D e

u2 P‰: (2.23)

Proof. By the de�nition of the Penrose operator, for any tangent vector �eld X ,

xPX .eu2 x‰/ D xrX.e

u2 x‰/C 1

nX N� xD.e u

2 x‰/: (2.24)

By (2.20),

xrX.eu2 x‰/ D e

u2

hxrX x‰ C 1

2X.u/x‰

i

D eu2

hrX‰ � 1

2X � du �‰

i:

(2.25)

On the other hand, using (2.21) and (2.22) it follows that

xD.e u2 x‰/ D e

u2 xD x‰ C 1

2e� u

2 du �‰

D e� u2

hD‰ C n � 1

2du � ‰

iC 1

2e� u

2 du � ‰

D e� u2

hD‰ C n

2du �‰

i:

(2.26)

We get (2.23) by inserting (2.25) and (2.26) into (2.24) and using the fact that forany tangent vector �eld X and spinor �eld ˆ, the following relation holds

X N� x D euX �ˆ: �

2.3.6 Conformally covariant powers of the Dirac operator

Let us mention without details a recent generalization [GMP12] of the conformalcovariance of the Dirac operator. Namely, on a Riemannian spin manifold .M; g/of dimension n there exist conformally covariant differential operators L2kC1 ofodd order 2kC 1 for every k if n is odd, respectively for k < n=2 if n is even, withprincipal symbol equal to that ofD2kC1. Moreover, for k D 1 the expression of theoperator is explicit.

2.4. Spinors in classical geometrical contexts 73

Proposition 2.37. Let .M; g/ be a Riemannian spin manifold of dimension n � 3,and denote by Scal; Ric, and D the scalar curvature, the Ricci curvature, and theDirac operator of .M; g/. Let denote the Clifford multiplication. Then the oper-ator L3 de�ned by

L3 ´ D3 � .d.Scal//

2.n � 1/ � 2 ı Ric ı rn � 2 C Scal

.n� 1/.n� 2/D

is a natural conformally covariant differential operator:

enC3

2 uL3 x D L3.en�3

2 uˆ/

ifL3 is de�ned in terms of the conformal metric Ng D e2ug. The Ricci tensor is iden-ti�ed via g with an endomorphism of the cotangent bundle and acts on T�M ˝†M

in the above formula by Ric.˛ ˝‰/ ´ Ric.˛/˝‰.

These conformally covariant operators arise as residues of a certain meromor-phic family of pseudodifferential operators on M , the scattering operators for theDirac operator on some Poincaré-Einstein manifold X with conformal in�nity M .Since we do not de�ne these metrics in this book, we refer to Section 4 in [GMP12]for the proofs.

2.4 Spinors in classical geometrical contexts

In this section we obtain relations between the covariant derivatives acting on spinorson two Spinc manifoldsM and N , where N is

(1) a hypersurface ofM or

(2) a warped Riemannian product N D M �f B , where B is some other Spinc

manifold and f is a function on B or

(3) the total space of a Riemannian submersion N ! M with totally geodesicone-dimensional �bers.

2.4.1 Restrictions of spinors to hypersurfaces

Let .M n; g/ be a Spinc manifold with auxiliary line bundle L and let N be anoriented hypersurface of M endowed with the Riemannian metric gN induced byg. We denote by V the unitary vector �eld compatible with the orientations ofM and N which spans the normal bundle of N in M (compatibility means that.X1; : : : ; Xn�1; V / is a positively oriented basis of TM for every positively orientedbasis .X1; : : : ; Xn�1/ of TN ). Let � be the inclusion of N inM .


Lemma 2.38. There exists a Spinc structure on N canonically induced by that ofM , whose auxiliary line bundle is the restriction of L toN . For n even .resp. odd/,there exists an identi�cation of the pull-back bundle ��.†CM/ .resp. ��.†M//with†N such that the Clifford product on †N satis�es

X � ��‰ D ��.X � V �‰/; X 2 TN: (2.27)

The notation �� stands for the restriction of objects – spinors, bundles, etc. – fromM to N .

Proof. The isomorphism Cln�1 ! Cl0n given by Proposition 1.11 induces by com-plexi�cation an isomorphism Cln�1 ! Cl0n. This identi�es Spincn�1 with theinverse image of SOn�1 �U1 by �cW Spincn ! SOn �U1. We obtain in this waya complex representation � of Cln�1 on †n which satis�es �.ei/ D �n.ei � en/; forany 1 � i � n � 1.

For n odd, Theorem 1.28 shows that this representation is isomorphic to .�n�1;†n�1/.

For n even, � preserves the decomposition †n D †Cn ˚ †�

n (see the proof ofProposition 1.32) and its restriction to†C

n is isomorphic to .�Cn�1; †n�1/ since they

have the same dimension and �.!Cn�1/ D �n.!

Cn / D Id

†Cn(easy computation).

Themap .X1; : : : ; Xn�1/ 7! .X1; : : : ; Xn�1; V / identi�esPSOn�1N with a sub-

bundle P of ��.PSOnM/. Thus, the inverse image PSpinc

n�1N of P � ��.PU1

M/ by

�� W ��.PSpincnM/ �! ��.PSOnM � PU1

M/

de�nes a Spinc structure on N with auxiliary line bundle ��L.The algebraic discussion above shows that the spinor bundle associated with this

Spinc structure is isomorphic to ��.†M/ for n odd and to ��.†CM/ for n even, andthat the Clifford product satis�es (2.27). �

Given a connection A on PU1M (inducing a covariant derivative rA on †M ),

we obtain a connection ��A on ��.PU1M/ inducing, together with the Levi-Civita

connection of N , a covariant derivative, denoted r ��A, on the spinors of N . Let‰ D Œ Qu; � be a spinor ofM , where we have chosen the local section Qu ofPSpincnM

in such a manner that its projection by �1 on PSOnM be of the form

u D .X1; : : : ; Xn�1; V /:

Then ��‰ D Œ Qu; �� , and an easy calculation using (2.27) and Theorem 2.11implies the following result.


Theorem 2.39. The covariant derivative of a spinor‰ onM and that of its restric-tion to N are related by

��.rAX‰/ D r ��A

X .��‰/C 1

2II.X/ � .��‰/; X 2 TN � TM; (2.28)

where II is the second fundamental form of N .

2.4.2 Spinors on warped products

We treat here a special case of warped product, namely the Riemannian cone, whichwill be of particular interest for us in Chapter 8. Let .M n; g/ be a Riemannianmanifold and let

.N; gN / ´ .M � R�C; r

2g C dr2/

be the Riemannian cone overM . We denote by� WN ! M the canonical projectionand identify in the sequel the vector �elds on M and the projectable vector �eldson N . In particular, every vector X 2 TxM induces a vector �eld (also denotedby X) along the geodesic line fxg � R

�C. The following result is quite similar to

Lemma 2.38:

Lemma 2.40. A Spinc structure on the manifoldM endowed with an auxiliary linebundle L induces, in a canonical manner, a Spinc structure on N whose auxiliaryline bundle is the pull-back ��L of L. For n even, the spinor bundle ��.†M/ canbe canonically identi�ed with †N so that the Clifford product satis�es

1

rX � @r � .��‰/ D ��.X �‰/; X 2 TM: (2.29)

For n odd, the equation

1

rX � @r � .��‰/ D ˙��.X �‰/; X 2 TM: (2.30)

de�nes an identi�cation of ��.†M/ and †˙N as Cl0.N /-modules.

Proof. The oriented orthonormal frame bundle of N can be canonically identi�edwith the inverse image by � WN ! M of the extension PSOnC1

M of PSOnM toSOnC1. This identi�cation is given by

.Œux; A�; r/ 2 ��.PSOnC1M/ 7�!

�1r��u.x;r/; @r

�A 2 PSOnC1

N: (2.31)

By extension of the structure group ofPSpincnM to SpincnC1, we obtain an equivariantmorphism of principal bundles

� WPSpincnC1

M �! PSOnC1M � PU1

M;


whose inverse image by � yields a Spinc structure on N

�� WPSpincnC1N �! PSOnC1

N � ��.PU1M/;

with auxiliary line bundle ��L. Equation (2.29) follows directly from (2.31) andfrom the isomorphism of †n and †nC1 (resp. †

CnC1 for n odd) as Cln-representa-

tions described by Lemma 2.38. �

The covariant derivatives r and xr de�ned by the Levi-Civita connections of gand gN satisfy the warped product formulas (see [O’N83], p. 206)

xr@r@r D 0; (2.32)

xr@rX D xrX@r D 1

rX; (2.33)

xrXY D rXY � rg.X; Y /@r : (2.34)

LetA be a connection on PU1M and rA the covariant derivative induced on the

spinors onM . We denote by ��A the induced connection on PU1N ' ��PU1

M

and by xrA the covariant derivative induced by xr and ��A on the spinors on N .Using (2.29)–(2.34) and Theorem 2.11, we obtain directly the formula which relatesrA and xrA:

Theorem 2.41. Let‰ be a spinor �eld onM and ��‰ .resp. .��‰/˙/ the inducedspinor �eld on †N for n even .resp. †˙N for n odd/. Then

xrAX .�

�‰/ D ��rA��X

‰ � 1

2.��X/ �‰

�; X 2 TN .n even/;

xrAX .�

�‰/˙ D ��rA��X

‰ � 1

2.��X/ �‰

�; X 2 TN .n odd/:

andxrA@r.��‰/˙ D 0:

2.4.3 Spinors on Riemannian submersions

The third geometrical situation that we treat here is that of a Riemannian submer-sion � WS ! M , with totally geodesic �bers diffeomorphic to the circle S

1, where.M n; g/ is a Spinc manifold and .SnC1; Ng/ is an oriented Riemannian manifold.By de�nition, � is called a Riemannian submersion if for all s 2 S , the restric-tion of ��WTsS ! T�.s/M maps the orthogonal complement (with respect to Ng) ofKer.��/ isometrically onto .T�.s/M;g/.


For each s 2 S , let Vs 2 TsS be the unit vertical vector (i.e., contained inKer.��/) compatible with the orientations of M and S (see the beginning of thissection for the de�nition of compatibility). We obtain in this way a vector �eld on S ,called the standard vertical vector �eld.

For each vector Xx 2 TxM and for each point y 2 ��1.x/ we de�ne thehorizontal lift X�

y 2 Ker.��/?y � TyM of X at y to be the inverse image of Xby the above isomorphism. It is clear that every vector �eld X on M induces ahorizontal vector �eld X� on S that projects to X . The vector �elds on S obtainedin this way are called standard horizontal vector �elds.

Lemma 2.42. The �bers of � are circles of constant length, denoted by `.

Proof. Let r and xr be the covariant derivatives on .M; g/ and .S; Ng/; let X be avector �eld onM and let X� be its horizontal lift to S . Since X� projects to X andV onto 0, it follows that ŒX�; V � projects to 0, in other words it is vertical. On theother hand, Ng.ŒX�; V �; V / D � Ng.xrVX�; V / D Ng.X�; xrV V / D 0, so �nally

ŒX�; V � D 0: (2.35)

Let �t be the local group of diffeomorphisms of S induced by X�. Equation (2.35)shows that �t preserves the �bers of � and satis�es .�t /�V D V . Consequently,the restriction of �t to the �bers of � is an isometry for each (small) t , so the �bersare isometric, since X was arbitrary. �

By Lemma 2.42, the �ow of `2�V induces a free U1-action on S which turns �

into a principal U1-bundle. The horizontal distribution of S is invariant with respectto this action because of (2.35), so it de�nes a connection. We denote by F the2-form onM with the property that 2�iF is the projection onM of the curvatureform G of this connection (we identify u1 and iR, so that the curvature form can beseen as an imaginary-valued two-form on S ).

Let T be the �eld of endomorphisms on M de�ned by g.TX; Y / D F.X; Y /

and let i! 2 �1.S; iR/ be the connection form. As `2�V is the standard vertical

vector �eld induced by i 2 u1 onS , we deduce that!.`2�V / D 1, and consequently,

!.Z/ D 2�`

Ng.Z; V /; Z 2 TS .We now derive the relationship between xr and r. It is enough to compute

Ng.xrAB;C / for any (horizontal or vertical) standard vector �elds A;B; C . First,the Koszul formula shows that

Ng.xrX�Y �; Z�/ D g.rXY;Z/; X; Y; Z 2 �.TM/: (2.36)

Next, we have

Ng.xrX�V; V / D Ng.xrVX�; V / D � Ng.xrV V;X�/ D 0; X 2 �.TM/:


Because of (2.35), it only remains to compute Ng.xrX�Y �; V /, or, equivalently, tocompute xrX�V .

With the notations above, for any vector �elds X; Y onM

2�i Ng..TX/�; Y �/ D 2�ig.TX; Y /

D 2�iF.X; Y /

D G.X�; Y �/

D id!.X�; Y �/

D �i!.ŒX�; Y ��/

D �2�i`

Ng.ŒX�; Y ��; V /

D �4�i`

Ng.xrX�Y �; V /

D 4�i

`Ng.Y �; xrX�V /I

hence

xrX�V D `

2.TX/�; X 2 TM: (2.37)

Conversely, let � WS ! M be a principal U1-bundle with connection formi! 2 ƒ1.S; iR/ and curvature formG 2 ƒ2.S; iR/ projecting over 2�iF for some2-form F onM . De�ne a 1-parameter family of metrics on S by

gtS .X; Y / D g.��.X/; ��.Y //C t2!.X/!.Y /

and let r and xr t denote the covariant derivatives of the Levi-Civita connections ofg and gtS .

Denote by V t the unit vertical vector �eld on S de�ned by !.V t / D 1=t and,for X 2 TM , let X� denote its horizontal lift to TS .

From the very construction, the projection .S; gtS / ! .M; g/ is a Riemanniansubmersion. Each �ber has constant length 2�t , and is totally geodesic. Indeed, V t

has constant norm so gtS .xr tV tV

t ; V t / D 0. Moreover, from the invariance of thehorizontal distribution of the connection, one gets ŒX�; V t � D 0, for every X ,

gtS .xr tV tV

t ; X�/ D �gtS .V t ; xr tV tX

�/ D �gtS .V t ; xr tX�V

t / D 0:

We thus proved the following proposition.


Proposition 2.43. Let .M; g/ be an oriented Riemannian manifold. There existsa one-to-one correspondence between oriented Riemannian submersions S ! M

with totally geodesic one-dimensional �bers, on the one hand, and U1-connectionson principal U1-bundles S ! M , on the other hand. The parameter t de�ning themetric on the total space of the bundle is de�ned by the formula 2�t D `, where ìs the length of the �bers of the submersion. The the covariant derivative xr on Sand the curvature form G of the U1-connection are related by

gS .xrX�V; Y �/ D t

2iG.X�; Y �/; (2.38)

where V denotes the unit vertical vector �eld compatible with the orientations ofM and S .

Note that (2.38) is equivalent to

d.i!/ D G;

which is just the structure equation of the connection.We are now ready to consider the spinorial aspect of this geometrical situation.

For the sake of simplicity, we will restrict ourselves to the case where the dimensionofM is even, which is the only case to be considered in the sequel.

Lemma 2.44. If M and S are as above, then the Spinc structure on M inducescanonically a Spinc structure on S whose auxiliary line bundle is the pull back��L of L. The associated spinor bundle †S can be identi�ed with ��.†M/, andwith respect to this identi�cation, the Clifford multiplication is given by

X� � .��‰/ D ��.X �‰/; X 2 TM (2.39)

and

V � .��‰/ D i��.x‰/; (2.40)

wherex‰ ´ ‰C � ‰�:

Proof. We �rst point out that the oriented orthonormal frame bundle on S can becanonically identi�ed with the pull-back by� of the extensionPSOnC1

M ofPSOnM

to SOnC1. This identi�cation is given by

��.PSOnC1M/ �! PSOnC1

S;

.Œ.X1; : : : ; Xn/�.s/; a�; s/ 7�! .X�1 ; : : : ; X

�n ; V /sa:


Then, an argument similar to that used in the proof of Lemma 2.40 shows that thepull-back by � of PU1

M and of the extension to SpincnC1 of the structure group ofPSpincnM de�nes a Spinc structure on S .

For the second part of the lemma, we observe that the representation � of ClnC1on †n given by

�.ek/. / ´

8<:ek � for 1 � k � n

i x for k D nC 1

is equivalent to .�C; †nC1/ and thus de�nes an isomorphism between †S and��.†M/ satisfying (2.39) and (2.40). �

Let A be a connection on PU1M with connection form ˛ 2 �1.PU1

M; iR/ andlet rA be the covariant derivative that it de�nes on the spinors ofM . We denote by��A the induced connection on PU1

S ' ��PU1M (with connection form ��˛)

and by xrA the covariant derivative de�ned by xr and ��A on the spinors of S .Using (2.37)–(2.40) and Theorem 2.11 we deduce the formulas relating rA and xrA.

Theorem 2.45. Let ‰ be a spinor �eld onM and ��‰ the projectable spinor �eldthat it induces on †S . Then

xrAX�.�

�‰/ D ��rAX‰ � i`

4TX � x‰

�; X 2 �.TM/

and

xrAV .�

�‰/ D `

4��.F �‰/:

Proof. Let u D .X1; : : : ; Xn/ and be local sections of PSOnM and PU1M

respectively and let Qu be a local section of PSpincnM such that �. Qu/ D .u; /.The spinor ‰ can be written locally as ‰ D Œ Qu; �, where is a local functionofM with values in †n.

Let v D .X�1 ; : : : ; X

�n ; V / be the local frame of PSOnC1

S induced by u and letı D �� be the local section of PU1

S D ��PU1M induced by (which satis�es

� ı ı D ). We de�ne a local section

Qv ´ ��. Qu/

of the Spinc structure of S which projects over v � ı. By de�nition, we have

��.‰/ D Œ Qv; ı ��

2.5. The Schrödinger–Lichnerowicz formula 81

and Theorem 2.11 applied to ‰ and ��.‰/, together with (2.36)–(2.40) yield

xrAX�.�

�‰/ D Œ Qv; X�. ı �/�C 1

2

X

i<j

Ng.xrX�X�i ; X

�j /X

�i �X�

j � .��‰/

C 1

2

X

i

Ng.xrX�X�i ; V /X

�i � V � .��‰/C 1

2��˛.ı�X

�/.��‰/

D Œ Qv; X. / ı ��C 1

2

X

i<j

g.rXXi ; Xj /X�i � X�

j � .��‰/

� `

4

X

i

g.TX;Xi /X�i � V � .��‰/C 1

2˛. �X/.��‰/

D ��Œ Qu;X. /�C 1

2

X

i<j

g.rXXi ; Xj /Xi �Xj �‰

� i`

4

X

i

g.TX;Xi /Xi � x‰ C 1

2˛. �X/‰

�

D ��rAX‰ � i`

4TX � x‰

�:

The proof of the second relation is similar. �

2.5 The Schrödinger–Lichnerowicz formula

Definition 2.46 (Extension of h�; �i and r). i) Extend the natural Hermitian scalarproduct h�; �i on �.†M/ to sections of T�M ˝†M by

h�; �iW �.T�M ˝†M/ � �.T�M ˝†M/ �! C1.M;C/;�˛ ˝‰; ˇ ˝ˆ

�7�! g.˛; ˇ/h‰;î;

where the metric g extends to covectors by means of the isomorphism T�M ' TMinduced by g. Thus, for !, � 2 �.T�M ˝†M/, we get

h!; �i DnX

jD1h!.Xj /; �.Xj /i

for any orthonormal basis fX1; : : : ; Xng of TxM .

ii) Assume thatM is compact and de�ne r� to be the formal adjoint of r, i.e.,

r�W�.T�M ˝†M/ �! �.†M/

with hr�‚;îL2 D h‚;rîL2 for all ‚ 2 �.T�M ˝†M/ and ˆ 2 �.†M/.


Lemma 2.47. In local normal coordinates .X1; : : : ; Xn/ at x 2 M , we have

r�r‰ D �nX

jD1rXj

rXj‰

for all ‰ 2 �.†M/.

Proof. By the de�nition of r�, for any spinor �elds ‰;ˆ, we have

hr�r‰;îL2D hr‰;rîL2 D

nX

iD1hrXi

‰;rXiîL2 :

Consider the two vector �elds V1, V2 2 �.TM/ de�ned for all X 2 TM by

g.V1; X/C ig.V2; X/ D h‰;X �î:

Then

divV1 C idivV2 DnX

kD1g.rXk

V1; Xk/C i

nX

lD1g.rXl

V2; Xl/

DnX

kD1.Xkg.V1; Xk/ � g.V1;rXk

Xk//

C i

nX

lD1.Xlg.V2; Xl/ � g.V2;rXl

Xl //

DnX

kD1Xk.g.V1; Xk/C ig.V2; Xk//

DnX

kD1Xkh‰;Xk �î:

Hence,

nX

kD1hrXk

‰;rXkî D

nX

kD1XkhrXk

‰;î � hrXkrXk

‰;î

D divV1 C idivV2 �nX

kD1hrXk

rXk‰;î;

The sum in the last term is the divergence of a complex vector �eld. By integration,it gives the required condition for r� to be the formal adjoint of r. �

2.5. The Schrödinger–Lichnerowicz formula 83

Proposition 2.48. Let

zR ´ 1

2

nX

i;jD1Xi �Xj � RXi ;Xj

;

whereR is the spinorial curvature. The square of the Dirac operator is given by

D2 D r�r C zR:

Proof. Take normal coordinates at x 2 M . Then

D2 D� nX

iD1Xi � rXi

�� nX

jD1Xj � rXj

�

DnX

i;jD1Xi � Xj � rXi

rXj

D �nX

iD1rXi

rXiC

nX

i¤jD1Xi �Xj � rXi

rXj

D �nX

iD1rXi

rXiC

nX

i<j

Xi � Xj ��rXi

rXj� rXj

rXi

�

D r�r C 1

2

nX

i;jD1Xi �Xj � RXi ;Xj

D r�r C zR: �

Theorem 2.49 (Schrödinger–Lichnerowicz formula). Let Scal be the scalar curva-ture ofM . Then

D2 D r�r C 1

4Scal:

Proof. By Proposition 2.48, it is suf�cient to show that

zR D 1

4Scal:


By Corollary 2.8,

zR D 1

2

nX

i;jD1Xi � Xj � RXi ;Xj

D 1

2

nX

iD1Xi �

�X

j

Xj � RXi ;Xj

�

D 1

2

nX

iD1Xi �

�� 1

2Ric.Xi / �

�

D 1

4Scal: �

Remark 2.50. It is useful to point out that the Ricci identity (2.8),

� 1

2Ric.X/� D

nX

iD1Xi � RX;Xi

; X 2 �.TM/; (2.41)

can be recast as

�12Ric .X/� D ŒrX ;D�:

To see this, it is suf�cient to consider normal coordinates at some point so that

�12Ric.X/� D

nX

iD1Xi � RX;Xi

DnX

iD1Xi �

�rXrXi

� rXirX/

DnX

iD1.rX

�Xi � rXi

��Xi � rXi

rX�

D rXD � DrX :

Chapter 3

Topological aspects

In this chapter we discuss the interaction between spin structures and the topologyof the underlying manifold. There are two sides to this: one concerns existence,and the other classi�cation. The most natural tools to deal with these issues arebased on describing principal bundles using local trivializations and cocycle condi-tions. We will show that spin and Spinc structures can be described in terms of Cechcohomology groups with respect to the good cover associated to any triangulation.In particular, the corresponding cohomology groups are independent of the triangu-lation. We therefore do not need to worry about independence on the cover, since itfollows for free.

3.1 Topological aspects of spin structures

Let M be a smooth manifold of dimension n. A k-dimensional simplex is an em-bedding inM of the standard k-simplex

�k D f.x1; : : : ; xkC1/ 2 RkC1I x1 C � � � C xkC1 D 1; xj � 0 for all j g:

The hyperfaces of �k (i.e., the intersections of �k with some subspace fxj D 0g)are canonically identi�ed with�k�1. Inductively, a simplex de�nes other simplices(namely, its faces) of every lower dimension. The image of a simplex � W�k ,! M

is also the image of �ı� , where the permutation � 2 SkC1 acts on�kC1 by permut-ing the coordinates. We identify all these simplices with their common image inM .An orientation on a simplex is an equivalence class of such geometric realizationsmodulo the alternating group AkC1. A triangulation T ofM is a decomposition ofM into a union of n-simplices, each two of them intersecting along a sub-simplexor the empty set. Moreover, we require each vertex (i.e., 0-simplex) to belong toa �nite number of n-simplices in the triangulation. More precisely, T is the set ofn-simplices and of their faces of every dimension. A standard result of Whiteheadasserts that every smooth manifold admits triangulations.

86 3. Topological aspects

Consider a triangulation T of M , let Tl be the set of l-simplices, and Star.p/the open star of p 2 T0, de�ned as the union of the interiors of all simplices inthe triangulation containing p. After replacing the triangulation by its barycentricsubdivision if necessary, the sets

Sj ´ Star.pj /

form a good open cover S ofM , in the sense that all intersections are contractible.For every l-simplex � D .pj0

; : : : ; pjl/ 2 Tj , let

Star.�/ ´l\

sD0Sjs :

Then Si \ Sj is empty unless .pi ; pj / 2 T1 is a 1-simplex in T , in which caseSi \ Sj retracts onto the open segment .pi ; pj /. Similarly, Si \ Sj \ Sk is emptyunless .pi ; pj ; pk/ 2 T2 is a 2-simplex in T , in which case Si \ Sj \ Sk retractsonto the open triangle .pi ; pj ; pk/. More generally, for every l-simplex � , the openset Star.�/ retracts onto the interior of � .

We �x in the rest of this chapter the triangulation T of M and the associatedgood cover S . The results below will express certain topological properties ofM interms of cohomology groups associated to the cover; clearly then, those groups donot depend on the cover as well. It is of course well known that Cech cohomologycan be computed using any good open cover, but in order to be as self-contained aspossible we do not need to use this fact.

3.1.1 Cech cohomology and principal bundles

Let F be a sheaf of commutative groups overM (in this book, we in fact use onlysheaves of sections of locally trivial �ber bundles). We emphasize that the sheavesmust be commutative in order for the construction below to make sense. A Cechcochain of degree j is a collection of maps � associating to each ordered j -simplex.�;O/ in T a section in F over the contractible open set Star.�/. If we reversethe orientation, we require that �.�;O�1/ D ��.�;O/. Denote {C j .M;F/ the set ofcochains of degree j . The boundary map

@j W {C j .M;F/ �! {C jC1.M;F/

is de�ned by

.@j�/i0:::ij C1´

jX

lD0.�1/l�i0:::O{l :::ij :

We have omitted the obvious restriction map from the above formula.

3.1. Topological aspects of spin structures 87

When F is noncommutative, we can still extend this de�nition as follows, butonly for j D 0; 1:

.@0�/ij ´ ��1i �j ; .@1�/ijl ´ �ij�jl�li :

Henceforth we make the assumption that either F is commutative, or j � 1, andwe note that in these cases @j ı @j�1 D 0 (or 1 in the multiplicative notation). TheCech cohomology group {H j .M;F/ is de�ned for F commutative or for j D 1 asthe group

{H j .M;F/ ´ Ker @j= Im @j�1:

The 0th cohomology group {H 0 is just the group of global sections in F . When F

is noncommutative, we can still de�ne {H 1.M;F/ as the set of equivalence classesof 1-cocycles modulo the relation

�0 Š � ” there exists a 2 {C 0.M;F/ such that �0ij D a�1

i �ijaj :

Let G be a Lie group. A G-principal bundle over M is a locally trivial �berbundle PG ! M with a right action ofG which is free and transitive on each �ber.Let C1

G denote the sheaf of smooth G-valued functions onM .

Proposition 3.1. The set of isomorphism classes of principal G-bundles overM isin natural bijection with {H 1.M; C1

G /.

Proof. Let P be a G-principal bundle over M . Since the open sets Sj are con-tractible, there exist sections sj WSj ! P . Let �ij WSi \ Sj ! G be the uniquesmooth function such that sj D si�ij . Then clearly the functions �ij form a1-cocycle. By changing the local sections to s0

j D sjaj , aj WSj ! G we get anequivalent cocycle �0

ij D a�1i �ijaj . Conversely, any cocycle �ij can be used to

construct a G-principal bundle PG , de�ned as the trivial bundle Sj � G over Sjmodulo the identi�cations .x; g/ � .y; g�ij / for all x 2 Si ; y 2 Sj with x D y.This bundle has by construction trivializations over Sj , and the associated cocycleis �. �

The trivial bundle corresponds to a distinguished element in the set {H 1.M; C1G /,

denoted 0. When G is commutative, the set of equivalence classes of G-bundles istherefore a group, called the Picard group when G D U1. Recall that a sheaf ofcommutative groups over M is called soft if every germ over a closed set can berealized as the germ of a global section. Soft sheaves are acyclic (their cohomologyvanishes in degree j � 1). Typical examples are sheaves of smooth real functions,in particular {H j .M; C1

R/ D 0 for j � 1.

Proposition 3.2. For every j � 1 there exists a natural isomorphism

{H j .M; C1U1/ ! {H jC1.M;Z/:


Proof. Consider the short exact sequence 0 ! C1Z

! C1R

! C1U1

! 1, wherethe �rst map is the inclusion and the last the exponential x 7! exp.2�ix/. Thesheaf C1

Zconsists of locally constant functions on M with values in Z, so by a

slight abuse of notation we denote it by Z. Since {H j .M; C1R/ D 0 for j � 1, the

corresponding long exact sequence of cohomology groups splits into the short exactsequence

0 �! C1.M;R/=Z �! C1.M;U1/ �! {H 1.M;Z/ �! 0;

and isomorphisms

ıj W {H j .M; C1U1/ �! {H jC1.M;Z/; j � 1: �

For any Hermitian (complex) line bundle L, let ŒL� denote its representative in{H 1.M; C1

U1/ given by Proposition 3.1. The class ı1ŒL� 2 {H 2.M;Z/ is called the

�rst Chern class of L, and is denoted by c1.L/.

Remark 3.3. If �WG ! G0 is a homomorphism of Lie groups, we get the inducedmap

��W {H 1.M; C1G / �! {H 1.M; C1

G0/:

For anyG-principal bundle PGM , there exist aG0-principal bundle PG0M , uniqueup to isomorphism, and a bundle map PGM ! PG0M compatible with �. Thecorresponding cohomology class ŒPG0M� 2 {H 1.M; C1

G0/ is just ��ŒPGM�.

3.1.2 Lifting principal bundles via central extensions

A homomorphism of Lie groups �W zG ! G is an extension of G if � is onto, and acentral extension ifZ ´ Ker� is a central subgroup of zG. Let pWPGM ! M be aG-principal bundle overM . A zG-lift ofPG is a zG-principal bundle QpWP zGM ! M

together with a bundle map � WP zGM ! PGM compatible with �, in the sense that

�.x Qg/ D �.x/�. Qg/

for all x 2 P zGM , Qg 2 zG.Assume we have aG-principal bundle pWPGM ! M and a central extension �.

Let sj WSj ! PGM be local sections and � D .�ij / the associated C1G -valued

1-cocycle of transition maps. Lift �ij to some maps Q�ij WSi \ Sj ! zG so that� ı Q�ij D �ij (this is possible since Si \ Sj is contractible). Then the products ijk ´ Q�ij Q�jk Q�ki take values in Z D Ker � since .�ij / is @1-closed. Moreover,as Z commutes with zG, the 2-cochain obtained in this way is closed. Differentchoices of trivializations s0

i ´ sigi and of lifts Q�0ij D zij Qg�1

iQ�ij Qgj lead to the

cochain @1zC . Hence, aG-principal bundle PG and a central extension � de�nea cohomology class ı.PG/ 2 {H 2.M; C1

Z / independent of choices.

3.1. Topological aspects of spin structures 89

Proposition 3.4. Let pWPGM ! M be a G-principal bundle over M and let

�W zG ! G be a central extension with kernel Z � zG.

(1) There exists a lift

� WP zGM �! PGM �! M

compatible with � if and only if the class ı.PG/ vanishes in {H 2.M; C1Z /.

(2) If ı.PG/ D 0, the group {H 1.M; C1Z / acts freely and transitively on the set of

isomorphism classes of lifts of p via �.

Proof. (1) If a lift P zGM exists, let zsj be local sections over Sj projecting onto sj .Then zsj D zsi Q�ij for some zG-valued cocycle Q�ij covering �ij , so the class ı.PG/clearly vanishes. Conversely, let ı.PG/ D 0 in cohomology. In terms of arbitrarylifts Q�ij WSj ! zG of �ij this means that ı1 Q� D ı1z for some C1

Z -valued 1-cocyclezij . Since Z is central in G, the cochain Q�0

ij de�ned by Q�0ij ´ z�1

ijQ�ij forms

a cocycle. Let P zGM be the bundle de�ned by this cocycle. By de�nition, it istrivialized over Sj , so we can de�ne local bundle maps to PGM by sending thecanonical section Qsj to sj . Thesemaps are compatible on overlaps and automaticallycompatible with �, hence P zGM is a lift of PGM .

(2) Assume there exists a lift P zGM ofPGM . Fix local sections Qsj WSj ! P zGMand let sj WSj ! PGM be their projections. For every other lift P 1zGM , take

Qs1j WSj ! P 1QGM to be lifts of sj , and Q�1ij to be the associated cocycle. Then we

have �.. Q�ij /�1 Q�1ij / D 1, hence zij ´ . Q�ij /�1 Q�1ij de�nes a C1Z -valued 1-chain.

Since both Q�ij , Q�1ij are cocycles and Z commutes with zG, it follows that zij is a

cocycle. Different choices of Q�1ij lead to a cohomologous cocycle, hence we get a

map ˆ from the set of zG-lifts of PGM to {H 1.M; C1Z /.

In the reverse direction, {H 1.M; C1Z /, which is the group ofZ-principal bundles

onM , acts on the set of lifts of PGM by the Z cross-product:

PZM � P 1zGM 7�! P 1zGM �Z PZM

where the right-hand side is the quotient of the �bered product by the equivalencerelation .pz; q/ � .p; qz/ for all p 2 P 1zGM , q 2 PZM , z 2 Z. A direct applica-

tion of the de�nitions shows that the map {H 1.M; C1Z / 3 z 7! P zGM �Z z (using

the �xed lift P zGM ) is a right and left inverse of the map ˆ constructed above, sothe action is free and transitive. �


3.1.3 Stiefel–Whitney classes

To avoid introducing unnecessary notions, we restrict the discussion to degrees 1and 2. For i D 1; 2, the i th Stiefel–Whitney class wi of the tangent bundle ofM is a Z=2Z-cohomology class on M measuring (in the sense explained below)an obstruction to �nding n � i C 1 linearly independent sections in TM over thei -skeleton of T , i.e., the union of all closed simplices of dimension at most i . It isde�ned as follows. Choose n � i C 1 linearly independent sections in TM overthe .i � 1/-skeleton (this is possible for i D 2 since the space of .n � 1/-tuplesof linearly independent vectors in R

n is connected). For i D 1 or n � 3, asso-ciate to each i -simplex � 2 Ti either 0 or 1, depending on whether these sectionscan (resp. cannot) be extended over � in a linearly independent way. In the spe-cial case i D 2; n D 2 we have a nonzero vector �eld along the 1-skeleton, andsince �1.R2 n f0g/ D Z we obtain an integer k� for each oriented triangle � in T .We associate to the open set Star.�/ the mod 2 residue k� C 2Z, thus obtainingan i -dimensional Z=2Z-valued Cech cochain. Since �0.GLn.R// D Z=2Z, thecochain w1 is closed, and its cohomology class is independent of the choices. Fori D 2 and n � 3 we have �1.GL

Cn .R// D Z=2Z, hence w2 is closed, while

for n D 2 any 2-cocycle is trivially closed. Hence, we get well-de�ned classesw1 2 {H 1.M;Z=2Z/ and w2 2 {H 2.M;Z=2Z/.

Lemma 3.5. AssumeM is orientable and letPGLCnM be the oriented frame bundle.

The cohomology class ı.PGLCnM/ 2 {H 2.M;Z=2Z/ corresponding to the central

extension

0 �! Z=2Z �! eGLCn

��! GLCn �! 1

is the second Stiefel–Whitney classw2. Here �W eGLCn ! GLC

n is the universal coverof GLC

n for n � 3, the two-folded cover sqWGLC2 ! GLC

2 for n D 2, respectivelyj � jW R

� ! R�C for n D 1.

Proof. Choose a section s in the oriented frame bundle in a closed, suf�ciently smallneighborhood of the 0-skeleton, and extend it smoothly over the 1-skeleton Sk1 us-ing the fact that GLC

n is connected. For each j , extend sjSk1\Sjto local sections

sj WSj ! PGLCnM and let �ij WSi \ Sj ! GLC

n be the transition maps. By con-struction, �ij D 1 over the 1-simplex Œpipj �, and �ij .pk/ D 1 if Œpipjpk � 2 T2.

Let Q�ij be the unique lift of �ij to eGLCn which equals 1 on Œpipj �. For each triangle

� D Œpipjpk� 2 T2, Q�ij .pk/ maps to 1 via �, thus it belongs to f˙1g D Ker �.Moreover, by continuity, Q�ij Q�jk Q�ki is constant and equals Q�ij .pk/ (we see this byevaluating at pk). We claim that this is precisely w2.Star.�// from the above de�-nition of w2. When n � 3, this is so because s is null-homotopic along the bound-ary of � if and only if �ij is null-homotopic on Œpjpk � relative to the boundary.

3.2. Topological classi�cation of Spinc structures 91

For n D 2, the parity of the homotopy class of s along @� equals Q�ij .pk/. Hence,with the above choices, ı.PGLC

nM/ and w2 are represented by the same cocy-

cle. �

Proposition 3.6. A spin structure exists on M if and only if the Stiefel–Whitneyclasses w1 and w2 of TM vanish. In this case, the set of spin structures (up toisomorphism) is a free and transitive {H 1.M;Z=2Z/-module.

Proof. The vanishing of w1 means that the frame bundle of M is trivial over the1-skeleton, which is clearly equivalent to the orientability of M and hence to theexistence of the oriented frame bundle PGLC

nM . The necessary and suf�cient

condition for the existence of a spin structure follows from Proposition 3.4 andLemma 3.5. The classi�cation of spin structures is contained in Proposition 3.4. �

In particular, 2-connected manifolds (i.e., �1.M/ D �2.M/ D 1) always admitspin structures.

3.2 Topological classi�cation of Spinc structures

Any Spinc structure (De�nition 2.19) induces a morphism of principal bundles

Spincn //

�c

��

PSpincnM

$$❍❍❍

❍❍❍❍

❍❍

�

��

M:

SOn // PSOn

::✉✉✉✉✉✉✉✉✉

Conversely, such a diagram de�nes a Spinc structure, the auxiliary bundle beingobtained from PSpincnM using the group morphism sqW Spincn ! U1.

Proposition 3.7. A Spinc structure on M exists if and only if w1.M/ D 0 andthe second Stiefel–Whitney class w2.M/ is the reduction modulo 2 of an integerclass. If non-empty, the set of Spinc structures is acted upon freely and transitivelybyH 2.M;Z/.

Proof. The bundle PSOnM exists if and only if w1 D 0. Consider the centralextension

1 �! U1 �! Spincn �! SOn �! 1:


By Proposition 3.4, there exists a Spincn-lift of PSOnM if and only if the

cohomology class ı.PSOnM/ vanishes in {H 2.M; C1

U1/. Since the subgroup

Spinn � Spincn surjects over SOn, the lifts of the transition maps Q�ij in the de�nitionof ı.PSOnM/ can be chosen with values in Spinn, hence the resulting f˙1g-valued2-cocycle @1 Q� is precisely a representative of w2. Consequently, ı.PSOn

M/ D{�w2 2 {H 2.M; C1

U1/, where {W f˙1g ! U1 is the inclusion. The existence part

follows from Lemma 3.8 below. The classi�cation of Spincn structures (when theyexist) follows directly from Proposition 3.4. �

Lemma 3.8. A class w 2 {H 2.M; f˙1g/ maps to 0 2 {H 2.M; C1U1/ via {� if and

only if it is the reduction modulo 2 of an integer class in {H 2.M;Z/.

Proof. The group homomorphism

expW R �! U1; k 7�! exp.�ik/

leads to an inclusion of short exact sequences of commutative groups

0 // 2Z //

��

Zexp //

��

f˙1g //

{

��

1

0 // 2Z // R exp// U1 // 1:

Consider the sheaves of smooth functions on M with values in the above groups.Since the sheaf C1

Ris soft, the commutative diagram induced in cohomology has

the form

H 2.M; 2Z/ // H 2.M;Z/exp //

��

H 2.M;Z=2Z/ı //

{�

��

H 3.M; 2Z/

1

��0 // H 2.M; C1

U1/

ı// H 3.M; 2Z/ // 0:

Thus, the class Œw2� 2 H 2.M; f˙1g/ is mapped to 0 2 H 2.M; C1U1/ if and only if

it lives in the image of exp�WH 2.M;Z/ ! H 2.M; f˙1g/. �

Proposition 3.9. Let M be orientable and PU1M ! M a U1-principal bundle.

A Spinc structure over the bundle PSOnM �M PU1

M exists if and only if the reduc-tion modulo 2 of c1.PU1

M/ equals the Stiefel–Whitney class w2. If non-empty, theset of such Spincn structures is a free and transitive {H 1.M;Z=2Z/-module.

3.3. Spin structures in low dimensions 93

Proof. The classi�cation of lifts (when they exist) follows again from Proposi-tion 3.4. Assume PSpincnM exists and consider the Spincn-valued 1-cocycle Q�ijassociated to local trivializations of PSpincnM over the open cover S . Over the con-tractible sets Si \ Sj lift Q�ij to a Spinn � U1-cochain . Q ij ; zij /. Then @ Q and@z are both f˙1g-valued 2-cocycles, and their sum is 0 since Q�ij is a 1-cocycle.It follows that .z2ij / is a 1-cocycle, hence it de�nes a U1-principal bundle PU1

M .Note that this bundle, called the determinant bundle of PSpincnM , is induced fromPSpincnM using the group homomorphism sqW Spincn ! U1. Its �rst Chern class

c1.PU1/ is de�ned as @.fij /, where .fij / 2 {C 1.M; C1

R/ is a logarithm of .z2ij /, i.e.,

exp.2�ifij / D z2ij . The mod 2 reduction of any integer k is just exp.i�k/. Hencethe mod 2 reduction of the integer class c1.PU1

/ is precisely @ exp.�ifij / D @.zij /,which represents w2.

Conversely, assume that w2 is the mod 2 reduction of the �rst Chern class of aU1-bundle L. By Proposition 3.7, there exists a principal Spincn bundle PSpincnM

lifting PSOnM . Let PU1be the determinant bundle of PSpincnM . We have seen in

the �rst part of the proof that c1.PU1/ � w2 mod 2, hence by hypothesis we have

c1.L/ � c1.PU1/ � 0 mod 2. By exactness of the top row in the cohomology

diagram from Lemma 3.8, c1.L/ � c1.PU1/ D 2c0 for some integer 2-class c0.

Let E be a U1-bundle with c1.E/ D c0 and de�ne a new Spincn-bundle by

P 0Spincn

M ´ PSpincnM �U1E:

The determinant bundle of P 0Spincn

M has the �rst Chern class c1.PU1/ C 2c0 D

c1.L/, thus it is isomorphic to L as required. �

3.3 Spin structures in low dimensions

3.3.1 Dimension 1

Every 1-dimensional manifold M is orientable and has w2 D 0 by dimensionalreasons, hence it is spin. By parametrizing using arc-length, every connected Rie-mannian manifold in this dimension is isometric either to the circle S

1 with a con-stant multiple of its standard metric, or to a (possibly in�nite) interval inside thereal line. The group Spin1 is f˙1g and the spinor representation is the canon-ical (i.e., non-trivial) representation of this group on C. For the real line thereexists just one spin structure, while on S

1 there are two such structures becauseH 1.S1;Z=2Z/ D f˙1g. The orthonormal frame bundle of S

1 is identi�ed withS1. The two spin structures are covering spaces of S

1; one consists of two copiesof S

1 (the so-called trivial spin structure), while the second one is the two-sheetedcover

S1 �! S

1; z 7�! z2:


In dimension 1 the spinor bundle is thus of rank 1 and trivial (every complexvector bundle on S

1 is trivial since it admits a non-zero real section). Let X be thepositively oriented unit vector �eld on M . Since the volume element !C D iX

de�ned in (1.10) acts as 1 on the spinor representation, it follows that Clifford mul-tiplication by X on spinors equals �i . The vector �eld X is clearly parallel withrespect to the Levi-Civita connection onM , hence every local lift zX to PSpin1

M ishorizontal. The Dirac operator D is therefore a scalar differential operator. Exceptfor the case of the non-trivial spin structure on the circle, the lift ofX exists globallyonM and it trivializes the spinor bundle. In this trivialization, D D �iX . In partic-ular, the Dirac operator of the trivial spin structure on S1 is �i@t if we parametrizethe circle by z D eit . The spectrum of this operator consists of the integers k 2 Z,with eigenspinors ekit . Theorem 4.35 states in this case that L2.S1/ is spanned bythis orthogonal system of functions, thus we recover the classical decomposition ofperiodic functions in Fourier series.

For the non-trivial spin structure, the Levi-Civita connection on spinors is �at,but has non-trivial holonomy �1, because the lift zX satis�es zX.tC2�/ D e�i zX.t/.Hence a trivialization of the spinor bundle is provided by the global spinor whichequals s D Œ zX; eit=2� above eit 2 S

1. In this trivialization, D D �iX C 12, and so

the spectrum of D is 12

C Z; in particular D is invertible, unlike for the trivial spinstructure.

The non-trivial spin structure on S1 is the restriction of the unique spin structure

on R2, as explained below.

3.3.2 Dimension 2

Every oriented surfaceM with a Riemannian metric g inherits an almost complexstructure J , de�ned as follows. For 0 ¤ X 2 TxM , JX is the unique vector ofthe same length as X , orthogonal to X and such that fX; JXg is a positively ori-ented frame. This almost complex structure is integrable, and thus de�nes a Kählerstructure, because of the existence of local isothermal coordinates .x; y/ such thatg D ef .x;y/.dx2 C dy2/. Indeed, the atlas consisting of oriented isothermal chartsis a holomorphic atlas, since the transition maps are oriented and conformal. IfMis compact, then its tangent bundle is stably trivial, thus the second Stiefel–Whitneyclass w2 vanishes. If M is non-compact, then its holomorphic tangent bundle istrivial, so w2 D 0 as well. Thus in both casesM is spin.

Let us examine in more detail some particular cases. OnR2 there exists a unique

spin structure PSpin2R2 ! PSO2

R2. By �xing the canonical frame on TR

2, thebundles PSpin2

R2 and PSO2

R2 are both identi�ed with the trivial bundle R

2 � S1,

and the covering map becomes the square z 7! z2. LetX be the unit tangent vectorto the circle, so that .X; JX/ is an orthonormal frame in TR

2 along S1. Its lift to

3.3. Spin structures in low dimensions 95

the spinor bundle is not a closed loop, hence the restriction to the boundary of thespin structure is the connected (or non-trivial) spin structure on the circle.

More generally, consider an oriented compact surface M of genus g with aconnected boundary. The spin structure induced along the boundary @M is againthe non-trivial one. Indeed, let N be the closed surface obtained by gluing a diskalong @M . By Proposition 3.6, every spin structure onM is the restriction of a spinstructure from N , since H 1.M;Z=2Z/ and H 1.N;Z=2Z/ are both isomorphicto .Z=2Z/2g .

Now let M be a compact oriented surface of negative Euler characteristic, letp 2 M and endowM n fpg with a complete hyperbolic metric of �nite volume. Itwas proved in [Bär00] that the spectrum ofD onM n fpg is discrete, as in the caseof compact manifolds. The key remark is that the continuous spectrum is relatedto the zero-modes of the Dirac operator along the boundary, and that the inducedDirac operator along the boundary is invertible. Moreover, in [Mor08] it is provedthat the eigenvalues of D grow according to the usual Weyl law (valid for closedsurfaces): the number of eigenvalues smaller in absolute value than � is asymptot-ically cj�.M/j�2, where c is a universal constant. The same fact is actually validfor more general complete metrics in every dimension, under the hypothesis thatthe induced Dirac operator along the model boundary is invertible.

Closed oriented surfaces of genus g are Kähler. By the results of Section 6.2, thespinor bundle decomposes as †CM D L, †�M D L ˝ ƒ0;1M , where L is anyof the 22g square roots of the canonical bundle K D ƒ1;0M . Moreover, the Diracoperator is self-adjoint, and its positive component isDC D x@. The Riemann–Rochformula in this case computes the Euler characteristic of L to be 0, since Lmust beof degree g � 1 D deg.TM/=2:

�.L/ D deg.L/C 1 � g D 0:

The Euler characteristic of L coincides with the index of DC (see Theorem 4.32),which by the Atiyah–Singer index theorem 3.10 is a Pontryagin number, so it van-ishes in dimension 2.

3.3.3 Dimensions 3 and 4

Every orientable 3-manifoldM is parallelizable, in particular it is a spin manifold.To see that TM is trivial one must in fact check that the Stiefel–Whitney classesw2 and w3 vanish (of course, the vanishing of the former implies the existence ofspin structures). The proof involves some non-elementary ingredients of algebraic-topological nature (Steenrod squares), so we will not attempt to reproduce it here.

Along the same lines, the second Stiefel–Whitney class of an oriented 4-manifoldis the reduction mod 2 of an integer class, thus every oriented 4-manifold admitsSpinc structures.


Let us stress that the above statements are true regardless of whetherM is com-pact or not.

3.4 Examples of obstructed manifolds

The �rst dimension in which there exist non-spin orientable manifolds is 4. Thestandard example is CP2; we have

H 2.CP2;Z/ ' Z; H 2.CP2;Z=2Z/ ' Z=2Z;

and w2.CP2/ corresponds to 1 under this isomorphism. In higher complex dimen-sions, CPn admits exactly one spin structure if n is odd, and none if n is even.

An obstruction to the existence of spin structures on compact manifolds is pro-vided by the yA-genus, a certain characteristic class with rational coef�cients whichwe review below. This obstruction is a direct consequence of the index theorem forthe Dirac operator, due to Atiyah and Singer.

Let .M n; g/ be a closed Riemannian manifold, and R 2 �.ƒ2M ˝End.TM//

its curvature tensor. We de�ne

tr.Rk/ 2 �.ƒ2kM/

as the End.TM/-trace of the composition R ı � � � ı R of k endomorphisms withvalues in the (commutative) even exterior algebra (by skew-symmetry, for k odd theabove trace is zero). The result of the trace is a closed differential form of degree2k, whose cohomology class is independent of the choice of the metric g on M .This allows one to de�ne tr.F.R// for any polynomial (or even formal series) F inone variable. Let

F.X/ ´ X=2

sinh.X=2/be such a series, and set

yA.R/ ´ exp�12tr�log

�F� 1

2�iR��

:

Formally one can interpret this differential form as the square root of the determinantof F.R=.2�i//. One could also describe the cohomology class of yA.R/ in termsof a polynomial in the Pontryagin classes ofM with rational coef�cients.

The index of the Dirac operator DC onM is by de�nition

Ind.DC/ D dimKer.DC/ � dimCoker.DC/ 2 Z:

Both quantities are �nite by Theorem 4.32 below. Although we recall certain ana-lytical aspects of the theory of elliptic operators in Chapter 4, a full treatment of theindex theorem would be too far from the spirit of the present book. We limit our-selves to stating the index theorem, referring to the books of Lawson and Michel-son [LM89] or Berline, Getzler, and Vergne [BGV92] for the proof.

3.4. Examples of obstructed manifolds 97

Theorem 3.10 (Atiyah–Singer index theorem). The index of the Dirac operatorDC

onM equals the integral of yA.R/ overM .

Thus if the top-dimensional component of the rational cohomology class yA.R/is not integer, we obtain an obstruction to the existence of spin structures on M .More geometrical consequences of Theorem 3.10 appear in Chapter 5.

Chapter 4

Analytical aspects

We use in this book, in an essential way, some properties of the Dirac and Laplaceoperators which are typical for more general elliptic differential operators. In thischapter we provide a self-contained treatment of pseudo-differential operators oncompact manifolds without boundary, which allows us to prove the following facts.

• The space of harmonic spinors is �nite-dimensional.

• The Dirac operator has a discrete set of real eigenvalues of �nite multiplicity,accumulating to ˙1, such that the corresponding eigenspinors are smoothand form a complete set in L2.M;†/.

• For every k 2 N, every smooth spinor can be approximated in Ck norm by�nite sums of eigenspinors.

• The eigenvalues of the Laplacian form a discrete subset in Œ0;1/. The corre-sponding eigenspaces are of �nite dimension and consist of smooth functions.

• Every smooth function can be approximated in Ck norm by �nite sums ofeigenfunctions of the Laplacian.

These facts are nowadays well known, but we felt that assuming them without proofwould have obscured the analytic facet of spinors and of the Dirac operator. Thesefacts are also used in Chapter 12 in our self-contained proof of the Peter–Weyltheorem. The theory of pseudo-differential operators was developed by Calderónand Zygmund, and in its modern form by Kohn, Nirenberg and Hörmander, seefor instance [Hor87]. Our presentation follows closely Melrose’s yet unpublishedlecture notes [Mel05].

4.1 Fourier transform

A smooth function f W Rn ! C is said to belong to the Schwartz class if for any

multi-indices j D .j1; : : : ; jn/; k D .k1; : : : ; kn/ 2 Nn, there exists a constant

cj;k 2 R such that

jxj @xkf .x/j � cj;k :

100 4. Analytical aspects

Schwartz functions are clearly absolutely integrable, thus for every � 2 Rn the

integral

F.f /.�/ ´Z

Rn

e�ix��f .x/dx

is absolutely convergent (here x � � denotes the Euclidean scalar product in Rn).

By Lebesgue’s dominated convergence theorem, F.f /, called the Fourier trans-form of f , is continuous and bounded in absolute value by kf kL1 . Using integra-tion by parts, one can easily see thatF.f / is itself Schwartz. Sometimes F.f /willalso be denoted by Of .

Lemma 4.1. The Fourier transform of e�jxj2=2 is .2�/n=2e�j�j2=2.

Proof. On every horizontal strip a � =.w/ � b, the complex function e�w2is

bounded and decays exponentially as j<.w/j ! 1. Cauchy’s integral formula onthe contour f=.w/ D 0g [ f=.w/ D �g and the identity

Z

R

e�x2=2dx D .2�/12

imply the lemma. �

Setg.x/ ´ e�jxj2=2 and g".x/ ´ g."x/

for every " > 0. As above, one computes F.g"/ D "�n.2�/n=2g1=". In particular,Z

Rn

g.x/dx D .2�/n=2:

Lemma 4.2. The Fourier transform preserves the L2 inner product on Schwartzfunctions up to the multiplicative factor .2�/n.

Proof. Let f; g 2 S.Rn/. Since jg.�/j � 1 and lim"!0 g" D 1 pointwise, byLebesgue’s dominated convergence we have

h Of ; Ogi DZ

Rn

Of .�/ Og.�/d�

D lim"!0

Z

Rn

g".�/ Of .�/ NOg.�/d�

D lim"!0

Z

R3n

e�"2j�j2=2�i.x�x0/��f .x/g.x0/dxdx0d�

D .2�/n=2 lim"!0

"�nZ

R2n

g1=".x � x0/f .x/g.x0/dxdx0

4.2. Pseudo-differential calculus 101

(by changing the contour of integration as in Lemma 4.1)

D .2�/n=2 lim"!0

Z

R2n

g.Y /f .x/g.x � "Y /dxdY

D .2�/nhf; gi

(again by dominated convergence and by evaluating the integral of g). �

LetL2.Rn/ denote the completion ofS.Rn/with respect to theL2 inner product.It follows from the above lemma that F extends as a bounded operator on L2.Rn/.Moreover, the adjoint of F is the operator {F de�ned by

{F.f /.�/ DZ

Rn

eix��f .x/dx;

and Lemma 4.2 implies that {FF D .2�/nI . In precisely the same way, we getF {F D .2�/nI and hence .2�/�n=2F is an isometry of L2.Rn/.

4.2 Pseudo-differential calculus

4.2.1 Symbols

Fix n 2 N. The Euclidean space Rn is the interior of a manifold with boundary Rn

de�ned by radial compacti�cation. More precisely, de�ne

Rn ´�RnG.0;1/� S

n�1�=�;

where � identi�es 0 ¤ x D r� 2 Rn, � 2 S

n�1, with .1=r; �/. We shall usethe boundary-de�ning function � ´ .1C r2/�1=2, where r D jxj is the Euclideandistance to the origin.

Let V be a real vector space (in practice, V D 0, Rn, or R

2n). For any com-pact manifold X , with or without boundary, let C1

V .X/ denote the space of smoothfunctions a.v; x/ on V � X such that for every k 2 N

dim.V / and every differentialoperator P on X ,

[email protected]; x/j � ck;P ; .v; x/ 2 V �X: (4.1)

If X has non-empty boundary, then restriction to @X evidently de�nes a surjectivemap

C1V .X/ �! C1

V .@X/:

If � is any vector �eld on X , it also de�nes a map

�W C1V .X/ 7�! C1

V .X/:


By iterating � followed by restriction to @X , we get a map into formal power series

C1V .X/ �! C1

V .@X/JtK: (4.2)

When � is transverse to @X , we obtain in this way the Taylor series at the boundarywith respect to �.

One particular case is obtained when X is a point; we denote by C1V the space

of functions on V with bounded partial derivatives of every order. At the otherextreme, when V D 0 we recover C1.X/, the space of smooth functions on thecompact manifold X .

Definition 4.3. The space of symbols of order 0 on Rn with coef�cients in V ,

denoted S0.V I Rn/, is the set C1

V .Rn/. For z 2 C, the space of symbols of

order z on Rn with coef�cients in V , denoted Sz.V I R

n/, is

Sz.V I Rn/ D ��zS0.V I R

n/ D .1C jxj2/z=2S0.V I Rn/:

Clearly the second de�nition of S0, obtained by setting z D 0, agrees with the�rst. The space of symbols of order 0 forms an algebra, while in general

Sz � Sw � SzCw :

Let us also de�ne S�1.V I Rn/ as the subset of S0 consisting of symbols which

vanish at the boundary of V � Rn in Taylor series, or equivalently the kernel of themap (4.2) when � is the radial vector �eld @�.

Remark 4.4. For V D 0, the space S�1.0I Rn/ is the same as the Schwartz space

S.Rn/.

Lemma 4.5. For every z 2 C the intersection

1\

jD0Sz�j .V I R

n/

equals S�1.V I Rn/.

Proof. For z D 0 this is just the de�nition, since the �rst j Taylor coef�cients ofa symbol of order �j are 0. In general, note that for every z 2 C, multiplicationby ��z preserves S�1.V I R

n/. �

4.2.2 Asymptotic summation

Lemma 4.6. Let .aj /j�0 be a sequence of complex numbers. There exists a smoothfunction f W R ! C whose Taylor series at x D 0 is given by

f .x/ �1X

jD0ajx

j :

4.2. Pseudo-differential calculus 103

Of course, when the series is absolutely convergent, one could de�ne f as itssum.

Proof. Let �W R ! R be a non-negative, smooth function with compact support in.�1; 1/, and identically 1 on Œ�1

2; 12�. For j � 1, take bj ´ maxfj; 4jaj j2=j g and

set

f .x/ ´1X

jD0ajx

j�.bjx/:

The sum is locally �nite near every x ¤ 0 because�.bjx/ D 0whenever jxj > 1=j .By direct computation, one checks that for all p � 0 there exists a constant cp suchthat for every j > 2p we have the uniform bound in x 2 R

jaj j [email protected]�.bjx//j � cpj p

2j:

This means that the series is absolutely convergent in the Cp topology for everyp � 0, hence the sum of the series is smooth. �

Lemma 4.7. Let z 2 C and consider a sequence sj 2 Sz�j .V I Rn/, j � 0. Then

there exists s 2 Sz.V I Rn/ such that for every k � 0

s �kX

jD0sj 2 Sz�k�1.V I R

n/:

The symbol s is called the asymptotic sum of the (generally divergent) seriesP1jD0 sj .

Proof. We will prove more generally that for every compact manifold with bound-ary X and for every sequence aj 2 C1

V .@X/, there exists f 2 C1V .X/ with Taylor

series f �P1jD0 aj �

j , where � is a boundary-de�ning function for @X ; in otherwords, the map (4.2) is onto. The proof is based on a diagonal subsequence proce-dure adapted to the proof of Lemma 4.6. We claim that there exists a countable setof differential operators A on X so that in the de�nition of C1

V .X/ it is enough tocheck inequality (4.1) forP 2 A. Such a sequence can be obtained as follows. Takea �nite number of vector �elds U1; : : : ; Um which generate TxX at each x 2 X ,including at boundary points. Then every differential operator P on X can be writ-ten as a polynomial in U1; : : : ; Um with coef�cients in C1.X/. By compactness ofX , the desired countable set can thus be chosen to be the monomials in the variablesU1; : : : ; Um.

The set of multi-indices k appearing in (4.1) is also countable; therefore,we must choose the constants bj as in Lemma 4.6 to satisfy a countable set ofinequalities. In other words, we have a double-index sequence

aj;m ´ supfj@kmv Pmaj .v; x/jI v 2 V; x 2 Xg


and we would like the series

fm.�/ ´1X

jD0aj;m�

j�.bj�/

to be absolutely convergent in Cp norm for all m;p. This is achieved by settingbj ´ maxfbj;1; : : : ; bj;j g, where bj;m ´ maxfj; 4jaj;mj2=j g is de�ned in terms ofaj;m as in Lemma 4.6. �

4.3 Pseudo-differential operators

Let V D R2n and a D a.x; x0; �/ 2 Sz.R2nI R

n/. We wish to de�ne a linearoperator

A D OpaWS.Rn/ �! S.Rn/

de�ned as

Af .x/ D .2�/�nZ

R2n

ei.x�x0/��a.x; x0; �/f .x0/dx0d�: (4.3)

The motivation for such a de�nition is the Fourier transform expression of a differ-ential operator,

P.x; i�1@x/f D .2�/�nZ

R2n

ei.x�x0/��P.x; �/f .x0/dx0d�;

where P depends polynomially on �. The integral (4.3) is absolutely convergent for<.z/ < �n. For other z, one must perform �rst the x0 integration, whose result isbounded from above by every power of � D .1C j�j2/�1=2.Lemma 4.8. Let a 2 Sz.R2nI R

n/ and f 2 S.Rn/. Then

ua;f .x; �/ ´Z

Rn

ei.x�x0/��a.x; x0; �/f .x0/dx0

is uniformly of order O.1C j�j2/�j for every j > 0.Proof. Since the integral in x0 is uniformly convergent, we have a priori

ua;f .x; �/ D O.��<.z//:

Write a D .1 C j�j2/b with b D ��2a 2 Sz�2. Integrating by parts, for everyc 2 Sw , we get

u�kc;f D �iu @c

@x0k

;f� iu

c; @f

@x0k

D O.��<.w//:

Thus u�c;f D O.��<.w// and by the lemma follows by induction. �

4.3. Pseudo-differential operators 105

We remark that directly from the de�nition,

@xjua;f D ui�ja;f ; xjua;f D iu@�j

a;f C iua;� 0jf ;

so by iteration we get ua;f 2 S�1.RnI Rn/. This implies that Af 2 S.Rn/, so Ade�nes a linear operator on S.Rn/.

Let z‰z.Rn/ be the space of linear operators on S.Rn/ de�ned by the oscillatingintegral (4.3) in terms of symbols in Sz.R2nI R

n/. De�ne also the subspaces

‰zl .Rn/ D fOpaI a D a.x; �/ 2 Sz.RnI R

n/g;and

‰zr .Rn/ D fOpaI a D a.x0; �/ 2 Sz.RnI R

n/g:

If <.z/ < �n, the integral (4.3) is absolutely convergent; thus, the operator Ais de�ned by a continuous integral kernel:

Af .x/ DZ

Rn

�A.x; x0/f .x0/dx0;

�A.x; x0/ D .2�/�n

Z

Rn

ei.x�x0/��a.x; x0; �/d�:

If <.z/ < �n�k, then �A.x; x0/ is of class Ck . In fact, �A.x; x0/ is smooth outsidethe diagonal fx D x0g:Lemma 4.9. Let � be a cut-off function as in the proof of Lemma 4.6. For everysymbol a 2 Sz.R2nI R

n/, the kernel

�.x; x0/ ´ .1� �.jx � x0j//�A.x; x0/

is smooth on R2n.

Proof. We will prove that �.x; x0/ belongs to C1R2n , in other words, all its partial

derivatives are bounded. Since �.jx � x0j/ 2 C1R2n , the product b.x; x0; �/ ´

.1 � �.jx � x0j//a.x; x0; �/ is also a symbol in Sz.R2nI Rn/. Using �.x; x/ D 0

write

b.x; x0; �/ DZ 1

0

@tb.x0 C t .x � x0/; x0; �/dt

D .x � x0/ �Z 1

0

.@xb/.x0 C t .x � x0/; x0; �/dt;

(4.4)

therefore

b DnX

jD1.x � x0/jbj ; bj 2 Sz.R2nI R

n/:


Integrating by parts with respect to �j , we see that .x � x0/jbj de�nes the sameoperator as �@�j bj 2 Sz�1.R2nI R

n/. By iterating this argument, we conclude that�.x; x0/ is the integral kernel of an operator de�ned by symbols of arbitrarily loworder, hence it is smooth. �

LetR denote the space of integral kernels �.x; x0/ such that

.x; x0/ 7�! �.x; x � x0/

belongs to S�1.RnI Rn/.

Proposition 4.10. The space of integral kernels of operators in each of the threespaces z‰�1.Rn/, ‰�1

r .Rn/, and ‰�1l .Rn/ isR.

Proof. It is rather clear that for every a.x; x0; �/ 2 S�1.R2nI Rn/ the associated

integral kernel is Schwartz as a function of x � x0 and has bounded derivativesin the variables xj . It is thus enough to prove that the right and left quantizationmaps Opr;Opl from S�1.RnI R

n/ to R are onto. That this is indeed the case is aconsequence of the fact that the Fourier transformation is an isomorphism from S

to itself. �

An operator with smooth integral kernel is called smoothing; its action clearlyextends from S.Rn/ to spaces of functions with less regularity, for instance, toL1.Rn/.

Proposition 4.11. For every z 2 C t f�1g, the spaces of operators z‰z.Rn/,‰zr .R

n/, and ‰zl .Rn/ are equal.

Proof. A priori, it is clear that ‰zr � z‰z . We will prove that

z‰z.Rn/ � ‰zr .Rn/C z‰z�1.Rn/:

Let � be a cut-off function and let a 2 Sz.R2nI Rn/. Write as in (4.4)

a.x; x0; �/� a.x0; x0; �/ DnX

jD1.x � x0/jaj ; aj 2 Sz.R2nI R

n/:

Integrating by parts with respect to �j , the terms in the right-hand side de�ne opera-tors of order z� 1. Hence Op.a/ D Op.a.x0; x0; �//COp.b/, Op.b/ 2 z‰z�1.Rn/.By iterating this argument, z‰z.Rn/ � ‰zr .R

n/C z‰z�k.Rn/. By asymptotic summa-tion, there exists a symbol b 2 Sz.RnI R

n/ such that Op.a/�Op.b.x0; �// µ C 2z‰�1.Rn/. Proposition 4.10 shows that z‰�1.Rn/ D ‰�1

r .Rn/, thus z‰z.Rn/ D‰zr .R

n/. The equality with ‰zr .Rn/ is analogous. �

4.4. Composition of pseudo-differential operators 107

Definition 4.12. A pseudo-differential operator of order z 2 C onRn is an operator

de�ned by an oscillatory integral (4.3) for some symbol in Sz.R2nI Rn/.

For every pseudo-differential operator A, there exist unique symbols al.x; �/,resp. ar.x0; �/ whose quantization Op de�nes A. It is moreover clear from Proposi-tion 4.11 that for every � 2 R

n n f0g,

limr!1

r�zal.x; r�/ D limr!1

r�zar.x; r�/:

This common limit is called the principal symbol of A, denoted �z.A/. The prin-cipal symbol is radially constant, hence it de�nes an element in C1

Rn.Sn�1/. From

Proposition 4.11, the principal symbol of Op.a.x; x0; �// equals �z.A/.x; �/ Dlimr!1

r�zal.x; x; r�/.

4.4 Composition of pseudo-differential operators

Let A, A0 be two pseudo-differential operators on Rn of orders z; z0 2 C. We claim

that the composition A ı A0WS ! S is again pseudo-differential, and

�zCz0.AA0/ D �z.A/�z0.A0/:

Indeed, represent A as the left quantization of the symbol a.x; �/, and A0 as theright quantization of a0.x0; �/. Their composition is

AA0f .x/ D .2�/�2nZei.x�y/��a.x; �/ei.y�x0/�� 0

a0.x0; � 0/f .x0/dx0d� 0dyd�:

Formally, the integral in y is the Fourier transform of the constant function 1, whichis the delta distribution in the variable � � � 0. This can be computed without usingdistributions. To be able to reverse the order of integration, introduce a Gaussianweight as in Section 4.1. Namely, set

g.y/ ´ e�jyj2=2; and g".y/ ´ g."y/

for every " > 0. Then

AA0f .x/

D .2�/�2n lim"!0

Zg".y/e

iy�.� 0��/eix��ix0�� 0a.x; �/a0.x0; � 0/f .x0/dx0d� 0dyd�:


Now the integral is absolutely convergent in y, and is computed by exchanging theorder of integration, as in Section 4.1:

AA0f .x/ D .2�/�3n=2 lim"!0

"�nZ

g1=".� � � 0/eix��ix0�� 0

a.x; �/a0.x0; � 0/f .x0/dx0d� 0d�

D .2�/�3n=2 lim"!0

Ze�jX j2=2eix��ix0�.�C"X/

a.x; �/a0.x0; � C "X/f .x0/dx0dXd�

D .2�/�nZeix��ix0�a.x; �/a0.x0; �/f .x0/dx0d�:

Thus AA0 is a pseudo-differential operator de�ned by the symbol a.x; �/a0.x0; �/.In particular, the space of smoothing operators forms a bilateral ideal inside the

pseudo-differential calculus, in the sense that ‰z ı‰�1 D ‰�1 ı‰z D ‰�1.

4.5 Action of diffeomorphisms on pseudo-differentialoperators

Let � � Rn be an open set, and ˆW� ! �0 � R

n a diffeomorphism. For everysymbol a.x; x0; �/ of order z with support in K � K � R

n with K compact in �0,let A be the associated pseudo-differential operator on S .

Proposition 4.13. The pull-back of A by the diffeomorphism ˆ is again pseudo-differential of order z, and its principal symbol is

�z.ˆ�A/.x; �/ D �z.A/.ˆ.x/;

tDˆ�1x �/:

Proof. By de�nition, ˆ�Af D ˆ�1 ı A.f ıˆ/. In coordinates,

ˆ�Af .x/ D .2�/�nZ

R2n

ei.ˆ.x/�x0/��a.ˆ.x/; x0; �/f .ˆ�1.x0//dx0d�

D .2�/�nZ

R2n

ei.ˆ.x/�ˆ.y//��a.ˆ.x/; ˆ.y/; �/f .y/Jˆ.y/dyd�;

where Jˆ.y/ D detDyˆ. Write ˆ.x/ � ˆ.y/ D Ux;y.x � y/ for some ma-trix Ux;y obtained as in (4.4) via Taylor expansion near x D y. This implies thatUx;x D Dxˆ is invertible, hence Ux;y is also invertible for x suf�ciently close toy, say for jx � yj < " whenever x 2 K.

Recall that outside the diagonal the integral kernel of a pseudo-differentialoperator is smooth; such a kernel clearly pulls back viaˆ to a smooth kernel, henceto a smoothing operator in ‰�1. Thus it is enough to analyze what happens near

4.6. Pseudo-differential operators on vector bundles 109

the diagonal. By multiplying a with a suitable cut-off function .x � y/, we canassume that Ux;y is invertible wherever a.x; y; �/ ¤ 0. Hence we write

ˆ�Af .x/ D .2�/�nZ

R2n

eiUx;y.x�y/��a.ˆ.x/; ˆ.y/; �/f .y/Jˆ.y/dyd�

D .2�/�nZ

R2n

ei.x�y/� tUx;y�a.ˆ.x/; ˆ.y/; �/f .y/Jˆ.y/dyd�

D .2�/�nZ

R2n

ei.x�y/�„a.ˆ.x/; ˆ.y/; tU�1x;y„/

f .y/Jˆ.y/ detU�1x;ydyd„:

Thus ˆ�A is a pseudo-differential operator with symbol

a.ˆ.x/; ˆ.y/; tU�1x;y„/f .y/Jˆ.y/ detU

�1x;y :

The principal symbol is computed by extracting the leading asymptotic term ina.ˆ.x/; ˆ.x/; tDxˆ

�1„/. �

4.6 Pseudo-differential operators on vector bundles

Let A be a m � k matrix of pseudo-differential operators with symbol a.x; x0; �/,a D .aij /, acting from S.Rn/˝ Rm to S.Rn/˝ Rk . Let also

G0 2 C1Rn ˝ GLm and G 2 C1

Rm ˝ GLk ;

i.e., both G0 and G are square matrices of functions on Rn with bounded partial

derivatives. ThenG ıAıG0 is again a matrix of pseudo-differential operators, withthe symbol

G.x/a.x; x0; �/G0.x0/: (4.5)

Let E;F ! M be complex vector bundles over a compact manifold M .A smoothing operator on .M IE;F / is an operator

RW C1.M;E/ �! C1.M; F /

de�ned by a smooth integral kernel r.x; x0/jdx0j 2 C1.M �M IE� � F ˝�M /,where �M is the bundle of 1-densities onM :

Rf .x/ DZ

M

r.x; x0/f .x0/jdx0j:

The notationE� �F means the tensor product of the bundles overM �M obtainedby pulling back F from the �rst copy ofM , respectively E� from the second copy.


Definition 4.14. LetM be a compact manifold without boundary, and E;F ! M

vector bundles of respective rank m, k. A pseudo-differential operator

AW C1.M;E/ �! C1.M; F /

is called pseudo-differential of order z 2 C if there exist

• a smoothing operator RW C1.M;E/ ! C1.M; F /,

• a �nite cover ofM D U1[� � �[Up by domains of charts �j WUj ! Vj � Rn,

• an associated partition of unity �j , and a family of smooth functions withcompact support �0

j WUj ! R (equal 1 on supp.�j /), such that �j�0j D �j ,

• trivializations j ; 0j of E;F over Uj , and

• for each j , am�kmatrixAj of pseudo-differential operators of order z on Vj ,such that

A D RCpX

jD1�0��

j Aj .��1j /��j :

From the invariance of the class of pseudo-differential operators under diffeo-morphisms (Proposition 4.13) and linear transformations, it follows immediatelythat a pseudo-differential operator on M is of the form in De�nition 4.14 in anysystem of charts, with respect to any local trivializations of E and F . Moreover, itis clear from the local composition results that pseudo-differential operators form acalculus, in the following sense:

‰z1.M IE 0; E/ ı‰z2.M IE 00; E 0/ � ‰z1Cz2.M IE 00; E/

where z1; z2 2 C t f�1g (by de�nition, ‰�1 denotes the space of smoothingoperators).

The principal symbol of a pseudo-differential operator A 2 ‰z.M IE;F / isde�ned once we choose a chart �WU ! R

n and local trivializations of E and Fover U . Using (4.5), �z.A/ is in fact a smooth section over U � R

n in the bundleE� ˝ F , i.e., it does not depend on the trivializations. By Proposition 4.13, if weidentify U � R

n with T�U using the chart �, the resulting principal symbol

�z.A/ 2 C1.T�U n f0g; E� ˝ F /

is independent of � altogether. Here a bundle over U is pulled back to T�U usingthe bundle projection. Moreover �z.A/ is homogeneous of degree z along the �bersof T�U .

4.6. Pseudo-differential operators on vector bundles 111

Thus to each pseudo-differential operator A 2 ‰z.M IE;F /, we associate asection �z.A/ in ��.E� ˝ F / over T�M n f0g, homogeneous of degree z. Thismap is compatible with composition of operators and of symbols:

�z1Cz2.AB/ D �z1

.A/�z2.B/

for every A 2 ‰z1.M IE 0; E/, B 2 ‰z2.M IE 00; E 0/.For every manifoldM , let

S�M �! M

denote the bundle of .n � 1/-dimensional spheres associated to T�X , with �ber.T�pX n f0g/=RC over p 2 X . By �xing a Riemannian metric on M , this sphere

bundle is identi�ed with the unit sphere bundle in T�M . Homogeneous functionsof degree z on T�M n f0g are in bijection with smooth functions on S�M .

Lemma 4.15. For every z, the principal symbol map

�z W‰z.M IE;F / �! C1.S�M;��.E� ˝ F //

is onto.

Proof. By using a partition of unity, it is enough to prove the surjectivity of the prin-cipal symbol map from scalar pseudo-differential operators on R

n onto compactlysupported smooth functions on S�

Rn � @.Rn � xRn/. This follows for operators

of order 0 from the de�nition of symbols as smooth functions on the compacti�ca-tion of T�

Rn and from the fact that for every manifold X with boundary @X , the

restriction map C1c .X/ ! C1

c .@X/ between the spaces of compactly-supportedfunctions is onto. For symbols of order z 2 C, the statement is reduced to the casez D 0 by multiplying with .1C j�2j/�z=2. �

Every smooth function f onM de�nes a multiplication operator on C1.M;E/.This operator is pseudo-differential of order 0, since in local coordinates and trivi-alizations it is given by the 0-order symbol f .x/, independent of �. More generally,every differential operator is pseudo-differential (of the same degree), with polyno-mial symbol.

If A is any operator of order z, the commutator ŒA; f � is pseudo-differentialof order z by the composition theorem. Its principal symbol is the commutatorŒ�z.A/; f � D 0 (because multiplication by f commutes with �berwise morphismsform E to F ). Thus, ŒA; f � 2 ‰z�1.

Proposition 4.16. Let A 2 Diff1.M IE;F / be a �rst-order differential operator oforder 1 onM . Let f 2 C1.M/ de�ne multiplication operators on C1.M;E/ andC1.M; F /. Then the principal symbol of A evaluated on .x; dxf / 2 T�

xM equalsthe commutator 1

iŒA; f � evaluated at x (the commutator is a 0-order differential

operator, hence a bundle morphism).


Using this proposition we can compute the principal symbols of the spin Diracoperator D and of the de Rham differential. Namely,

�1.D/.x; �/ D 1

i� �W†xM �! †xM

and

�1.d/.x; �/ D 1

i�^Wƒ�

xM �! ƒ�xM;

where � � denotes Clifford multiplication by �.

4.7 Elliptic operators

Definition 4.17. An operator A 2 ‰z.M IE;F / is called elliptic if for everyp 2 M and � 2 S�

pM , the map

�z.A/.�/WEp �! Fp

is invertible.

If A is elliptic, then of course the two bundles must have the same rank.Examples of elliptic differential operators are the spinDirac operatorD, the de RhamDirac operator d C ı where ı is the adjoint of d with respect to some Rieman-nian metric, and the Laplace operator on differential forms, � D dı C ıd (seeLemma 2.28).

Proposition 4.18. Let A 2 ‰z.M IE;F / be an elliptic pseudo-differential opera-tor. Then there exists G 2 ‰�z.M IF;E/ such that

AG � IF D R0 2 ‰�1.M IF /and

GA � IE D R 2 ‰�1.M IE/

Such an operator G is called a parametrix for A.

Proof. Find �rst G0 2 ‰�z.M IF;E/ with ��z.G0/.x; �/ D .�z.A/.x; �//�1.

This is possible by ellipticity and by surjectivity of the principal symbol map.Then �0.AG0/.x; �/ D IFx , so �0.AG0 � IF / D 0. Therefore, there existsR1 2 ‰�1.M; F / with AG0 � IF D R1. Choose G1 2 ‰�z�1.M IF;E/ withsymbol ��z�1.G1/ D ��z.A/�1��1.R1/ (again, this is possible since �z.A/ isinvertible and of the symbol map is surjective). Then R2 ´ A.G0 C G1/ � IF 2‰�2.M IF /. Continuing this procedure we get a sequence Gj 2 ‰�z�j .M IF;E/with A.G0 C � � � C Gj / � 1F 2 ‰�j�1.M IF /. Let G 2 ‰�z.M IF;E/ denoteany asymptotic sum of the series

P1jD0Gj . ThenR

0 ´ AG�1F 2 ‰�1.M IF /;

4.8. Adjoints 113

in other words,G is a right inverse forAmodulo smoothing operators. Similarly, weconstruct a left inverseG0 2 ‰�z.M IF;E/withR00 ´ G0A�IE 2 ‰�1.M IE/.ThenG0R0 �G0AG D G0 andGR00 �G0AG D G, and since the smoothing opera-tors form an ideal, G �G0 is smoothing and hence G is itself a left inverse modulosmoothing operators. �

It is clear from the proof that the parametrix G is unique up to ‰�1.M IF;E/.

4.8 Adjoints

Lemma 4.19. LetM be a compact manifold, A 2 ‰z.M IE;F /, and �M a volumedensity on M . There exists a pseudo-differential operator A� 2 ‰z.M IF �; E�/such that for every e 2 C1.M;E/ and f 2 C1.M; F �/ we have

Z

M

hAe; f i�M DZ

M

he; A�f i�M :

Clearly the operator A� is unique if it exists. For differential operators, A� isthe formal adjoint obtained by integration by parts.

Proof. The adjoint of a smoothing operator with integral kernel �.x; y/ exists andis also smoothing with kernel .x; y/ 7! �.y; x/�, where � denotes the adjointendomorphism. Using partitions of unity as in De�nition 4.14, it is enough to showthe existence of adjoints for a matrix of pseudo-differential operators with sym-bol a.x; y; �/, with compact support in x; y. The adjoint of Op.a.x; y; �// is justOp.a.y; x; �/�/. �

Hence, in particular, the principal symbol map is a �-morphism, in the sensethat is commutes with the operation of taking adjoints.

4.9 Sobolev spaces

To understand the topology of the Fréchet space of smooth functions on a compactmanifoldM we need to introduce some Hilbert spaces. The main one is L2.

Definition 4.20. Let E ! M be a vector bundle with a Hermitian metric, and �Ma density onM . The L2 inner product on C1.M;E/ is

he; e0iL2 DZ

M

he.x/; e0.x/i�M .x/:

The space L2.M;E/ of square-integrable sections in E is de�ned as the closure ofC1.M;E/ with respect to the L2 inner product.


SinceM is compact, we obtain an equivalent L2 inner product, hence the sameL2 space, if we start with a different inner product on E and a different density.

Theorem 4.21. Every operator A 2 ‰z.M IE;F / with <.z/ � 0 is bounded withrespect to the L2 norms on C1.M;E/ and C1.M; F /.

As a consequence, A extends as a Hilbert space operator

AWL2.M;E/ �! L2.M; F /:

Proof. The statement is clear for smoothing operators; using Fubini’s formula wecan check immediately that if K is de�ned by a continuous integral kernel �.x; y/,then

kKekL2.M;F / � supx;y2M

k�.x; y/kVol.M/ kekL2.M;E/:

Thus it remains to check the statement locally in Rn for operators with symbol of

compact support in x; y and acting on the trivial bundles E D Rk , F D Rm. Sinceboundedness of A is equivalent to boundedness for each coef�cient Aij , we canassume that A is a scalar operator. The condition of compact support implies thatthere exists 2 C1

c .Rn/ with ıA D A D A ı , where denotes the operator

of multiplication by on S.Rn/.DecomposeA into an operator with symbol a.x; �/ (i.e., a is independent of x0)

and an operator S with smooth kernel, i.e., A D Op.a.x; �//CS . We deduce A D Op.a/ C S . The operator S is de�ned by a compactly supported smoothintegral kernel, hence it is clearly L2-bounded. The operator of multiplication by is also clearly bounded. To �nish the proof, it is enough to show that a scalaroperator of order z, <.z/ � 0, with left symbol a.x; �/ with compact support in x,is bounded in L2 sense.

Assume that A 2 ‰z.Rn/ with <.z/ < �n. Then the symbol a.x; �/ is abso-lutely integrable in �, so for every f; g 2 S.Rn/,

jhAf;AgiL2 j � .2�/�2n supx2Rn

�Z

Rn

ja.x; �/jd��2

h Of ; Ogi:

Since the Fourier transform is an L2 isometry up to a dilation (Lemma 4.2), theabove inequality implies that A is bounded as an operator in L2 with domain S ,thus it extends to a unique bounded operator.

Assume now that z D 0. Let c > sup.x;�/2T�Rn ja.x; �/j2 (such a c < 1exists since a is of order 0). By the de�nition of symbols, ��a is a strictly positivesmooth function onR

n� xRn. The square root .c�ja.x; �/j2/1=2 is also smooth withcompact support onR

n�xRn, hence it de�nes a strictly positive symbol b0 of order 0.Let B0 2 ‰0 be an operator with symbol b0. By replacing it with .B0 C B�

0 /=2 if

4.9. Sobolev spaces 115

necessary, we can assume B�0 D B0. Since �0.B20 / D �0.c � A�A/, there exists

R1 2 ‰�1 withB20 D c � A�ACR1:

Let r1 D ��1.R1/ be the principal symbol of R1. Since B0, c, and A�A are self-adjoint, it follows that r1 is real. Choose C1 2 ‰�1 to be an operator with principalsymbol b1 D �b�1

0 r1=2. Again, C1 can be assumed to be self-adjoint. Set

B1 ´ B0 C C1

Then B21 � B20 belongs to ‰�1 and its principal symbol is �r1. It follows thatB21 D c � A�ACR2 for some self-adjoint R2 2 ‰�2. Continuing in this way, we�nd Bn 2 ‰0 with

B2n � .c � A�A/ D RnC1 2 ‰�n�1:

Let � 2 S.Rn/. Then to bound the L2 norm of A� we write

kA�k2L2 D hA�A�; �iL2 D ck�k2

L2 C hRnC1�; �iL2 � kBn�k2L2 :

The term containing Bn is negative, while the term containing RnC1 is bounded interms of j�j2 by the �rst part of the proof.

Finally, for arbitrary z with <.z/ � 0, let A0 be the operator with symbola.x; �/.1 C �2/�z=2, and D D Op..1 C �2/z=2/. By the composition theorem,A0D D A. The operator D is clearly L2 bounded because after conjugation withthe Fourier transform it becomes the multiplication operator by the bounded func-tion .1C �2/z=2. The operator A0 was shown above to be bounded. �

Let A 2 ‰z.M IE;F / be a pseudo-differential operator. De�ne an inner prod-uct on C1.M;E/ by

h�; iA ´ h�; iL2 C hA�;A iL2 :

If <.z/ � 0, by Theorem 4.21 the resulting norm is equivalent to the L2 norm.

Lemma 4.22. Let A 2 ‰z.M IE;F /, A0 2 ‰z0.M IE;F 0/ be two pseudo-differ-

ential operators with injective principal symbols, such that <.z/ D <.z0/ > 0.Then the norms k � k2A and k � k2A0 are equivalent.

Proof. In �nite dimensions, a linear map L between Euclidean vector spaces isinjective if and only if L�L is an isomorphism. Accordingly, if we choose metricsonM;E; F; F 0 in order to de�ne adjoints, the operators

U ´ A�A and U 0 ´ .A0/�A0

are elliptic. Let G be a parametrix for U , so UG D I �R with R 2 ‰�1.M;E/.Then U 0 D UGU 0 C RU 0 D U.GU 0/ C R0 where R0 2 ‰�1.M;E/ and


GU 0 2 ‰2z0�2z.M;E/. By the hypothesis and Theorem 4.21, GU 0 is boundedon L2; since R0 is also bounded, we see that k � k2A dominates k � k2A0 . The reverseinequality is analogous. �

Based on this lemma, for every k � 0 we de�ne the Sobolev space H k.M;E/

as the completion of C1.M;E/ with respect to the inner product h�; �iA, for somepseudo-differential operator A 2 ‰z.M IE;F /, <.z/ D k, with injective principalsymbol. The Hilbert space H k itself is independent of the choice of A, while theinner product clearly depends on A.

With this de�nition, it is useful to show that the Sobolev spaces are subspacesof L2.

Lemma 4.23. For every k � 0,H k.M;E/ � L2.M;E/.

Proof. Fix A 2 ‰z.M IE;F / and let fj be a Cauchy sequence for the norm k � k2A.Then clearly the sequence fj is also Cauchy with respect to the L2 norm. Letf 2 L2.M;E/ denote its limit, and denote by F D Œ.fj /j�1/� its limit inH k . Weclaim that the map F 7! f is injective. To prove injectivity, assume that f D 0 and�x " > 0. Then there exists j1 such that for i; j � j1 we have kA.fi � fj /k2 < ".For j � j1 �xed we write

" > kA.fi � fj /k2L2 D kAfik2L2 � 2<hfi ; A�Afj i C kAfj k2L2;

so, since fi ! f D 0 as i ! 1, we have lim sup kAfik2L2 < ". Since " was

arbitrary, Afi ! 0, so the limit F of .fj /j�1 in H k is 0. �

A pseudo-differential operatorA 2 ‰z.M IE/, <.z/ � 0 is called non-negativeif for every � 2 C1.M;E/ one has

hA�; �i � 0:

In this case, we clearly have A� D A, which implies that z 2 R. A non-negativeoperator is called positive if there exists some c > 0 such that

hA�; �i � ck�k2; � 2 C1.M;E/:

For every elliptic positive operator of order k, the norm kA�k2L2 dominates �2,

hence it is equivalent to the Sobolev norm k � kA.

Lemma 4.24. Let k � 0 be an integer and A 2 ‰k.M IE;F / an elliptic, non-negative pseudo-differential operator. Then the operator

I C AWH k.M;E/ �! L2.M;E/

is an isomorphism of Hilbert spaces.

4.9. Sobolev spaces 117

Proof. We already know that I C A is continuous as an operator from H k to L2.In fact, by de�ning the inner product onH k usingA, the non-negativity ofA impliesthat I C A increases norms, in the sense that

k.I C A/�k2L2 � k�k2A: (4.6)

Immediately this implies that the range of I C A is closed. Let G be a parametrixfor I C A, so G.I C A/ D I C R, .I C A/G D I C R0 with smoothing errorterms R;R0. Then every � 2 H k which is mapped to 0 by 1 C A must satisfy� C R� D 0, thus � is smooth. By the non-negativity of A, this implies that� D 0. Now, let 2 L2.M;E/ be orthogonal to the range of I C A. Then must be orthogonal to the range of I C R0, in particular ? .I C R0/� for all� 2 C1.M;E/. This can be interpreted as � ? .I C .R0/�/ , and since smoothsections are dense in L2, .I C .R0/�/ D 0. Hence D �.R0/� is smooth.But a smooth section orthogonal to the range of I C A must be in the kernel ofI C A, hence it must be 0. Thus I C A is continuous and bijective. The inverseis clearly continuous thanks to (4.6), or alternately thanks to the bounded inversetheorem. �

The above lemma can be extended to every elliptic positive operator of realnon-negative order k instead of I C A; the modi�cations in the proof are minor.

Elliptic non-negative pseudo-differential operators are easily constructed;for instance, for any B 2 ‰k=2.M;E; F / with injective symbol, the operator

A ´ B�B

is both elliptic and non-negative.

Lemma 4.25. Let A 2 ‰z.M IE;F / be an elliptic pseudo-differential operatorwith k ´ <.z/ � 0. Assume that

AWH k.M;E/ �! L2.M; F /

is a linear isomorphism. Then the inverse operator

A�1WL2.M; F / �! H k.M;E/

belongs to ‰�z.M IF;E/.

Proof. Choose a parametrix G withGA D I CR,AG D I CR0 for some smooth-ing errors R;R0. Multiplying by A�1 we get G D A�1 CRA�1 D A�1 C A�1R0

as operators on L2. This yields

A�1 D G �RA�1 D G � A�1R0 D RA�1R0 � RG CG:


The last two terms are in ‰�1, resp. ‰�z . We claim that the operator RA�1R0 isgiven by a smooth integral kernel �.x; x0/. For every x; x0 2 M de�ne �.x; x0/ Dhr.x; �/; A�1r 0.�; x0/i. Since A�1 is bounded in L2, � is continuous. Applyingthe same reasoning for the smoothing operators PR, resp. R0P 0 where P; P 0 arearbitrary differential operators onM , we see that P�P 0 is again continuous, so � issmooth. �

Thus, the Sobolev spaces H k are isomorphic to L2, the isomorphism and itsinverse being both given by pseudo-differential operators. From this we deduce thatfor k > k0 > 0we haveH k.M;E/ � H k0

.M;E/ � L2.M;E/. For k > 0, de�neH�k.M;E/ as the dual of the Hilbert spaceH k.M;E/, i.e., the space of continuouslinear functionals on H k.M;E/. It clearly does not depend on the choice of normonH k.M;E/. It follows that for all k; k0 2 R, k > k0 we have canonical inclusions

H k.M;E/ ,�! H k0.M;E/:

Hence, every pseudo-differential operator B 2 ‰z.M IE;F / acts boundedlyfrom H k.M;E/ to H k�<.z/.M; F / for every k 2 R. Moreover, ‰�1.M IE;F /maps H k.M;E/ into C1.M; F /.

Proposition 4.26 (Elliptic Regularity). Let A 2 ‰z.M IE;F / be elliptic, andk 2 R. If � 2 Hh.M;E/ for some h 2 R such that A� 2 H k.M; F /, then� 2 H kC<.z/.M;E/. If A� 2 C1.M; F /, then � 2 C1.M;E/.

Proof. Let G 2 ‰�z.M IE;F / be a parametrix,

GA D I CR:

Then � D GA� � R�. Since R is smoothing, R� is smooth and so it belongs toevery Sobolev space. If A� 2 H k , then from the mapping properties of pseudo-differential operators it follows that GA� 2 H kC<.z/. Assume now A� to besmooth. Then GA� is smooth by the de�nition of pseudo-differential operators asacting on smooth functions; sinceR� is also smooth, we have � 2 C1.M; F /. �

4.10 Compact operators

Fix a Hilbert space H and let B.H/ be the algebra of bounded linear endomor-phisms of H . Let F denote in this section the bilateral ideal of operators of �niterank. Let C.H/ denote the norm-closure of F in B.H/. It is again a bilateral ideal.

Definition 4.27. A compact operator on H is an operator which can be approxi-mated in norm by �nite-rank operators.

Clearly, the space of compact operators is just the ideal C.H/. Alternately, anoperator in B.H/ is compact if and only if it maps bounded sets into relativelycompact sets in H .

4.10. Compact operators 119

Lemma 4.28. Every smoothing operator R 2 ‰�1.M; F / is compact as an oper-ator on L2.M; F /.

Proof. By the Stone–Weierstraß theorem, every continuous function r.x; x0/ onM � M can be C0-approximated by �nite sums of terms of the form f .x/g.x0/(such a term de�nes a rank 1 operator). The operator norm of R is bounded by(in fact, equivalent to) the sup norm of its integral kernel r . Thus R can be approx-imated in norm by �nite-rank operators. �

Definition 4.29. A bounded linear operator AWH ! H 0 between Hilbert spaces iscalled Fredholm if

(1) dimKer.A/ < 1 and

(2) dimCoker.A/ < 1, where the cokernel of A, Coker.A/, is the linear spaceH 0=A.H/.

The second condition implies that the range of A is closed in H 0. Indeed, afterfactoring A through H=Ker.A/, we can assume that A is injective. Let V be a�nite-dimensional complement in H 0 to A.H/, i.e., H 0 D V ˚ A.H/. Every�nite-dimensional subspace in H 0 is a Hilbert space. Then the linear map

ˆWV ˚H �! H 0; .v; x/ 7�! v C A.x/;

is bounded and bijective between Hilbert spaces, hence by the open mapping theo-rem it maps the closed subspaceH bijectively onto another closed subspace, whichis precisely A.H/.

Proposition 4.30. An operator AWH ! H 0 is Fredholm if and only if it admits aninverse GWH 0 ! H modulo compact operators, i.e.,

GA D IH CR; and AG D IH 0 CR0

with R 2 C.H/, R0 2 C.H 0/.

Definition 4.31. The index of a Fredholm operator AWH ! H 0 is the difference

Ind.A/ ´ dimKer.A/� dimCoker.A/ 2 Z:

Theorem 4.32. LetA 2 ‰k.M IE;F / be an elliptic (pseudo-)differential operatoron a compact manifoldM ,

AW C1.M;E/ �! C1.M; F /:

Then A has �nite-dimensional kernel and cokernel.


Proof. Let G 2 ‰�k.M IF;E/ be a parametrix for A. By elliptic regularity, thekernel of

AWH k.M;E/ �! L2.M; F /

is precisely the kernel of AW C1.M;E/ ! C1.M; F /. Let now 2 L2.M; F / beorthogonal to A.H k.M;E//. Then ? AG� for every � 2 C1.M; F /, whereAG D I CR0, R0 2 ‰�1.M; F /. This means that .I C R0/ is orthogonal to �,and since � was arbitrary we have D �R0 2 C1.M; F /. This shows that theorthogonal complement A.H k.M;E//? is Ker.A�/, which consists of smooth sec-tions thanks to elliptic regularity. Thus L2.M; F / decomposes into the orthogonaldirect sum

L2.M; F / D A.H k.M;E//˚ Ker.A�/; Coker.A/ ' Ker.A�/ � C1.M; F /:

Hence every 2 L2.M; F / which is orthogonal to the �nite-dimensional spaceCoker.A/ is of the form A.�/ with � 2 H k.M;E/. By elliptic regularity, if is smooth, so is �. Let f1; : : : ; fm be an orthonormal basis consisting of smoothsections in the �nite-dimensional space Ker.A�/. The proof is done by noting thatevery f 2 C1.M; F / can be L2-orthogonally decomposed as

f D a1f1 C � � � C amfm C f 0;

with aj D hf; fj iL2 and f 0 ? Ker.A�/, f 0 2 C1.M; F /. �

The proof also shows that the kernel and cokernel of

AWH s.M;E/ �! H k�s.M; F /

are independent of s 2 R and consist of smooth sections.

Corollary 4.33. The eigenspaces of any elliptic operator A 2 ‰k.M IE;F / withk > 0 are �nite dimensional.

Proof. If k > 0, then for every � 2 C the operator A � �I is elliptic. ApplyTheorem 4.32 to A � �I . �

Lemma 4.28 is in fact valid for operators of negative order.

Lemma 4.34. Let A 2 ‰z.M IE;F /, <.z/ < 0. Then

AWL2.M;E/ �! L2.M; F /

is compact.

Another way of stating this is that the canonical inclusion

H k.M; F / ,�! L2.M; F /

is compact; this is known as the Rellich lemma.

4.11. Eigenvalues of self-adjoint elliptic operators 121

Proof. Since operators in ‰iy act boundedly on L2 for y 2 R, ‰�1 consists ofcompact operators, and the compact operators form an ideal inB.L2/, it is enough toprove the claim for one elliptic operator in ‰k.M;E/, k < 0. Since multiplicationby cut-off functions is bounded on L2, we can localize the proof to a relativelycompact domain� � R

n, for scalar operators. It is enough to show that the operatorOp..1 C j�j2/k=2/ is compact from L2.�/ to L2.Rn/. By the Cauchy–Schwarzinequality, Of is smooth and there exist c0; c1 depending on � such that for everyf 2 L2.�/,

k Of kC0 � c0kf kL2and k@�j Of kC0 � c1kf kL2

(for instance, c0 D vol.�/1=2). Let .fj /j�1 be a sequence in L2.�/, kfj kL2 � 1.Since Op..1C j�j2/k=2/f is the Fourier transform of .1C j�j2/k=2 Of , it is enoughto show that the sequence .1C j�j2/k=2 Ofj admits a convergent subsequence.

Take a dense countable subset p1; p2; : : : in Rn (like the points with rational

coordinates). Since Ofj .pk/ is bounded by c0, by the local compactness of C wecan extract a subsequence such that Ofjs .p1/ is convergent. Extract then a sub-subsequence such that the evaluations at p2 are also convergent, and so on. Thediagonal procedure applied to this nested sequence of sub-sequences produces asubsequence . Ofi/i2I with Of .pk/ convergent for all k. Since the derivatives of Ofj areuniformly bounded by c1 and the pk’s are dense, it follows that Ofi .�/ is convergentto some u.�/ for all � 2 R

n. By the Lebesgue dominated convergence theorem, thisfunction is square-integrable on every ball BR.0/ in R

n (i.e., it is locally L2). Thus,after relabeling the sequence, it follows that Ofj is a Cauchy sequence in L2.BR.0//for every R. On the other hand, the norm of Ofj .1 C j�j2/k=2 in L2.Rn n BR.0//is uniformly bounded above by c0.1 C R2/k=2, hence small for large R since k isnegative. This means that Ofj .1C j�j2/k=2 is a Cauchy sequence in L2.Rn/. �

4.11 Eigenvalues of self-adjoint elliptic operators

We recall that a pseudo-differential operator A 2 ‰k.M;E/ is called here self-adjoint ifM is endowed with a volume density, E with a Hermitian metric, and

hAf; f 0iL2 D hf; Af 0iL2 ; f; f 0 2 C1.M;E/:

By continuity, the same identity holds for f 2 H s.M;E/, f 0 2 H k�s.M;E/. Thisis not the usual de�nition of self-adjoint unbounded operators, but for our purposesit is adequate.


Theorem 4.35. LetA 2 ‰k.M;E/ be elliptic and self-adjoint, with k > 0. ThenAhas �nite-dimensional eigenspaces, with discrete eigenvalues accumulating to˙1.Moreover, the space L2.M;E/ decomposes as an orthogonal Hilbert direct sum ofthe eigenspaces of A.

Proof. Let f1; : : : ; fm be an orthonormal basis in Ker.A/. It consists of smoothsections in E, so the integral kernel

p.x; x0/ ´mX

jD1fj .x/˝ fj .x

0/

is smooth and de�nes the orthogonal projector onto Ker.A/ in L2.M;E/. Hencethe operator A C PKerA is pseudo-differential, elliptic, self-adjoint and invertible.It follows from Lemma 4.25 that B ´ .A C PKerA/

�1 belongs to ‰�k.M;E/.The operator B is bounded on L2.M;E/ by Theorem 4.21, and in fact compact byLemma 4.34. It is also clearly self-adjoint.

HenceB shares the properties of every self-adjoint compact operator on aHilbertspace: the eigenspaces of B are �nite dimensional, mutually orthogonal, and theyspan a dense subspace of L2. The eigenvalues are real, and they accumulate to 0.Since B is invertible, 0 is not an eigenvalue.

Every eigenspace of B of eigenvalue � is also an eigenspace of AC PKerA, ofeigenvalue ��1. Moreover, both A and A C PKerA preserve the orthogonal split-ting L2 D Ker.A/ ˚ Ker.A/?, hence they have essentially the same eigenspaceswith the same eigenvalues, with the exception of Ker.A/, which is an eigenspace ofeigenvalue 1 for AC PKerA, resp. 0 for A. The conclusion follows. �

Since the Dirac operator D is elliptic and self-adjoint, we obtain directly

Corollary 4.36. Let .M; g/ be a closed Riemannian spin manifold and †M thespinor bundle. There exists an orthonormal basis in the Hilbert space L2.M;†M/

consisting of smooth eigenspinors. Each eigenspace is �nite dimensional, and theeigenvalues accumulate to ˙1.

The discrete set of eigenvalues ofD considered with their multiplicities is calledthe spectrum of D.

Similarly, the Laplace operator� D d�d on C1.M/ is self-adjoint and elliptic.Thus we deduce

Corollary 4.37. Let .M; g/ be a closed connected Riemannian manifold. Thereexists an orthonormal basis in the Hilbert spaceL2.M/ consisting of smooth eigen-functions of�. Each eigenspace is �nite dimensional, and the eigenvalues accumu-late to 1. The kernel of � is the space of constant functions onM .

4.11. Eigenvalues of self-adjoint elliptic operators 123

Only the last statement deserves some explanation. If �f D 0, then by inte-grating against f we get

0 D h�f; f iL2 D hd�df; f iL2 D hdf; df iL2 D kdf k2L2.M;ƒ1M/

and so df D 0, which is equivalent to f D constant.

Corollary 4.38 (Sobolev embedding). Let f be a smooth function (resp. a smoothspinor) on a compact Riemannian (resp. spin) manifold .M; g/. Let

f DX

fj

be the orthogonal L2 decomposition of f into eigenfunctions of �, resp. eigen-spinors. Then the above series converges in Ck norm for every k � 0.

Proof. Let feig be an orthonormal basis ofL2 consisting of eigensections ofD D �

orDwith eigenvalue �i , andwrite f DPaiei . Then hDf; eii D hf;Dei i D �iai ,

so the decomposition of Df in the basis feig is Df DPai�iei . It follows that

the L2 decomposition of f into eigenmodes of D converges in every H s norm.We now prove that the Ck norm is dominated by any H s norm with s > k C n=2

where n D dimM . This can be checked locally, thus for compactly supportedfunctions on R

n. In that setting, it is a consequence of the fact that .1C j�j2/k�s2 is

square-integrable under our hypothesis k � s < �n=2. �

Corollary 4.39. Let .M; g/ be a closed connected Riemannian manifold. Thenevery square-integrable function f 2 L2.M; �g/ of mean 0 with respect to theRiemannian volume density �g (i.e.,

RMf �g D 0) belongs to the image of the

Laplacian. If f belongs to the Sobolev space H k , k � 0, then f D �u withu 2 H kC2 unique up to an additive constant. If f is smooth, then the solution u isalso smooth.

Proof. The conditionRM f �g D 0 is equivalent to f being L2-orthogonal to the

constant function 1. But we have seen in Corollary 4.37 that Ker.�/ is spannedby the constant functions. Thus f decomposes as a series f D

P1jD1 fj�j in

terms of an orthonormal basis of eigenfunctions �j for �, of eigenvalues �j > 0.Set u ´

P1jD1 �

�1j fj�j . Since the sequence �j is increasing, the series de�ning u

is absolutely convergent, hence it de�nes an L2 function. The regularity propertiesfollow from Proposition 4.26. �

Part II

Lowest eigenvalues

of the Dirac operator

on closed spin manifolds

Chapter 5

Lower eigenvalue bounds on Riemannian closed spin

manifolds

In this chapter we start by recalling the following fundamental result of A. Lich-nerowicz: on a closed (compact without boundary) Riemannian spin manifold thereexists a topological obstruction to the existence of metrics with positive scalar cur-vature. This result is obtained as a corollary of a (non-optimal) eigenvalue estimatefor the Dirac operator. We then derive Friedrich’s (optimal) inequality, which canbe seen as the spinorial Cauchy–Schwarz inequality. In fact, we shall see that thedefect in such an equality is measured by the Penrose operator. We also give nec-essary conditions for the existence of “special spinor �elds” satisfying the limitingcase of this inequality. Finally, we show that the conformal covariance of the Diracoperator and that of the Yamabe operator lead to a relation between their �rst eigen-values.

5.1 The Lichnerowicz theorem

An important consequence of the Schrödinger–Lichnerowicz formula is the follow-ing gap phenomenon in the spectrum of the Dirac operator [Lic63].

Theorem 5.1. On a compact Riemannian spin manifold with positive scalar cur-vature Scal, the Dirac operator has no eigenvalues in the closed interval

h� 1

2

pScal0;C

1

2

pScal0

i;

where Scal0 denotes the in�mum of the scalar curvature.

Proof. If r� denotes the formal adjoint of the Levi-Civita connection with respectto the natural Hermitian scalar product h�; �i on the spinor bundle, then applying theSchrödinger–Lichnerowicz formula

D2 D r�r C 1

4Scal; (5.1)

128 5. Lower eigenvalue bounds on Riemannian closed spin manifolds

to a spinor �eld‰, taking its scalar product with‰, and integrating over the compactmanifoldM with respect to the volume element �g , yields

Z

M

jr‰j2�g DZ

M

jD‰j2�g � 1

4

Z

M

Scalj‰j2�g : (5.2)

Let � be any eigenvalue of D and ‰ a corresponding eigenspinor. Then sincejr‰j2 � 0, (5.2) yields

�2 � 1

4Scal0:

If equality holds, then the scalar curvature is constant andr‰ � 0. Hence,D‰ D 0

and

0 D �2 D 1

4Scal0 D 1

4Scal;

which contradicts the curvature assumption. Therefore,

�2 >1

4Scal0 (5.3)

which concludes the proof. �

A fundamental consequence of this result is the famous Lichnerowicz theorem.Before stating this result, let us recall from Chapter 4 that the analytical index of anelliptic operator D is de�ned as

IndD D dimKerD � dimCokerD (5.4)

(by Theorem 4.32, both quantities in the right-hand side are �nite). Since the Diracoperator D is elliptic and formally self-adjoint, its analytical index is zero.As we have seen, on an even-dimensional manifold, under the action of the volumeelement, the spinor bundle splits into the direct sum of positive and negative spinors.The corresponding splitting of the Dirac operator is given byD D DCCD�, whereDC (resp. D�) interchanges positive (resp. negative) and negative (resp. positive)spinors. Since .DC/� D D�, the analytical index of the elliptic operator DC isgiven by

IndDC D dimKerDC � dimKerD�:

By the Atiyah–Singer index theorem 3.10, the analytical index of DC is equalto the yA-genus ofM (see Section 3.4 for the de�nition), hence

Theorem 5.2. On a compact Riemannian spin manifold with non-vanishingyA-genus, there is no metric with positive scalar curvature.Proof. By Theorem 5.1, if the scalar curvature is positive, then

Ker.DC/ D Ker.D�/ D f0g;

hence IndDC D 0. �

5.2. The Friedrich inequality 129

The importance of this result resides in the surprising fact that a weak geometri-cal assumption (Scal > 0) imposes severe restrictions on the topology of the man-ifold. Important extensions of positive scalar curvature obstructions were obtainedby Hitchin, Gromov and Lawson, and Stolz [Hit74], [GL80b], [GL80a], [GL83],[Sto90], and [Sto92]. More recently, in a series of papers by Ammann, Dahl, andHumbert [ADH09a], [AH08], and [ADH09b], the authors obtained some re�nedtopological obstructions. See also [Gin09] for a short survey of these results.

We are ready now to give a uniform approach, based on the use of the Pen-rose operators introduced in Section 2.3, yielding to basic optimal re�nements ofinequality (5.3). For metrics with positive scalar curvature, three families of com-pact spin manifolds are of interest: Riemannian, Kählerian, and quaternion-Kählermanifolds. It turns out that on the spectral level, via the twistor construction, thethree structures are related. The round sphere, the complex projective space (of oddcomplex dimension), and the quaternionic projective space are the model spaces ofsuch families.

We will show that in the three cases, if �1 denotes the lowest eigenvalue of theDirac operator, then the ratio �21=Scal0 for the given manifold is at least equal to thecorresponding ratio for the model space.

It is worth noting that the size of the gap in the spectrum of the Dirac operator in-creases with the rigidity of the geometry of the compact Riemannian spin manifold.As we have seen in Section 2.3, it is suf�cient to determine the Penrose operatorsof manifolds with special holonomy group.

5.2 The Friedrich inequality

The next step is to improve the Lichnerowicz inequality (5.3) so that equality couldbe achieved. We have the following result of Th. Friedrich [Fri80].

Theorem 5.3. On a compact Riemannian spin manifold any eigenvalue � of theDirac operator satis�es

�2 � n

4.n� 1/Scal0: (5.5)

Moreover, if equality holds, then the manifold is Einstein.

Remark 5.4. Of course inequality (5.5) is trivial if Scal0 � 0, so it is interestingonly for the case Scal0 > 0. In this case, denote by

�21 ´ �21.Mn; g/

the lowest eigenvalue ofD2 on a compact Riemannian spin manifold .M n; g/ withpositive scalar curvature and let �1 be the ratio

�21Scal0

:


Since

�21.Sn; g0/ D n2

4

(see [Sul79]), equation (5.5) can be recast as

�1.Mn; g/ � �1.S

n; g0/; (5.6)

where .Sn; g0/ denotes the standard n-dimensional sphere with its canonical struc-ture.

Proof. The �rst proof we give here, due to Sylvestre Gallot, is based on the Cauchy–Schwarz inequality, like for the corresponding problem for the Laplacian on func-tions and differential forms; see [Lic58] and [GM75]. For an arbitrary spinor �eld‰ 2 �.†M/ we have

jD‰j2 DˇˇnX

iD1ei � rei

‰

ˇˇ2

�� nX

iD1jei � rei

‰j�2

D� nX

iD11jrei

‰j�2

� n

nX

iD1jrei

‰j2

D njr‰j2:

(5.7)

This inequality combined with the Schrödinger–Lichnerowicz formula implies that

1

n

Z

M

jD‰j2�g �Z

M

jr‰j2�g DZ

M

jD‰j2�g � 1

4

Z

M

Scalj‰j2�g ;

whence �1� 1

n

� Z

M

jD‰j2�g � 1

4

Z

M

Scalj‰j2�g :

For an eigenspinor ‰ 2 �.†M/, it follows that

�2Z

M

j‰j2�g � 1

4

n

n � 1

Z

M

Scalj‰j2�g � 1

4

n

n � 1Scal0Z

M

j‰j2�g :

For the last statement, see Proposition 5.11 below. �

5.2. The Friedrich inequality 131

Remark 5.5. If ‰ 2 �.†M/ is an eigenspinor for which equality holds, that is

�20 D n

4.n � 1/Scal0;

then, for all X 2 �.TM/, the spinor �eld ‰ satis�es the twistor equation

rX‰ C 1

nX � D‰ D 0;

which specializes to the Killing equation

rX‰ C �0

nX �‰ D 0;

since D‰ D �0‰.

To see this, note that if equality is achieved in (5.7), then by Cauchy–Schwarz,the two vectors are proportional, i.e.,

ei � rei‰ D ej � rej‰; i; j 2 f1; : : : ; ng;

which implies, for all i 2 f1; : : : ; ng,

D‰ D nei � rei‰;

i.e.,

rei‰ C 1

nei � D‰ D 0:

For further considerations, it is useful to prove inequality (5.5) by consideringan optimal decomposition of the gradient of a spinor. From Section 2.3, we knowthat the Penrose (or twistor) operator P , the complement of the Dirac operator inthe Levi-Civita covariant derivative, is locally given by

PW�.†M/r�! �.T�M ˝†M/

��! Ker � �.T�M ˝†M/;

‰ 7�!nX

iD1e�i ˝ rei

‰ 7�!nX

iD1ei ˝

�rei

‰ C 1

nei � D‰

�;

hence,

PX‰ ´ rX‰ C 1

nX � D‰; X 2 �.TM/;‰ 2 �.†M/:


Note that by de�nition, the Penrose operator satis�es

nX

iD1ei � Pei

‰ D 0; ‰ 2 �.†M/;

and recall that, see (2.18),

jr‰j2 D jP‰j2 C 1

njD‰j2: (5.8)

It is worth noting that (5.8) can be easily established as follows. Since Cliffordmultiplication by vectors is skew-symmetric, one has

jP‰j2 DnX

iD1hPei

‰;Pei‰i

DnX

iD1

DPei

‰;rei‰ C 1

nei � D‰

E

DnX

iD1hPei

‰;rei i

DnX

iD1

Drei

‰ C 1

nei � D‰;rei

‰E

D jr‰j2 � 1

njD‰j2:

Combining (5.8) with (5.2), we obtainZ

M

jP‰j2�g D n � 1n

Z

M

jD‰j2�g � 1

4

Z

M

Scalj‰j2�g ; (5.9)

which implies (5.5). If equality holds in (5.5), i.e., if there exists a spinor �eld ‰such that D‰ D �0‰, with �20 D n

4.n�1/Scal0, then

rX‰ D ��0nX � ‰; X 2 �.TM/:

Remark 5.6. In fact the Dirac and the twistor operators are given by the composi-tion of the covariant derivative with the projections on the Spinn-irreducible com-ponents of �.T�M ˝†M/. This is a special case of a general approach developedin [SW68]. In [Feg87] it is shown that any such differential operator is conformallycovariant (see Section 2.3.5, where this fact is proved explicitly for the Dirac andtwistor operators). We refer to [Bra97] for further results in this direction, based ona systematic use of representation theory.

5.3. Special spinor �elds 133

5.3 Special spinor �elds

We now examine some necessary conditions for the existence of certain specialspinor �elds which arise naturally in the setup of eigenvalue estimates for the Diracoperator.

Definition 5.7. A non-trivial spinor ‰ 2 �.†M/ is called

(i) a twistor-spinor if P‰ D 0, i.e., for all X 2 �.TM/,

rX‰ C 1

nX � D‰ D 0I

(ii) a real Killing spinor if for some non-trivial real-valued function f , for allX 2 �.TM/,

rX‰ C f

nX �‰ D 0: (5.10)

Remark 5.8. In De�nition 5.7 of real Killing spinors the function f is supposed tobe real. In fact, A. Lichnerowicz showed that if the function f is complex-valued,then it should be either real or purely imaginary. In the second case, such spinorsare called imaginary Killing spinors. Manifolds with imaginary Killing spinorshave been classi�ed by H. Baum in the case where the function is constant andby H. B. Rademacher for a general function. We refer to the book [BFGK91] forsuch classi�cations. In the real case, the function f is always constant (cf. Propo-sition 5.11 below).

The notion of a twistor-spinor, as an element in the kernel of the twistor op-erator, was introduced by A. Lichnerowicz, who showed that if such a spinor hasno zero, then it can be conformally related to an imaginary Killing spinor. Sincethen, twistor-spinors and their zeros have been extensively studied by many peo-ple including Th. Friedrich, K. Habermann, and W. Kühnel and H.B. Rademacher;see [BFGK91].

For a spinor �eld ‰ 2 �.†M/ one can de�ne its associated vector �eld by

X‰ ´ i

nX

jD1h‰; ej �‰iej :

The following result explains the terminology of Killing spinors.


Lemma 5.9. For a real Killing spinor‰, the associated vector �eldX‰ is aKillingvector �eld, i.e., £X‰

g D 0, where £X‰denotes the Lie derivative in the direction

of X‰.

Proof. For arbitrary vector �elds X , Y , and Z one has

.£Xg/.Y; Z/ ´ Xg.Y;Z/� g.£XY;Z/� g.Y; £XZ/

D g.rXY;Z/C g.Y;rXZ/ � g.ŒX; Y �; Z/ � g.Y; ŒX;Z�/D g.rYX;Z/C g.Y;rZX/

since r is metric and torsion free. On the other hand, by the de�nition of the asso-ciated vector �eld, in normal coordinates at the point x 2 M it holds that

rYX‰ D i

nX

jD1ŒhrY‰; ej �‰iej C h‰; ej � rY‰i�ej

D �i fn

nX

jD1h‰; .ej � Y � Y � ej / �‰iej

since ‰ is a real Killing spinor. Now, for any vector �elds Y and Z, we get

.£X‰g/.Y; Z/

D �i fn.h‰; .Z � Y � Y � Z/ �‰i C h‰; .Y � Z � Z � Y / �‰i/

D 0: �

Another important property of Killing spinors is the following:

Lemma 5.10. The square length j‰j2 of a real Killing spinor‰ is constant. Hence,a non-trivial Killing spinor has no zeros.

Proof. For this, we prove that for any vector �eld X , the Lie derivative of j‰j2 inthe direction of X is zero. By (5.10), it follows

X j‰j2 D hrX‰;‰i C h‰;rX‰i

D �fn.hX �‰;‰i C h‰;X �‰i/

D 0: �

We now examine some necessary conditions for the existence of real Killingspinors.

5.3. Special spinor �elds 135

Proposition 5.11. Let ‰ 2 �.†M/ be a real Killing spinor on a Riemannian spinmanifold .M n; g/. Then the manifold is Einstein with positive scalar curvatureScal0 and the function f is constant, given by

f 2 D �20 D n

4.n� 1/Scal0:

If .M; g/ is complete, thenM compact.

Proof. Since ‰ is a Killing spinor, for all X; Y 2 �.TM/ we have

rY‰ D �fnY � ‰;

rXrY‰ D � 1nX.f /Y � ‰ � f

nY � rX‰ � f

n.rXY / �‰

D � 1nX.f /Y � ‰ C f 2

n2Y �X �‰ � f

n.rXY / �‰;

rYrX‰ D � 1nY.f /X �‰ C f 2

n2X � Y �‰ � f

n.rYX/ �‰;

and

RX;Y‰ D � 1n.X.f /Y � Y.f /X/ �‰ C f 2

n2.Y � X � X � Y / �‰:

On the other hand, using the Ricci identity (2.41) we get

�12Ric.X/ � ‰ D � 1

n

X

i

.X.f /ei � ei �‰ � ei .f /ei �X �‰/

C f 2

n2

X

i

.ei � ei �X � ei � X � ei / �‰:

Using the fact that ei � ei D �1 andPi ei �X � ei D .n� 2/X , we further have

�12Ric.X/ �‰ D � 1

n.�nX.f / �‰ � df �X �‰/

C f 2

n2.�nX � .n � 2/X/ �‰

D X.f /‰ C 1

ndf � X � ‰ � 2.n � 1/

n2f 2X �‰:

(5.11)


Note that for a k-form ˛ and any spinor ‰,

h˛ �‰;‰i D .�1/k.kC1/

2 h˛ � ‰;‰i:

Hence, for any vector X and any spinor ‰,

hX �‰;‰i D �hX �‰;‰i;

so the function hX � ‰;‰i is purely imaginary. For X ´ grad f , then it followsthat

D� 1

2Ric.X/ � ‰ C 2.n� 1/

n2f 2X � ‰;‰

ED X.f /j‰j2 C 1

nhdf �X �‰;‰i

D df .X/j‰j2 � 1

njdf j2 j‰j2

D�1 � 1

n

�jdf j2 j‰j2:

The left-hand side being purely imaginary, we get jdf j2 D 0, since by Lemma 5.10,j‰j2 6D 0. This yields

X.f /‰ D 1

ndf �X �‰ D 0; X 2 �.TM/;

which when combined with (5.11), implies that for all X 2 �.TM/

Ric.X/ D 4.n� 1/n2

f 2X:

Thus 4.n�1/n2 f 2 D Scal0

n, by the de�nition of the scalar curvature. The last statement

follows directly from Myers theorem. �

It should be mentioned that real Killing spinors �rst appeared in mathemati-cal physics in the context of supergravity and then in superstring theories. In ourcontext, they appear as eigenspinors associated with the minimal eigenvalues ofthe Dirac operator. The isometry groups of compact manifolds with Killing spinorshave special properties; see [Mor00]. The classi�cation of compact manifoldscarrying real Killing spinors has been obtained by C. Bär and will be discussedin detail in Chapter 8 (cf. Theorem 8.37).

5.4. The Hijazi inequality 137

5.4 The Hijazi inequality

In the remaining part of this section we give a qualitative improvement of The-orem 5.3 based on the conformal covariance of the Dirac operator; cf. [Hit74]and [Hij86b].

Theorem 5.12. On a compact Riemannian spin manifold .M n; g/ any eigenvalue� of the Dirac operator satis�es

�2 � n

4.n� 1/ supuinfM

�Scal e2u

�; (5.12)

where Scal is the scalar curvature of the metric Ng D e2ug.

Proof. We use the notations and the facts concerning conformal covariance of theDirac operator established in Section 2.3.5. For ametric Ng D e2ug, in the conformalclass of g, the associated Dirac operators are related by the identity

xD.e� .n�1/2 u x‰/ D e� .nC1/

2 uD‰:

Note that, by (2.19),

D‰ D �‰ H) xD x D �e�u x ; with ˆ ´ e� .n�1/2 u‰: (5.13)

With respect to the metric Ng D e2ug, apply (5.9) to the spinor �eld x satisfy-ing (5.13) to get

n

n � 1

Z

M

j xP x j2� Ng

DZ

M

j xD x j2� Ng � n

4.n � 1/

Z

M

Scal j x j2� Ng

DZ

M

��2 � n

4.n� 1/Scal e2u�e�2u j x j2� Ng

� 0: �

If equality holds in (5.12), then

xP x � 0; with xD x D �e�u x ;

that isxr xX x C �e�u

nxX N� x D 0; X 2 �.TM/;

which, for � 6D 0, by Proposition 5.11 implies that the function u is constant, hencethe spinor �eld ‰ D e

n�12 uˆ is a real Killing spinor.


Definition 5.13 (Yamabe operator). On a Riemannian manifold .M n; g/ ofdimension n � 3, the Yamabe operator (also called conformal Laplacian) Lg isthe second-order operator de�ned (see [Bes87], for example) as

Lg D 4n� 1

n� 2�C Scal:

This operator, like the Dirac operator, is conformally covariant in the followingsense: for g and a conformally related metric Ng D e2ug, the associated Yamabeoperators Lg and L Ng are related by

L Ng D e� nC22 uLge

n�22 u: (5.14)

The conformal Laplacian is related to the Yamabe problem: on a compact Rie-mannian manifold .M n; g/, is there a metric conformal to g whose scalar curvatureis constant? Indeed, Scal D Lg .1/ (here 1 is the constant function), hence for in-stance g is conformally scalar-�at if we can solve the equation Lgv D 0, for somepositive smooth function v.

We now show that the Yamabe operator is intimately related to the Dirac oper-ator.

Corollary 5.14. On a compact Riemannian spin manifold .M n; g/ with positivescalar curvature, any eigenvalue � of the Dirac operator satis�es the followingproperties:

(a) for n � 3 (see [Hij86b]),

�2 � n

4.n� 1/�1.Mn; g/; (5.15)

where �1.M n; g/ stands for the lowest eigenvalue of the scalar conformalLaplacian Lg ;

(b) for n D 2 (see [Bär92b]),

�2 � 2��.M 2/

Area.M 2; g/; (5.16)

where �.M 2/ is the Euler characteristic.

In case of equality in (a) or (b), the eigenspinors associated with � are real Killingspinors.

Proof. (a) It is clear that �1.Mn; g/ � infM Scal, and equality is achieved if and

only if the scalar curvature is constant. Let uWM ! R be an arbitrary function.The scalar curvature transforms according to the formula (cf. [Bes87])

Scal e2u D Scal C 2.n� 1/�u � .n � 1/.n� 2/jduj2: (5.17)

5.4. The Hijazi inequality 139

For n � 3, de�ne the positive function h by

h4

n�2 D e2u: (5.18)

Then (5.17) simpli�es to

Scalh4

n�2 D h�1Lgh: (5.19)

By the maximum principle, one knows that an eigenfunction h1 associated with thelowest eigenvalue �1 of Lg can be chosen to be positive. Moreover, it is known that�1 is positive if and only if there exists a metric of positive scalar curvature in theconformal class of g. Thus h1 > 0 and

h�11 Lgh1 D �1 > 0:

Now, take u1 associated with h1 by (5.18). From (5.19) we get

Scal e2u1 D Scalh4

n�2

1 D h�11 Lgh1 D �1:

It can be shown that if Scal e2u1 is constant, then it equals supu infM .Scal e2u/;

see [Hij91]. Estimate (5.15) thus follows from (5.12).

(b) For n D 2, the transformation formula (5.17) of the scalar curvature underconformal change of the metric becomes

Scal e2u D Scal C 2�u:

Now for any function u we can write

infM .Scal e2u/ � 1

Area.M 2; g/

Z

M

Scal e2u�g

D 1

Area.M 2; g/

Z

M

.Scal C 2�u/�g

� 1

Area.M 2; g/

Z

M

Scal �g

D 4��.M 2/

Area.M 2; g/:

by Stokes and the Gauß–Bonnet formula. Let u1 be a smooth function satisfying

2�u1 D 1

Area.M 2; g/

Z

M

Scal �g � Scal

(the function on the right-hand side has zero integral overM so belongs to the imageof the Laplacian by Corollary 4.39). Then we �nally get

Scal e2u1 D Scal C 2�u1 D 1

Area.M 2; g/

Z

M

Scal �g D 4��.M 2/

Area.M 2; g/;

and we conclude as before by (5.12). �


Remark 5.15. For �1 > 0, with the help of the conformal covariance of the Diracoperator and of the Sobolev embedding theorem for pseudo-differential operators,J. Lott [Lot86] established the existence of a qualitative conformal lower bound forthe eigenvalues of the Dirac operator.

A second corollary of Theorem 5.12 could be obtained by comparing�1.M n; g/

to the Yamabe number, denoted by �.M n; Œg�/. This is the scalar conformal invari-ant de�ned by

�.M n; Œg�/n�2

infNg2Œg�

Z

M

Scal � Ng

.Vol.M; Ng//.n�2/=n

n�3infh>0

Z

M

hLg.h/�g

khk22n=.n�2/

:

By a direct application of the Hölder inequality, we get the following theorem.

Theorem 5.16. On a compact Riemannian spin manifold .M n; g/ with positivescalar curvature and dimension n � 2, any eigenvalue � of the Dirac operatorsatis�es

�2 .Vol.M n; g//2n � n

4.n� 1/�.Mn; Œg�/:

In the case n D 2, by the Gauß–Bonnet formula, one gets � D 4��.M/, where�.M/ is the Euler–Poincaré characteristic number. We point out that Bär’s proofof (5.16), see [Bär92b], uses modi�ed connections which correspond to themodi�edconnections xrf (see Proposition 5.11) associated with a conformal class of metrics,on which the original proof of (5.16) is based; see [Hij86b] and [Hij91].

We also mention that inequality (5.15) is also true for the second-order differen-tial operators introduced in [HL88], and there is a similar inequality for the twistoroperator [Lic89]. Moreover, we point out that extensions of (5.15) to other �rst orderdifferential operators and vanishing results were obtained in [BH97] and [BH00].

5.5 The action of harmonic forms on Killing spinors

Wewill now show that the topology and the geometry of manifolds with real Killingspinors are quite special. The �rst step was made in [Hij86b], and then extendedby A. Lichnerowicz [Lic87]. Moreover, we point out that extensions of (5.15) toother �rst order differential operators and vanishing results were obtained in [BH97]and [BH00].

5.5. The action of harmonic forms on Killing spinors 141

Theorem 5.17. (a) If ‰ is a real Killing spinor on a compact n-dimensionalRiemannian spin manifold M and ˛ 2 �k.M/ an arbitrary harmonic form ofdegree k ¤ 0, n, then

˛ �‰ D 0: (5.20)

(b) IfM admits a non-trivial Killing spinor, then there exists no non-trivial parallelform ˛ 2 �k.M/ for k … f0; ng.Proof. (a) TheKilling spinor‰ is an eigenspinor associated with the smallest eigen-value �0 of the Dirac operator D. Using now the formulas

d DnX

iD1ei ^ rei

; and ı D �nX

iD1ei ³ rei

;

and the compatibility of the covariant derivative with Clifford multiplication to-gether with the identity

nX

iD1ei � ˛ � ei D .�1/k�1.n � 2k/˛;

for any ˛ 2 �k.M/, we get

D.˛ �‰/ DnX

iD1ei � rei

.˛ �‰/

DnX

iD1ei � ..rei

˛/ �‰ C ˛ � rei‰/

D ..d C ı/˛/ �‰ CnX

iD1ei � ˛ �

�� 0

nei � ‰

�

D ..d C ı/˛„ ƒ‚ …

D0

/ �‰ C .�1/k�0�1� 2k

n

�

„ ƒ‚ …´�

˛ �‰„ƒ‚…

´'

;

since on compact manifolds �˛ D 0 ” d˛ D ı˛ D 0. Now j�j < j�0jimplies by Friedrich’s inequality (5.5)

˛ �‰ D 0:

(b) Let ˛ be a parallel k-form. Then by (5.20), ˛ � ‰ D 0. Now for any vector�eld X ,

0 D rX .˛ �‰/ D . rX˛„ƒ‚…D0

/ � ‰ C ˛ � rX‰:

Since ‰ is a Killing spinor, for all X 2 �.TM/, one has that

˛ � X � ‰ D 0:


On the other hand, by Proposition 1.13, that

˛ �X D .�1/k.X � ˛ C 2X ³ ˛/;

and so for any vector �eld X

.X ³ ˛/ � ‰ D 0;

which by induction on k, shows that ‰ D 0. �

This theorem has immediate topological consequences. For example, if thereis a non-trivial real Killing spinor on a compact manifold, then the �rst de Rhamcohomology group is zero. Since the Kähler form �.X; Y / ´ g.JX; Y / of aKähler manifold is parallel, compact Kähler manifolds of real dimension n � 4

carry no real Killing spinors.

5.6 Other estimates of the Dirac spectrum

Theorem 5.17 together with Remark 5.4 show that Friedrich’s inequality (5.5) can-not be sharp on manifolds with geometric structures which support parallel forms.Sharp improvements of (5.5) on Kähler and quaternionic Kähler manifolds havebeen obtained by Kirchberg [Kir86] and [Kir90], Hijazi [Hij94], by Kramer, Sem-melmann, and Weingart [HM95b], [KSW99], [KSW98a], and [KSW98b], and byHijazi and Milhorat [HM97]. They will be treated in detail in Chapters 6 and 7.

Another recent improvement of Friedrich inequality in this direction wasobtained by Alexandrov, Grantcharov, and Ivanov [AGI98]. They proved that theexistence of a parallel 1-form on a compact Riemannian spin manifold .M n; g/,n � 3, implies that every eigenvalue � of the Dirac operator satis�es the inequality

�2 � n � 14.n� 2/

Scal0: (5.21)

The universal covering space of the manifolds appearing in the limiting case is alsodescribed.

5.6.1 Moroianu–Ornea’s estimate

In this section we discuss the generalization of the above result obtained by A. Mo-roianu and L. Ornea [MO04], who showed that (5.21) can be derived from the exis-tence of a harmonic 1-form of constant length. Note that until now, this is the uniqueimprovement of Friedrich inequality which does not assume a holonomy reduction.

5.6. Other estimates of the Dirac spectrum 143

Theorem 5.18 ([MO04]). Inequality (5.21) holds on any compact spin manifold

.M n; g/; n � 3;

admitting a non-trivial harmonic 1-form � of unit length. The limiting case is ob-tained if and only if � is parallel and the eigenspinor ‰ corresponding to the small-est eigenvalue of the Dirac operator satis�es

rX‰ C �

n � 1 .X �‰ � hX; �i� � ‰/ D 0:

The condition that the norm of the 1-form being constant is essential, in thesense that the topological constraint alone – the existence of a non-trivial harmonic1-form – does not allow any improvement of Friedrich’s inequality.

Indeed, Bär and Dahl [BD04] have constructed on any compact spin manifoldM n and for every positive real number ", a metric g" onM with the property thatScalg" � n.n � 1/ and such that the �rst eigenvalue of the Dirac operator satis-

�es �21.D"/ � n2

4C ". This construction clearly shows that no improvement of

Friedrich’s inequality can be obtained under purely topological restrictions.

Proof of Theorem 5.18. Let � be a 1-form of unit length and let ‰ be an arbitraryspinor �eld onM . Consider the “twistor-like” operator

T W�.TM ˝†M/ �! �.†M/;

de�ned as follows:

TX‰ D rX‰ C 1

n � 1X � D‰ � 1

n� 1hX; �i� � D‰ � hX; �ir�‰:

A simple calculation yields

jT‰j2 D jr‰j2 � 1

n � 1jD‰j2 � jr�‰j2 C 2

n � 1hD‰; � � r�‰i: (5.22)

From now on we will suppose that � is harmonic,M is compact with positive scalarcurvature Scal, �g is the volume element, and ‰ is an eigenspinor of the Diracoperator D of M corresponding to the least eigenvalue (in absolute value), say �.We let feig, i D 1; : : : ; n denote a local orthonormal frame onM .


The harmonicity of � implies the useful relation

D.� �‰/ D �� D‰ � 2r�‰: (5.23)

Indeed,

D.� �‰/ DnX

iD1ei � rei

.� �‰/

DnX

iD1ei � .rei

�/ �‰ C ei � � � rei‰

D .d� C ı�/ �‰ C ei � � � rei‰

D �� nX

iD1ei � rei

‰ � 2hei ; �irei‰

D �� D‰ � 2r�‰:

Taking the square norm in (5.23) yields

jD.� �‰/j2 D j� � D‰j2 C 4jr�‰j2 � 4hD‰; � � r�‰i: (5.24)

By integration of (5.22) overM , using (5.24) to express the last term in the right-hand side of (5.22), and the Schrödinger–Lichnerowicz formula, we get

Z

M

jT‰j2�g DZ

M

°n� 2

n� 1jD‰j2 � 1

4Scal j‰j2 � n � 3

n � 1 jr�‰j2

� 1

2.n � 1/.jD.� �‰/j2 � j� � D‰j2/±�g :

(5.25)

The term in the last bracket of the integrand is clearly positive since, due to thechoice of � to be minimal, we have from the classical Rayleigh inequality that

�2 �

Z

M

jDˆj2�gZ

M

jˆj2�g

5.6. Other estimates of the Dirac spectrum 145

for every ˆ. In particular, for ˆ D � �‰ this reads

Z

M

jD.� � ‰/j2�g � �2Z

M

j� �‰j2�g

D �2Z

M

j‰j2�g

DZ

M

jD‰j2�g

DZ

M

j� � D‰j2�g :

Thus (5.25) gives

Z

M

�n � 2n � 1�

2 � 1

4Scal

�j‰j2�g

DZ

M

hjT‰j2 C n � 3

n � 1 jr�‰j2i�g C 1

2.n� 1/

Z

M

ŒjD.� �‰/j2 � j� � D‰j2��g

� 0;

which immediately implies the �rst statement of Theorem 5.18.

The limiting case. Suppose now that equality is reached in (5.21) for the eigen-value � with corresponding eigenspinor ‰. Then T‰ D 0. Contracting with ei :

nX

iD1ei �rei

‰C �

n � 1

nX

iD1ei �ei �‰� �

n � 1

nX

iD1ei �hei ; �i� �‰�

nX

iD1ei �hei ; �ir�‰ D 0;

gives � � r�‰ D 0, so r�‰ D 0 (for n > 3 this follows directly from the vanishingof the integral

RM

n�3n�1 jr�‰j2�g ). Thus ‰ satis�es the Killing type equation

rX‰ D aX � ‰ � ahX; �i� �‰; a D � �

n � 1:

In order to show that � is parallel, we �rst compute the spin curvature operatorRY;X D ŒrY ;rX �� rŒY;X� acting on ‰. We have successively

1

arYrX‰ D rYX �‰ C aX � .Y � hY; �i�/ �‰ � hrYX; �i� �‰

� hX; �irY � �‰ � hX;rY �i� �‰ � ahX; �i� � .Y � hY; �i�/ �‰


and

1

aRY;X‰ D a.X � Y � Y �X/ �‰ � a.hY; �iX � � � hX; �iY � �/ �‰

C .hY;rX�i � hX;rY �i/� �‰ C .hY; �irX� � hX; �irY �/ �‰C a.hY; �i� � X � hX; �i� � Y / � ‰

D a.X � Y � Y �X/ �‰ C 2a.hY; �i� �X � hX; �i� � Y / � ‰C .hY;rX�i � hX;rY i/� �‰ C .hY; �irX� � hX; �irY �/ � ‰:

Using again the harmonicity of � , we easily check that

1

2aRic.X/ � ‰ D 1

a

nX

iD1ei � Rei ;X‰

D 2.n� 2/a.X � hX; �i�/ �‰ C 2� � rX� �‰ � 2hX;r��i‰:(5.26)

But r�� D 0, since � is closed and has unit length. Indeed, for any vector �eld Y ,

hY;r��i D d�.�; Y /C h�;rY �i D 0:

Hence, taking X D � in (5.26), we obtain Ric.�/ D 0. Then, as � is harmonic,the Bochner formula ensures that � is parallel. This completes the proof of Theo-rem 5.18. �

Since the 1-form � has to be parallel in the limiting case, we can apply The-orem 3.1 in [AGI98] to determine the universal cover of M . In fact, as provedin [AGI98], this is isometric to a Riemannian product R � N , where N is a spinmanifold carrying a real Killing spinor, hence can be described by Bär’s classi�ca-tion [Bär93] (see Theorem 8.37 below). Finally,M turns out to be a mapping torusof an isometry of N=� (a �nite quotient of N ).

5.7 Further developments

Recall that for a closed (compact without boundary) Riemannian spin manifold.M n; g/, the Schrödinger–Lichnerowicz formula yields by integration overM

Z

M

jD‰j2�g D 1

4

Z

M

Scalj‰j2�g CZ

M

jr‰j2�g : (5.27)

As we have seen above, the �rst applications of this fundamental integral formulaare the Lichnerowicz vanishing theorem and the optimal eigenvalues lower boundsfor the Dirac operator. In this section we give an overview of further applications.For the proofs, we refer to the original papers or to N. Ginoux [Gin09] for a completesurvey of such applications with a brief presentation of the ideas of the proofs.

5.7. Further developments 147

5.7.1 The energy–momentum tensor

Recall that the lower bounds in question are based on the observations that

jr‰j2 � 1

njD‰j2; ‰ 2 �.†M/; (5.28)

together with an argument based on the conformal covariance of the Dirac oper-ator. As we have pointed out, inequality (5.28) is the spinorial Cauchy–Schwarzinequality.

Now we use the fact that, for any u; v 2 Rn, with u a unit vector, one has that

jvj � jhv; uij. Hence, for any spinor �eld‰ and for any unit vector �eldX , onZ‰ ,the complement set of zeros of ‰, one has

jrX‰jj‰j D jX � rX‰j

j‰j

�ˇˇDX � rX‰

j‰j ;‰

j‰jEˇˇ

� <DX � rX‰

j‰j ;‰

j‰jE;

whencejr‰j2 � jQ‰j2j‰j2; (5.29)

where the energy–momentum tensorQ‰ is the symmetric 2-tensor associated withthe quadratic form de�ned on Z‰ by

Q‰.X/ D <DX � rX‰;

‰

j‰j2E:

Assume now that ‰ is an eigenspinor of D corresponding to the eigenvalue �. Viathe Schrödinger–Lichnerowicz formula, inequality (5.29) translates to (see [Hij95])

�2 � infZ‰

�14Scal C jQ‰j2

�: (5.30)

Note that the set of zeros of ‰ is contained in an .n� 2/-dimensional submanifold(see [Bär99]) and, by the Cauchy–Schwarz inequality,

jQ‰j2 � .TrQ‰/2

nD �2

n;

inequality (5.30) implies (5.5). N. Ginoux and G. Habib showed (see [GH10]Ex. 6.4) that Heisenberg manifolds satisfy equality in (5.30), but not in (5.5).


We also point out that it has been observed by G. Habib [Hab07] that inequal-ity (5.30) is still true for any 2-tensor (see also [GH08], [HR09], and [GH10] forfurther results on foliated spin manifolds).

It is also straightforward to study the conformal behavior of the energy–momen-tum tensor in order to get the inequality (see [Hij95])

�2 � 1

4�1 C infZ‰

.jQ‰j2/; (5.31)

which by Cauchy–Schwarz implies inequality (5.15). Finally, it is interesting tomention that the energy-momentum tensor associated with a spinor �eld appearsnaturally in the study of the variation of the eigenvalues of the Dirac operator undermetric variation (see Bourguignon and Gauduchon [BG92]).

5.7.2 Witten’s proof of the positive mass theorem and applications

A classical survey on the Yamabe Problem and on Witten’s spinorial proof [Wit81]of the positive mass theorem is the paper byLee and Parker [LP87] (see also [PT82]).The theorem reads:

Let .M n; g/ be an asymptotically �at spin manifold of dimension n � 3 (whosemass m.g/ is well de�ned), with non-negative scalar curvature, then m.g/ � 0.Moreover,m.g/ D 0 if and only if themanifold .M n; g/ is isometric to the Euclideanspace.

We shall recall the de�nition of the mass in the next section. Roughly speaking,Witten’s idea is to observe that the mass of an asymptotically �at manifold can beseen as the limit at in�nity of a boundary term in (5.27), when applied to compactmanifolds with boundary and for a suitable choice of an asymptotically constantspinor �eld.

The conclusion is based on the fact that one can solve the problem of existenceof a harmonic spinor �eld ‰ (i.e., D‰ D 0) which is asymptotically constant.The equality case is then easily deduced from the fact that such spinor �elds areparallel, hence the manifold has a maximum number of parallel spinor �elds, i.e.,.M n; g/ D .Rn; h�; �i/.

Since we have not developed in this book the necessary analytic tools on asymp-totically Euclidean spaces, we will present at the end of this section the proof ofAmmann and Humbert, which has the advantage of using only elliptic theory oncompact manifolds.

We now present some other applications of Witten’s argument which rely on acareful study of the boundary term and adapted boundary value problems. To de�ne


the Dirac–Witten operator and the hypersurface Dirac operator, we consider restric-tions of spinor �elds to oriented hypersurfaces and the Gauß spinorial formula (seeLemma 2.38 and Theorem 2.39, in the special case of spin manifolds).

Let M be an .n C 1/-dimensional Riemannian spin manifold with non-emptyboundary @M . Denote by h�; �i its scalar product and by r its corresponding Levi-Civita connection on the tangent bundle TM . We �x a spin structure (and so acorresponding orientation) on the manifoldM and we denote by†M the associated

spinor bundle, which is a complex vector bundle of rank 2ŒnC1

2 �, and by

W Cl.M/ �! EndC.†M/

the Clifford multiplication, de�ned on the Clifford bundle Cl.M/. As we haveseen in Chapter 2, there exist a natural Hermitian metric and a compatible spino-rial Levi-Civita connection on the spinor bundle †M , also denoted by h�; �i and r.The (classical) Dirac operator D onM is given locally by

D DnC1X

iD1 .ei /rei

;

where fe1; : : : ; enC1g is a local orthonormal frame of TM .The boundary hypersurface @M is also an oriented Riemannian manifold, with

the orientation and the metric induced from the ambient space. Since its normalbundle is trivial, the Riemannian manifold @M is also a spin manifold and so wehave a corresponding spinor bundle†@M , a Clifford multiplication @M , a spinorialLevi-Civita connection r@M , and an intrinsic Dirac operatorD@M . It is not dif�cultto check (see for example Lemma 2.38 or [Bur93], [Tra93], [BFGK91], [Bär98],[HMZ01a], [HMZ01b], and [Mon05]) that the restriction of the spinor bundle ofM to its boundary is related to the intrinsic Hermitian spinor bundle †@M by

S ´ †M j@M Š

8<:†@M if n is even,

†@M ˚†@M if n is odd:

For any spinor �eld‰ 2 �.S/ on the boundary hypersurface @M and for any vector�eld X 2 �.T@M/, de�ne on the restricted bundle S the Clifford multiplication S

and the connection rS by (see Theorem 2.39)

S.X/‰ ´ .X/ .N/‰

and

rSX‰ ´ rX‰ � 1

2 S.AX/‰ D rX‰ � 1

2 .AX/ .N/‰;


where N denotes the outward unit normal vector �eld on @M and A ´ �II is theWeingarten tensor. Then, S and rS satisfy the same compatibility relations as rand , together with the additional relation

rSX . .N /‰/ D .N/rS

X‰:

Taking into account the relation between the Hermitian bundles S and †@M , onecan see that

rS Š

8<:

r@M if n is even,

r@M ˚ r@M if n is odd

and

S Š

8<: @M if n is even,

@M ˚ � @M if n is odd.

On the space of smooth sections ‰ 2 �.S/, we have a Dirac operator DS

associated with the connection rS and the Clifford multiplication S, locally givenby

DS‰ DnX

jD1 S.ej /rS

ej‰ D n

2H‰ � .N/

nX

jD1 .ej /rej‰;

where fe1; : : : ; eng is a local orthonormal frame tangent to the boundary @M andH D tr.A/=n is its mean curvature function. In this formula, the last term

nX

jD1 .ej /rej

is called Dirac–Witten operator and DS the hypersurface Dirac operator.In the particular case where the �eld‰ 2 �.S/ is the restriction of a spinor �eld

‰ 2 �.†M/ onM , one gets

DS‰ � n

2H‰ D � .N/D‰ � rN‰:

With the above identi�cations, the hypersurface Dirac operator is related to theintrinsic Dirac operator D@M of the boundary by

DS Š

8<:D@M if n is even,

D@M ˚ �D@M if n is odd.


Note that we always have the anticommutativity property

DS .N/ D � .N/DS

and so, when @M is compact, the spectrum of DS is symmetric with respect tozero and coincides with the spectrum of D@M if n is even, and with Spec.D@M / [�Spec.D@M / if n is odd.

With this, we are now able to give the formula corresponding to (5.27) in the caseof a compact spin manifold M with boundary. In fact, the Green formula appliedto (5.27) is precisely

Z

M

�jr‰j2 C 1

4Scalj‰j2 � jD‰j2

��g D �

Z

@M

h .N/D‰ C rN‰;‰i�g

DZ

@M

hDS‰ � n

2H‰;‰i�g :

In terms of the Penrose operator (see (2.18)), one has thatZ

M

�jP‰j2 C 1

4Scalj‰j2 � n � 1

njD‰j2

��g

DZ

@M

DDS‰ � n

2H‰;‰

E�g :

(5.32)

5.7.2.1 The Dirac–Witten operator. As a �rst application, consider the case of aspace-like spin hypersurface .M n; g/ on a Lorentzian manifold zM , whose metric gsatis�es the Einstein �eld equations and whose energy–momentum tensor satis�esthe dominant energy condition. Then the square of the eigenvalues of the Dirac–Witten operator satisfy lower bounds of type (5.5), (5.30), and (5.31) (see [HZ03]).We also refer to [Mae08] for the case of bounded domains with smooth boundaryand to [CHZ12] for the case of higher codimensions.

5.7.2.2 Compact spin manifolds with boundary. In what follows, we will seethat the fundamental formula (5.32) can be exploited in two directions: solve ap-propriate boundary value problems for the Dirac operator in order to control thesign of the left-hand side or the right-hand side of (5.32).

The spectrum of the classical Dirac operator on compact Riemannian spin man-ifolds with non-empty boundary can be studied under four different boundary con-ditions (see [HMR02]):

(1) the global Atiyah–Patodi–Singer (APS) condition associated with the spectralresolution of the intrinsic Dirac operator on the boundary hypersurface;

(2) the local condition associated with a chirality (CHI) operator on the manifold(for example, if its dimension is even or if it is a space-like hypersurface in aLorentzian manifold);


(3) the Riemannian version of the so-called (local) MIT bag condition; and

(4) a global boundary condition obtained by a suitable modi�cation of the APScondition (mAPS).

These four conditions satisfy ellipticity criteria and three of them, APS, CHI,and mAPS, make D a self-adjoint operator and so the corresponding spectra arereal sequences tending to C1 and �1. Under the MIT condition, the spectrum ofthe Dirac operator is an unbounded discrete set of complex numbers with positiveimaginary part. Finally one has, under the four boundary conditions, the same lowerbound (5.5) in terms of the in�mum of the scalar curvature as in the closed case,provided that the mean curvature of the boundary hypersurface is non-negative (notethat under the APS and the CHI conditions a different proof of this fact is givenin [HMZ01a] and [HMZ01b].)

It is interesting to note that the four conditions behave differently with respectto the equality case. In fact, such an equality is never achieved for the APS andMITconditions, it occurs for the CHI boundary condition if and only if the manifold is ahalf-sphere, and it is achieved for the mAPS condition if and only if the manifold hasa non-trivial real Killing spinor �eld and the boundary is a minimal hypersurface.For example, all domains enclosed in a sphere by embedded minimal hypersurfaceshave the same �rst eigenvalue for the Dirac operator under the mAPS condition.

Here, we also mention that under the MIT bag boundary condition, a lowerbound for the eigenvalues of the Dirac operator on a compact domain of a Rie-mannian spin manifold is established in [Rau05], where the limiting case is char-acterized by the existence of an imaginary Killing spinor. Moreover, in [Rau06]the Hijazi-type inequality on compact Riemannian spin manifolds is proved undertwo boundary conditions: the condition associated with a chirality operator, and theRiemannian version of the MIT bag condition. The limiting case is characterized asbeing a half-sphere for the �rst condition, whereas the equality cannot be achievedfor the second.

Finally, we point out that regularity for a class of boundary value problems for�rst-order elliptic systems, with boundary conditions determined by spectral de-compositions, under weaker differentiability conditions on the coef�cients is provedin [BC05]. Fredholm properties for Dirac-type equations with these boundary con-ditions are also established. Moreover, boundary value problems for linear �rst-order elliptic differential operators of order one are examined in [BB12], wherethe underlying manifold may be noncompact, but the boundary is assumed to becompact. A symmetry property, satis�ed by Dirac type operators, of the principalsymbol of the operator along the boundary is required.


5.7.2.3 The hypersurface Dirac operator. With the help of equation (5.32), onecan obtain lower bounds for the eigenvalues of the hypersurface Dirac operator interms of the Yamabe number, the energy–momentum tensor, and the mean curva-ture; see [Zha98], [Zha99], and [HZ01a]. In the limiting case, the hypersurface isan Einstein manifold with constant mean curvature.

Using the identi�cations given in [Bär98] for the restriction of the spinor bun-dle to submanifolds, generalizations of [HZ01a] were obtained (see [HZ01b]) forthe submanifold Dirac operator. In particular, one has optimal lower bounds forthe submanifold Dirac operator in terms of the mean curvature and other geometricinvariants, such as the Yamabe number or the energy–momentum tensor. In the lim-iting case, the submanifold is Einstein if the normal bundle is �at (see also [GM02]for further extensions).

Now, let† D @M be the boundary of a compact .nC1/-dimensional Riemanni-an spin manifoldM with non-negative scalar curvature. If the mean curvatureH of† is also non-negative, then the lowest non-negative eigenvalue �1 of the intrinsichypersurface Dirac operator DS satis�es (see [HMZ01a])

�1 � n

2H0; (5.33)

whereH0 D inf†H . This estimate improves previous ones. In particular, it is valideven if the scalar curvature of the boundary is negative. In fact, if the Einstein tensorof the manifold M is non-negative, then by the Gauß formula and the Cauchy–Schwarz inequality one has

n

2H0 � n

4.n � 1/R†0 ;

where R†0 is the in�mum of the scalar curvature of †. Note that the inequality canbe strict (for example, take † to be a revolution torus in R

3). It then became clearthat one can get subtle information on a spin manifold via extrinsic invariants.

A spinorial simple proof for the classical Alexandrov theorem can be deducedfrom the equality case in (5.33), where corresponding spinor �elds are the restric-tions to the boundary of parallel spinor �elds; see [HMZ01a]. Another applicationof the equality case is to prove that minimal compact hypersurfaces bounding acompact domain admitting a parallel spinor are totally geodesic.

Inequality (5.33) can be improved by using, as in the intrinsic setup, the confor-mal covariance of the hypersurface Dirac operator. We need three key ingredients.First, consider the Yamabe problem for manifolds with boundary, cf. [Esc92]: if Bdenotes the linear scalar operator which relates the mean curvatures of two metricsin the same conformal class on the boundary †, then the eigenvalue problem


´Lu D 0 on �;

Bu � �u D 0 on † D @�

has smooth solutions. Moreover, if the ambient scalar curvature is non-negative, the�rst eigenvalue �1.B/ ofB is a �nite number (whose sign is conformally invariant).In addition, it is shown that �1.B/ is positive if and only if there exists on� a metricin the conformal class with zero scalar curvature and positive mean curvature.

The second key ingredient is to �nd appropriate conformal local elliptic bound-ary conditions. Finally, a suitable choice of the metric in a conformal class anda corresponding spinor �eld in the Schrödinger–Lichnerowicz type formula (5.32)yield the following result.

Theorem 5.19 ([HMZ02]). Let † be a compact hypersurface of dimension n � 2

of a Riemannian spin manifoldM bounding a compact domain�. The lowest non-negative eigenvalue �1 of the intrinsic Dirac operator on † satis�es

�1 � n

2�1.B/;

where �1.B/ is the �rst eigenvalue of the boundary linear operator B on †, actingon functions f de�ned on � with Lf D 0, where L is the conformal Laplacian ofM . Moreover, if equality holds, then � is conformally equivalent to a Riemannianspin manifold with nontrivial parallel spinors and the eigenspace corresponding to�1 is isomorphic to the space of restrictions to † of these parallel spinors.

Using the Hölder inequality, one can establish a conformal extrinsic lower boundfor the non-negative eigenvalues of the intrinsic Dirac operator on the boundary †.

While in the original scalar Reilly inequality, the Ricci curvature of the ambientmanifold is assumed to be non-negative, in the spinorial set-up it is suf�cient toassume the non-negativity of the ambient scalar curvature. If the ambient scalarcurvature is bounded from below by a negative constant (normalized to be the scalarcurvature of a unit hyperbolic space) we have the following result.

Theorem 5.20 ([HMR03]). Let† be the boundary of an .nC1/-dimensional com-pact domain in a Riemannian spin manifold whose scalar curvature is bounded frombelow by �n.nC 1/. Assume that the inner mean curvature H � 1. Let �1 be the�rst eigenvalue of the intrinsic Dirac operator DS. Then

j�1j � n

2inf†

pH 2 � 1:

The limiting case is then characterized by the existence on the domain of imagi-naryKilling spinors. As a consequence, a spinorial proof of the (hyperbolic) Alexan-drov theorem could be obtained; see [Mon99].


5.7.2.4 Ammann–Humbert version of Witten’s positive mass theorem. A niceadaptation of Witten’s spinorial proof was provided by Ammann and Humbertin [AH05]. Consider a compact, locally conformally �at Riemannian manifold.M; g/ of positive Yamabe invariant (i.e., there exist metrics in the conformal classof g having positive scalar curvature).

Recall the de�nition of the Yamabe operator:

Lg D 4n� 1

n� 2�C Scal:

If g has positive scalar curvature, Lg is strictly positive, hence invertible. Thanksto the conformal covariance of Lg (see (5.14)), for every other metric conformal tog the corresponding conformal Laplacian is invertible. By the results of Chapter 4,there exists a bounded self-adjoint operator G on L2.M/, pseudo-differential oforder �2, which is a right and left inverse of Lg . As such, there exists a smoothfunction �G , the Schwartz kernel of G, de�ned on the complement of the diagonalinM �M , which is annihilated by Lg . This function has an asymptotic expansionnear the diagonal, and the mass at a point p 2 M is de�ned as the constant term inthis expansion. We give below a more elementary description of this invariant.

Fix p 2 M and choose a metric g in the given conformal class which is �at ina neighborhood U of p. In polar coordinates with respect to this �at metric near p,assuming n D dim.M/ � 3, the function r2�n is harmonic on U . Choose a cut-offfunction � with support on U and which equals 1 near p. Then

f ´ r2�n�.r/ 2 C1.M n fpg/

satis�es Lgf D 0 near p, because Scal D 0 on U . It follows that Lgf is asmooth function on M . Since Lg is invertible in C1.M/ (by Theorem 4.32 andLemma 4.24), we can solve the equation Lgu D Lgf for a unique u 2 C1.M/.Let � D �p 2 C1.M n fpg/ be the difference

� ´ f � u:

This function is annihilated by Lg (outside p, where it is not de�ned). Near p itbehaves like

� D r2�n Cm.p/CO.r/ D r2�n.1Cm.p/rn�2 CO.rn�1//:

Up to a positive multiplicative constant, the mass at p is by de�nition the constanttermm.p/. Since only the sign ofm.p/ is of interest to us, we shall ignore normal-ization constants.


Theorem 5.21 ([AH05]). Let .M; g/ be a compact connected locally conformally�at spin manifold of positive Yamabe invariant, of dimension n � 3, such that g is�at near p 2 M . Then the mass m.p/ is non-negative, and is equal to zero if andonly ifM is conformal to the standard sphere.

Proof. As in the construction of the solution f for Lg , we can build a harmonicspinor onM n fpg as follows. Choose a Euclidean coordinate system near p, andlet 0 be a constant (i.e., parallel) spinor onU in these coordinates, of norm 1. Thenby direct computation,

.x/ ´ r�nx � 0is a harmonic spinor on U n fpg, i.e., it is annihilated by the Dirac operator D.It follows that D.� / is a smooth spinor on M (its support does not contain thesingular point p for ). For any metric g0 of positive scalar curvature in the con-formal class of g, the Dirac operator is positive by the Schrödinger–Lichnerowiczformula (5.1), hence it is invertible on the space of smooth spinors (Theorem 4.32and Lemma 4.24). Using the conformal covariance of D with respect to confor-mal changes of metric, we deduce that one can solve uniquely the equation D Q DD.� / for a smooth spinor Q onM . It follows that

� ´ � � Q

is a harmonic spinor onM n fpg, of the form

� D r�nx � 0 CO.1/

near p, where the error term is in fact a smooth spinor.The function � lives in the null-space of Lg . This suggests a conformal factor

(singular at p) such that the resulting metric is scalar-�at:

Ng ´ �4

n�2g; Scal Ng D L Ng.1/ D �� 2.nC2/n�2 Lg.�/ D 0:

By conformal covariance, the spinor

ˆ ´ �� 2.n�1/n�2 �

is harmonic (in the null space of the Dirac operator with respect to Ng). We nowgather the asymptotic behavior of the various functions, metrics, and spinors nearp in polar coordinates with respect to g:

g D dr2 C r2d�2I

�g D rn�1dr ^ �� I

� D r�n.x � 0 CO.rn//I


� D r2�n.1Cm.p/rn�2 CO.rn�1//I

Ng D�1C 4

n�2m.p/rn�2 CO.rn�1/

�.r�4dr2 C r�2d�2/I

� Ng D r�n�1.1CO.rn�2//�g I

kˆk2 D .1CO.rn�1//�1� 2.n�1/

n�2 m.p/rn�2 CO.rn�1/�I

@r D r2.1CO.rn�2//@r :

The vector �eld in the last line is the unit vector �eld (with respect to Ng) orthogonalto the circles fr D constantg:The idea of the proof consists in exploiting the fact thatˆ is harmonic on M n fpg together with the Schrödinger–Lichnerowicz formula,the fact that Ng is scalar-�at, and integration by parts on the compact manifolds withboundaryM� ´ fr � �g:

0 DZ

M�

h xD2ˆ;î� Ng

DZ

M�

hxr� xrˆ;î� Ng

DZ

M�

hxrˆ; xrî� Ng CZ

@M�

hxr@rˆ;î� Ng ;

which, by Stokes’ formula, yields

0 � �Z

M�

hxrˆ; xrî� Ng DZ

@M�

hxr@rˆ;î� Ng D 1

2

Z

@M�

@rkˆk2� Ng :

Using the asymptotic expansions for @r , kˆk2, and � Ng near p, we can write theintegrand @rkˆk2� Ng in the form

@rkˆk2� Ng D�

� 2.n� 1/

n � 2 m.p/CO.rn�2/�� :

After integration over the sphere of radius �, the leading term in the expansion as� ! 0 is non-positive if and only ifm.p/ � 0. Thus the massm.p/ is non-negative,as claimed.

The equalitym.p/ D 0 implies thatˆ is parallel onM nfpg. The existence of anonzero parallel spinor entails the vanishing of Ricci curvature, hence Ng is actually�at since by hypothesis g is locally conformally �at. Moreover .M n fpg; Ng/ iscomplete (from the form of Ng near r D 0) and has an Euclidean end. It follows thatM nfpg is an isometric quotient of R

n, and since it contains an embedded Euclideanend, it must be isometric to R

n. The inverse of the stereographic projection mapsthen .M; g/ conformally onto Sn. �


5.7.3 Further applications

We �nally mention several directions of active research during the last years. Fordetails we refer to the original articles.

• Conformal spin invariants were introduced by Ammann, Dahl, Humbert, andMorel in [ADH09a, ADH09b], [AH08], and [AHM06], who also obtainedseveral vanishing theorems. The non-compact case was studied by Großein [Gro06], [Gro11], and [Gro12].

• The theory of spin hypersurfaces was studied by Friedrich [Fri98],Morel [Mor05], Roth [Rot10] and [Rot11], and Lawn and Roth [LR10]and [LR11].

• Probably the most spectacular applications of Spinc geometry were obtainedin dimension 4 by means of the Seiberg–Witten equations, cf. Seiberg andWitten [Wit94], LeBrun [LeB97] [GL98], and [Fri00], and Ginoux [Gin04].

• Dirac operators on Lagrangian submanifolds have been studied by Hijazi,Montiel, and Urbano [HMU06]. Spinc geometry of Kähler manifolds andthe Hodge Laplacian on minimal Lagrangian submanifolds have been investi-gated by Nakad [Nak10], [Nak11a], and [Nak11b] and Nakad and byRoth [NR12].

We synthesize these and several other recent results in Table 2.


Table 2

Riemannian manifolds Dirac lower bounds in terms of the References

spin closed �rst eigenvalue of the Yamabe operator

and the energy–momentum tensor (EMT) [Hij95]

scalar curvature, the EMT, and other terms [FK01]

scalar curvature and the Weyl tensor [FK02b]

scalar curvature and the Ricci tensor [FK02a]

scalar curvature and Codazzi tensors [FK08]

spin compact foliated scalar curvature and the generalized EMT [Hab07], [HR09]

spin compact with boundary �rst eigenvalue of the Yamabe operator

and the EMT [HMZ01b]

scalar and mean curvatures [Rau05]

�rst eigenvalue of the Yamabe operator [Rau06]

(Dirac–Witten operator)

intrinsic and extrinsic curvatures and the EMT [Mae08]

embedded hypersurfaces of compact mean curvature [HMZ01a]

domains in Riemannian spin manifolds �rst eigenvalue

of the Steklov–Escobar operator [HMZ02]

compact space-like hypersurfaces (Dirac–Witten operator)

of Lorentzian spin manifolds scalar curvature and the EMT [HZ03]

compact space-like submanifolds (Dirac–Witten operator)

of pseudo-Riemannian spin manifolds intrinsic/extrinsic curvatures and the EMT [CHZ12], [GM02]

compact hypersurfaces intrinsic and extrinsic curvatures [HZ01a]

compact submanifolds [GM02], [HZ01b]

complete non-compact spin �rst eigenvalue of the Yamabe operator [Gro06], [Gro11]

of �nite volume

open spin (not necessary complete) conformal or topological invariants [Bär09]

Spinc compact without boundary �rst eigenvalue of the perturbed Yamabe [HM99]

operator and the EMT [Nak10], [Nak11a]

Spinc non-compact of �nite volume �rst eigenvalue

of the perturbed Yamabe operator and the EMT [Nak11b]

Chapter 6

Lower eigenvalue bounds on Kähler manifolds

This chapter is devoted to the proof of Kirchberg’s inequalities [Kir86] and [Kir90]which say that on a compact Kähler spin manifold, the square of the lowest eigen-values of the Dirac operator is not less than the in�mum of the scalar curvature timesan explicit constant depending on the dimension.

In fact, this constant depends on the parity of the complex dimension. We shallsee that, in many respects, the odd-dimensional case is quite different from the even-dimensional one. There are also different proofs of these inequalities. First, theodd-dimensional case was proved by Kirchberg [Kir86] using the decompositionof the spinor bundle under the action by Clifford multiplication of the Kähler formtogether with the notion of modi�ed connections. Then, in [Kir90] both inequalitieswere proved by introducing the Kählerian Penrose operators. Later, another proofof inequality (6.1), which does not rely on the decomposition (6.4) below, was givenin [Hij94].

Here we present the approach of P. Gauduchon [Gau95a], based on the system-atic algebraic setting introduced in Section 2.3.

6.1 Kählerian spinor bundle decomposition

Recall that a Kähler structure on a Riemannian manifold .M n; g/ of dimensionn D 2m with Levi-Civita connection r is a r-parallel almost complex structureJ compatible with the metric g in the sense that g.JX; J Y / D g.X; Y / for alltangent vectors X; Y . The associated Kähler form � is the parallel 2-form de�ned,for any tangent vector �elds X and Y , by

�.X; Y / ´ g.JX; Y / D �g.X; J Y /:

Observe that the volume form �g de�ning the orientation of .M 2m; g/ is related tothe Kähler form � by

�g D 1

mŠ� ^ � � � ^�„ ƒ‚ …m times

:

162 6. Lower eigenvalue bounds on Kähler manifolds

We have seen that a compact Kähler spin manifold .M 2m; g; J; / with posi-tive scalar curvature cannot carry (real) Killing spinors. In other words, on suchmanifolds any eigenvalue � satis�es

�2.M 2m; g; J; / >n

4.n� 1/Scal0;

where Scal0 denotes as usual the in�mum of the scalar curvature ofM . This showsthat if a compact spin manifold has a Kähler structure, then Friedrich’s inequalitycan be improved. Indeed, we have the following result.

Theorem 6.1. On a compact Kähler spin manifold .M n; g; J; /, n D 2m, withpositive scalar curvature Scal, any eigenvalue � of the Dirac operator satis�es

�2 � mC 1

4mScal0; if m is odd, (6.1)

and

�2 � m

4.m� 1/Scal0; if m is even: (6.2)

The �rst proof of (6.1) could be found in [Kir86]. In the same paper, Kirchbergshowed that if equality in (6.1) is achieved, then the manifold is of odd complexdimension. The idea of the proof is to decompose the spinor bundle as the sumof eigenspaces associated to the eigenvalues of the Kähler form and to de�ne, onan arbitrary eigenbundle, an appropriate modi�ed connection. We point out thata proof of this inequality is given in [Hij94] using a natural modi�ed connectionwhich corresponds to the one introduced by Friedrich.

The proof we shall give here is based on the use of the Kählerian twistoroperators which we shall introduce. This follows the general principle introducedin Section 2.3, cf. [Lic90] and [Hij94]).

At any point x inM , choose normal coordinates at x so that .rei/.x/ D 0, forall i 2 f1; : : : ; ng. All the computations will be carried out in such charts.

First, we need to establish some lemmas, most of which are due to Kirchberg;cf. [Kir86], [Kir88], and [Kir90].

Lemma 6.2. As an endomorphism of the spinor bundle, the Kähler form is givenlocally by

� D 1

2

nX

kD1ek � Jek D �1

2

nX

kD1Jek � ek:

6.1. Kählerian spinor bundle decomposition 163

Proof. By the de�nition of the 2-form �,

� D 1

2

nX

k;lD1�klek ^ el

D 1

2

nX

k;lD1g.Jek ; el/ek � el

D 1

2

nX

kD1ek � Jek : �

Lemma 6.3. For any vector �eld X ,

Œ�;X� D 2JX; and Œ�; JX� D �2X: (6.3)

Proof. Since J 2 D �Id, the second identity follows from the �rst. By linearity, itis suf�cient to take X D ek , for 1 � k � n, �xed. We have

� � ek D 1

2

nX

lD1el � Jel � ek

D �12

nX

lD1el �

�ek � Jel C 2g.Jel ; ek/

�

D Jek C 1

2

nX

lD1

�ek � el C 2ıkl

�Jel

D 2Jek C ek ��: �

Lemma 6.4. Under the action of the Kähler form, the spinor bundle splits into anorthogonal sum

†M DmM

rD0†rM; (6.4)

where each sub-bundle †rM is an eigenbundle of � with eigenvalue

i�r ´ i.2r �m/and has rank

�mr

�.

Anorthonormal local coframe fe1; : : : ; em; emC1; : : : ; e2mg is said to be adaptedif for all 1 � j � m, one has J.ej / D emCj and J.emCj / D �ej . The actionof J on 1-forms is de�ned by the identi�cation between 1-forms and vectors via themetric. Let

pC.ej / ´ Nzj ´ ej � iemCj2


and

p�.ej / ´ zj ´ ej C iemCj2

:

It is a straightforward veri�cation that fz1; : : : ; zm; Nz1; : : : ; Nzmg is a Witt basis withrespect to the complex extension of the Riemannian metric g to ƒ1M . In fact, onehas

g.zj ; zk/ D g. Nzj ; Nzk/ D 0;

and

g.zj ; Nzk/ D g. Nzj ; zk/ D 1

2ıjk:

The complex Clifford algebra ClxM is generated by the above Witt basis, subjectto the relations

zj � zk C zk � zj D Nzj � Nzk C Nzk � Nzj D 0 (6.5a)

and

zj � Nzk C Nzj � zk D �ıjk: (6.5b)

6.2 The canonical line bundle

Lemma 6.5. The vector subspace

Lx ´ f‰ 2 †xM I zj �‰ D 0; for all j 2 f1; : : : ; mgg

of †xM has complex dimension 1.

Proof. The vector space Lx is non-empty since there exists at least one spinor such that

‰ ´ z1 � z2 � � � zm � ¤ 0

(the Clifford action is faithful) and clearly ‰ 2 Lx . Suppose ‰1; ‰2 2 Lx . Thereexists an element a 2 ClxM such that ‰2 D a � ‰1. By (6.5), a can be written asa D Nz1 � bC c, where b and c do not contain Nz1. Then z1 anti-commutes with b andc, so

0 D z1 �‰2 D z1 � a � ‰1 D �b � ‰1 � Nz1 � z1 � b � ‰1 C z1 � c �‰1 D �b �‰1;

as z1 � ‰1 D 0. Thus ‰2 D c � ‰1, and c does not contain Nz1. Repeating thisargument, we may assume that c does not contain Nzj for any j . Thus

c D c0 C cj zj C cjkzj � zk C : : :

and clearly ‰2 D c �‰1 D c0‰1. So Lx is one dimensional. �

6.2. The canonical line bundle 165

Let us �x a point x 2 M and a non-zero element ‰ 2 Lx . Since every elementof ClxM can be written as a linear combination of

Nzj1� � � Nzjp � zk1

� � � zkq;

the set

fNzj1� � � Nzjr �‰I 1 � j1 < � � � < jr � m; 0 � r � mg;

spans †xM . It is actually a basis, for dimensional reasons.The Clifford action of � can be expressed in terms of the Witt basis as

� DmX

kD1ek � Jek

D i

mX

kD1.zk C Nzk/ � . Nzk � zk/

D �imX

kD1.2 Nzk � zk C 1/

D �im� 2imX

kD1Nzk � zk:

Proof of Lemma 6.4. Denote by L the complex line bundle whose �ber at eachx 2 M is Lx . For each r 2 f0; : : : ; mg, let †rM be the complex vector subspaceof †M spanned by

fNzj1� � � Nzjr �‰I 1 � j1 < � � � < jr � m; ‰ 2 Lg:

Consider a spinor ‰r ´ Nzj1� � � Nzjr � ‰ 2 †rM . By (6.5),

� �‰r D �i�mC 2

mX

jD1Nzj � zj

�� Nzj1

� � � Nzjr �‰r

D �im‰r C 2ir‰r

D i�r‰r : �

The above considerations show the spinor bundle†M is canonically isomorphicto L˝ .ƒ0;0 ˚ � � � ˚ƒ0;m/.


In order to gather some information on L, we have to examine more carefullythe restriction of the spin representation to the unitary group. More precisely, letzUm be the inverse image of Um in Spin2m and let E � †2m be the complex line ofspinors annihilated by .1; 0/-forms (in other words, E is the algebraic analog ofL).Then for every v 2 R

2m and 2 E,

.v C iJ v/ � D 0;

whenceJv � D iv � : (6.6)

Denote by �2m the spin representation and by � the projection Spin2m ! SO2m.We claim that for every a 2 zUm, the homomorphism �2m.a/ acts on E by a com-plex scalar whose square is the determinant of �.a/, viewed as a unitary matrix.This clearly implies that the square of the line bundle L, which is associated withthe representation E of zUm, is isomorphic to the line bundle associated with thedeterminant representation of Um on C, which is nothing else than the canonicalbundle K ´ ƒm;0M ofM .

Now, it is an elementary fact that Um is generated by rotations of complex linesof .R2m; J /, so zUm is generated (as multiplicative group) by products v � w in the(real) Clifford algebra Cl2m of unit vectors belonging to the same complex line of.R2m; J /. In order to prove the claim, it is thus suf�cient to take a D v �.˛vCˇJv/,with ˛2Cˇ2 D 1. Then �.a/.v/ D a �v �a�1 acts as the identity on the orthogonalcomplement of v and Jv, and is equal to .˛I C ˇJ /2 on the subspace spanned byv and Jv. Thus detC.�.a// D .˛C iˇ/2. On the other hand, using (6.6) we get forevery 2 E

�.a/ D v � .˛v C ˇJv/ � D v � .˛v C ˇiv/ � D �.˛ C iˇ/ :

Thus �2m.a/2. / D det.�.a// , so the claim is proved.

Thus, we obtained the so-called Hitchin representation of the spinor bundle ofany almost Hermitian manifold as

†M Š K12 ˝ƒ0;�; (6.7)

where K12 denotes a square root of the canonical bundle.

6.3 Kählerian twistor operators

Lemma 6.6. For any tangent vector �eld X and r 2 f0; : : : ; mg, one has

pC.X/ �†rM � †rC1M and p�.X/ �†rM � †r�1M: (6.8)

with the convention that †�1M D †mC1M D M � f0g.

6.3. Kählerian twistor operators 167

Proof. The proof is a consequence of Lemma 6.3. It is also possible to use theexplicit decomposition given in Lemma 6.4 to get the result. To see this, considera nontrivial spinor ‰r ´ Nzj1

� � � Nzjr � ‰ 2 †rM . By linearity, it is suf�cient toshow (6.8) for X D ek , for any k 2 f1; : : : ; mg. Lemma 6.4 and (6.5) imply thatNzk � ‰r D 0 if k 2 fj1; : : : ; jrg and Nzk � ‰r 2 †rC1M if k 62 fj1; : : : ; jrg.The second inclusion can be seen similarly. �

We now show that Lemma 6.6 implies that the image of the restriction of theDirac operator to sections of the subbundle †rM lies in the space of sections of†r�1M ˚†rC1M .

Associated with J there is an elliptic formally self-adjoint operator zD de�nedlocally by

zD DnX

iD1Jei � rei

D �nX

iD1ei � rJei

I

see [Hit74], [Mic80], and [Kir86]. Since the almost complex structure J is parallel,Lemma 6.3 yields the following lemma.

Lemma 6.7. The operators D and zD satisfy the relations

Œ�;D� D 2 zD; (6.9)

Œ�; zD� D �2D; (6.10)

Œ�;D2� D 0; (6.11)

zDD C D zD D 0; (6.12)

zD2 D D2: (6.13)

Proof. Identities (6.9) and (6.10) follow from (6.3). Relation (6.11) can be estab-lished by using the Schrödinger–Lichnerowicz formula and the fact that� is paral-lel. Equation (6.12) is then a consequence of the �rst three identities.

By (6.9),

2D2 D DŒ zD; �� D D zD�� D� zD (6.14)

and

2 zD2 D zDŒ�;D� D zD�D � zDD�: (6.15)

Equations (6.14), (6.15), and (6.12) yield

2 zD2 � 2D2 D zD�D C D� zD: (6.16)


On the other hand, by (6.9) and (6.12),

zD�D C D� zD D .� zD C 2D/D C .�D � 2 zD/ zD (6.17)

D 2D2 � 2 zD2:

Equations (6.16) and (6.17) imply (6.13). �

Lemma 6.6 and Lemma 6.7 immediately imply the following result.

Lemma 6.8. De�ne the two operators D� and DC by

p�.D/ ´ D� ´ 1

2.D C i zD/ (6.18a)

and

pC.D/ ´ DC ´ 1

2.D � i zD/: (6.18b)

We have the differential complexes

�.†mM/D��! � � � D��! �.†rM/

D��! �.†r�1M/D��! � � � D��! �.†0M/;

and

�.†0M/DC�! � � �

DC�! �.†rM/DC�! �.†rC1M/

DC�! � � �DC�! �.†mM/:

Now we are ready to de�ne Kählerian twistor operators. On a Kähler spin man-ifold .M 2m; g; J /, for each r 2 f0; : : : ; mg, the Kählerian twistor operator P.r/

is the composition of the restriction of the covariant derivative to the subbundle�.†rM/ with the projection on Ker r :

P.r/W�.†rM/r�! �.T�M ˝†rM/

�r�! Ker r ;

where r is the restricted Clifford multiplication de�ned by

r ´ �r ˚ C

r W �.T�M ˝†rM/ �! �.†r�1M/˚ �.†rC1M/;

.X;‰r/ 7�! p�.X/ �‰r ˚ pC.X/ � ‰r ;and �r is the projection on Ker r � �.T�M ˝†rM/.

Lemma 6.9. The Clifford multiplications ˙r satisfy the identities

Cr ı . �

r /� D 0;

�r ı . C

r /� D 0;

Cr ı . C

r /� D 2.r C 1/Id†rC1M ;

�r ı . �

r /� D 2.m � r C 1/Id†r�1M :

6.3. Kählerian twistor operators 169

Proof. For any spinor �eld ‰rC1 2 �.†rC1M/, one gets

Cr ı . C

r /�‰rC1 D C

r

��

nX

jD1ej ˝ p�.ej / �‰rC1

�

D �nX

jD1ej � pC.ej / � p�.ej / � ‰rC1

D �2mX

jD1zj � zj � ‰rC1

D .�i�Cm/ �‰rC1

D 2.r C 1/‰rC1:

The other relations are obtained similarly. �

Remark 6.10. Identities (6.19) immediately yield

r ı . r/� D 2.r C 1/ ProjCr C2.m� r C 1/ Proj�r ; (6.19)

where ProjCr and Proj�r are respectively the orthogonal projections on †rC1M and†r�1M , and by convention, Proj�0 D ProjCm D 0. Therefore, the linear map r ı . r/� has two different eigenvalues 2.r C 1/ and 2.m� r C 1/ for 0 < r < m

2

and m2< r < m, while for 2r D m, r D 0 or r D m, it has only one eigenvalue.

Using the general approach of Section 2.3 and relations (2.16) and (2.24), weimmediately get the following:

Proposition 6.11. For each r 2 f0; : : : ; mg and for any spinor �eld‰r 2 �.†rM/,the local expression of the Kählerian twistor operator P.r/ is given by

P.r/‰r D2mX

jD1ej ˝

hrej‰r C 1

2.r C 1/p�.ej / � DC‰r

C 1

2.m � r C 1/pC.ej / � D�‰r

i(6.20)

Moreover,

jr‰r j2 � 1

2.r C 1/jDC‰r j2 C 1

2.m � r C 1/jD�‰r j2:


Note that by the de�nition of the Kählerian twistor operator P.r/, one has

P.r/X W�.†rM/ �! �.†rM/; X 2 �.TM/; (6.21)

and, for any spinor �eld ‰r 2 �.†rM/,

nX

jD1ej � P.r/ej ‰r D 0: (6.22)

Moreover, we have the following lemma.

Lemma 6.12. For each r 2 f0; : : : ; mg and any spinor �eld ‰r 2 �.†rM/, theKählerian twistor operator P.r/ satis�es

nX

jD1J.ej / � P.r/ej ‰r D 0: (6.23)

Proof. Identities (6.3), (6.21), and (6.22) imply that

0 D � �� nX

jD1ej � P.r/ej ‰r

�;

DnX

jD1.ej ��C 2Jej / � P.r/ej ‰r

D i�r

nX

jD1ej � P.r/ej ‰r C 2

nX

jD1J.ej / � P.r/ej ‰r

D 2

nX

jD1J.ej / � P.r/ej ‰r : �

6.4 Proof of Kirchberg’s inequalities

Proof of Theorem 6.1. Consider the space

†�M ´ f‰ 2 �.†M/I D2‰ D �2‰g:

Since Œ�;D2� D 0, it follows that there exists an orthogonal decomposition of†�M as

†�M DmM

rD0†�rM; (6.24)

6.4. Proof of Kirchberg’s inequalities 171

where †�rM � †rM . In decomposition (6.24), take r to be the smallest integerfor which †�rM 6D f0g. Then 0 � r < m. Since ŒD2;D˙� D 0, then for 0 6D ‰r 2†�rM , one has D�‰r D 0, i.e., D‰r D DC‰r . Identities (6.22) and (6.23) implythat

Z

M

jP.r/‰r j2�g DZ

M

nX

jD1jP.r/ej ‰r j

2�g

DZ

M

nX

jD1.P.r/ej ‰r ;rej‰r /�g

DZ

M

nX

jD1

�rej‰r C 1

2.r C 1/p�.ej / � DC‰r ;rej‰r

��g :

Since Clifford multiplication by X or JX is skew-symmetric, it follows that

Z

M

jP.r/‰r j2�g DZ

M

jr‰r j2�g � 1

2.r C 1/

Z

M

jD‰r j2�g : (6.25)

Equation (6.25) combined with the Schrödinger–Lichnerowicz formula yields

�2 � 2.r C 1/

2r C 1

Scal04

; 0 � r < m: (6.26)

On the other hand, if one takes R to be the largest integer for which †�RM 6D 0,then as above one gets

�2 � 2.m�R/C 2

2.m�R/C 1

Scal04

; 0 < R � m: (6.27)

Note that if we denote by f .r/ the coef�cient of Scal04

in the right-hand side of (6.26),then the corresponding coef�cient in inequality (6.27) is f .m �R/. By de�nition,one has R D rCj , for some integer j , 1 � j < m. Since the function f is strictlydecreasing, it follows from (6.26) and (6.27) that the number

infr;jŒsup.f .r/; f .m� j � r//� D

8ˆ<ˆ:

f�m � 1

2

�D mC 1

m; if m is odd;

f�m � 2

2

�D m

m � 1; if m is even;

gives the optimal coef�cient. This proves Theorem 6.1. �


6.5 The limiting case

We will now study the limiting cases of (6.1) and (6.2). We assume Scal0 > 0,since otherwise the limiting case corresponds (by the Schrödinger–Lichnerowiczformula) to the existence of a parallel spinor and is settled by Theorem 8.1 below.

We start with the case where m D 2k C 1 is odd. Let ‰ 2 �.†�M/ be aneigenspinor and assume that �2 D mC1

4mScal0. From the above proof we see that if

this equality holds, Scal is constant, the projection ‰r of ‰ on †rM vanishes forr � k � 1 and r � k C 2, and

P.k/‰k D 0 and P.kC1/‰kC1 D 0: (6.28)

By (6.11),D2‰k D �2‰k and D2‰kC1 D �2‰kC1:

Now, by (6.12), D�‰k 2 �.†k�1M/ and D2.D�‰k/ D �2D�‰k . The argu-ment above shows that D�‰k D 0. Similarly, DC‰kC1 D 0. From (6.18) we thenget DC‰k D D‰k and D�‰kC1 D D‰kC1. From (6.20) and (6.28) we obtain

rX‰k C 1

mC 1p�.X/ � D‰k D 0 (6.29)

and

rX‰kC1 C 1

mC 1pC.X/ � D‰kC1 D 0; (6.30)

for all tangent vectors X . On the other hand, since ‰ D ‰k C ‰kC1 2 �.†�M/,we deduce

D‰k D �‰kC1 and D‰kC1 D �‰k:

Since x‰ D .�1/k‰k C .�1/kC1‰kC1, (6.29) and (6.30) yield

rX‰ C �

mC 1X �‰ C .�1/k �

mC 1JX � x‰ D 0 (6.31)

Definition 6.13. Let .M 4kC2; g; J / be a Kähler spin manifold of odd complexdimension m D 2k C 1. A Kählerian Killing spinor is a section ‰ of the spinorbundle ofM satisfying (6.31) for some real constant �.

We have proved the following result.

Theorem 6.14 ([Kir86]). If .M 4kC2; g; J / is a compact Kähler spin manifold sat-isfying the limiting case in Kirchberg’s inequality (6.1), thenM carries a non-trivialKählerian Killing spinor.

6.5. The limiting case 173

Consider now the case where m D 2k is even. Let ‰ 2 �.†�M/ be an eigen-spinor and assume that �2 D m

4.m�1/Scal0. From the proof of Theorem 6.1 we seethat if this equality holds, then Scal is constant, the projection ‰r of ‰ on †rMvanishes for r � k � 2 and r � k C 2, and

P.k�1/‰k�1 D 0; P.k/‰k D 0; P.kC1/‰kC1 D 0: (6.32)

Decomposing the equality D‰ D �‰ yields

DC‰k D �‰kC1; D�‰k D �‰k�1; DC‰k�1 C D�‰kC1 D �‰k ; (6.33)

andDC‰kC1 D 0; D�‰k�1 D 0: (6.34)

Since � ¤ 0, this shows that ‰k�1 and ‰kC1 cannot vanish simultaneously.By changing the orientation ofM if necessary, we may assume that ‰kC1 ¤ 0.

From (6.20) and (6.32) we obtain

rX‰kC1 C 1

mpC.X/ � D‰kC1 D 0; X 2 �.TM/:

Taking (6.33) and (6.34) into account, we arrive at the following result.

Theorem 6.15 ([Gau95a, Hij94], [Kir90], and [Lic90]). A compact Kähler spinmanifold of real dimension n D 2m D 4k is a limiting manifold if and only if.up to an orientation change/ there exists a nonzero spinor ‰kC1 2 �.†kC1M/

satisfying

rX‰kC1 D � 1n.X � iJX/ � D‰kC1 (6.35)

andD2‰kC1 D �2‰kC1:

Chapter 7

Lower eigenvalue bounds on quaternion-Kähler

manifolds

In this chapter we establish lower bounds for the square of the eigenvalues of theDirac operator on compact quaternion-Kähler spin manifolds with positive scalarcurvature. By de�nition, a Riemannian manifold is said to be quaternion-Kähler ifits restricted holonomy group is contained in Sp1 � Spm. This holonomy conditionmay also be characterized by the existence of a parallel 4-form� of special algebraictype (cf. (7.5) below), hence Friedrich’s inequality cannot be an equality in this case(cf. Chapter 5).

The model space of this family of manifolds is the quaternionic projective spaceHPm (cf. Section 13.3). From the computation of the Dirac spectrum of HPm

(cf. Section 15.4), it was conjectured that any eigenvalue � of the Dirac operatoron such a manifold should satisfy

�2 � mC 3

mC 2

Scal

4: (7.1)

(Note that since any quaternion-Kähler manifold is Einstein, the scalar curvatureScal is a constant). Several results in this direction were obtained in [HM95b]and [HM97], where the �rst eigenvalues estimates were given. Those estimates re-lied on the decomposition of the spinor bundle †M into parallel subbundles underthe action of the group Sp1 � Spm,

†M DmM

rD0†rM:

In [HM95b], these lower bounds were obtained using a twistor operator which is thequaternionic analogue of the Kählerian twistor operator introduced by K.-D. Kirch-berg in [Kir90].

Better estimates were given in [HM97], using the same approach as in theKähler case (cf. Chapter 6), based on the splitting of the bundle TM˝†rM into theorthogonal sumKer r˚.Ker r/?, where r is the restriction of the Clifford multi-plication to TM ˝†rM . The quaternion-Kähler twistor operator was introduced

176 7. Lower eigenvalue bounds on quaternion-Kähler manifolds

as the composition of the Levi-Civita covariant derivative r with the projection onKer r . Applying this operator to a convenient eigenspinor ‰ of D2, eigenvalueestimates were obtained by means of a Weitzenböck formula.

These results were not totally satisfactory, inequality (7.1) could only bededuced under the assumption that some eigenspinor corresponding to the �rsteigenvalue � of the Dirac operator has a non-trivial component in †rM , r D 0

or r � Œm=2� C 1 (this is always true in quaternionic dimension m D 2 andm D 3, [HM97]). In particular, this condition is satis�ed by HPm: there existsan eigenspinor associated with the �rst eigenvalue of the Dirac operator which hasa component in†0M (this can be easily deduced using the results of Section 15.4).

The lower bound (7.1) was proved byW.Kramer, U. Semmelmann, andG.Wein-gart, using a new and conceptually simple method; see [KSW99], [KSW98a],and [KSW98b]. It is based on the decomposition of the two isomorphic bundles.TM˝TM/˝†M ' TM˝.TM˝†M/ into parallel subbundles (correspondingto the irreducible components of the corresponding representation) under the actionof the group Sp1 � Spm. The result is proved with the help of aWeitzenböck formula,obtained by considering the isotypical parts of the two splittings of r2‰, for a con-venient choice of the eigenspinor ‰ ofD2, cf. [KSW99] and [KSW98a]. However,this method leads to cumbersome computations (see [Bra98] and [Bra05]).

Later, using a new point of view, U. Semmelmann and G. Weingart gave a moregeneral and systematic approach to state eigenvalue estimates for various naturaloperators and obtained inequality (7.1) as a special case; see [SW01] and [SW02].

Similar results were obtained recently by Y. Homma [Hom06] with a differentpoint of view, based on the generalization in that context of Branson’s “optimalBochner–Weitzenböck formulas for gradients.”

In order to study the limiting case in (7.1) (which cannot be derived from thetwo general systematic approaches quoted before), we will present here a simpleralternative to the proof given in [KSW99].

We will then describe the general method of Semmelmann and Weingart to pro-duce eigenvalues estimates for various natural operators.

We start by giving some preliminaries on quaternion-Kähler manifolds.

7.1 The geometry of quaternion-Kähler manifolds

An introduction to quaternion-Kähler manifolds could be found in [Bes87], Chap-ter 14, or in the survey [Sal99].

Definition 7.1. A Riemannian manifold .M 4m; g/ is said to be quaternion-Kählerif its bundle of oriented orthonormal frames PSO4m

M admits a reduction P to thesubgroup Sp1 � Spm � SO4m, compatible with the Levi-Civita connection r (thatis, such that r reduces to a connection on P ).

7.1. The geometry of quaternion-Kähler manifolds 177

The group Sp1 � Spm can be de�ned1 as the image of the homomorphism

˛WSp1 � Spm �! SO4m;

.q; A/ 7�! .x 7! Ax Nq/;(7.2)

where R4m D H

m, is viewed as a right vector space over H, the group Spm act-ing on it by .m � m/ quaternion matrices. Indeed, ˛ de�nes a two-fold coveringof the group Sp1 � Spm with Ker˛ D f.1; Idm/; .�1;�Idm/g. Hence, any (irre-ducible) representation of this group comes from an (irreducible) representation ofSp1 � Spm which factors through the quotient. Because Sp1 � Sp1 is just the groupSO4, we will only consider quaternion-Kähler manifolds of dimension 4m � 8.

Let Q be the 3-dimensional R-vector space spanned by the R-endomorphismsI; J;K of H

m de�ned respectively by

I W x 7�! �xi ; J W x 7�! �xj ; KW x 7�! �xk; (7.3)

so that they satisfy the same commutation rules as the standard basis of imaginaryquaternions .i ; j ;k/. The action of the group Sp1 � Spm on End.Hm/ leaves thevector spaceQ invariant, hence induces a representation of the group onQ, whichcan also be described as follows. Identify R

3 D spanfi; j; kg with Q, by i 7! I ,j 7! J , k 7! K, and endow the vector space Q with the natural Euclideanstructure which makes this isomorphism an isometry. The unitary representationof Sp1 � Spm on Q, such that Spm acts trivially and Sp1 acts through the two foldcovering

Sp1 �! SO3 ' SO.spanfi; j; kg/;q 7�! .x 7! qx Nq/;

(7.4)

obviously factors through the quotient, and de�nes the aforementioned representa-tion of the group Sp1 � Spm.

By de�nition, the quaternionic structure onM is given by the rank 3-bundle

QM D P�Sp1 �SpmQ:

Note that QM is globally parallel under the Levi-Civita connection r, since Q isinvariant under the group Sp1 � Spm.

Any local section � of the bundle P de�nes a local frame .J˛/˛D1;2;3 of thebundle QM by

J1x D Œ�.x/; I �;

J2x D Œ�.x/; J �;

J3x D Œ�.x/; K�:

1The notation Spm � Sp1 is also often used to emphasize that Sp1 acts by right multiplication.


This local frame is such that

(a) the metric g is Hermitian for the three almost complex structures J˛ ,˛ D 1; 2; 3;

(b) we haveJ˛ ı Jˇ D �ı˛ˇ Id C "123˛ˇ J ;

where

"123˛ˇ D

8ˆ<ˆ:

C1 if .˛; ˇ; / is an even permutation of .1; 2; 3/;

�1 if .˛; ˇ; / is an odd permutation of .1; 2; 3/;

0 otherwise;

(c) for any ˛ D 1; 2; 3, the covariant derivative rJ˛ is a linear combination ofJ1, J2 and J3.

Such a local frame is said to be standard. Note that for the natural Euclideanstructure onQM carried from the �breQ, any standard local frame is orthonormal.

The holonomy condition can also be characterized by the existence of a parallel4-form �, called fundamental (see [Bon67] and [Kra66]), de�ned locally by

� D3X

˛D1�˛ ^�˛; (7.5)

where the �˛ are the local 2-forms associated via the metric with the local almostcomplex structures J˛ of a standard local frame ofQM :

�˛.X; Y / D g.J˛X; Y /; X; Y 2 TM:

Any quaternion-Kähler manifold is Einstein (cf. [Ber66], [Ale68], or [Ish74]; twoproofs of this result may be found in [Bes87] Theorem 14.39), hence its scalarcurvature is constant.

Note that if the manifold is supposed to be connected and complete (this will bethe case in all what follows) then, by Myers’ theorem, the positivity of the scalarcurvature implies that the manifold is compact. Any such quaternion-Kähler mani-fold is said to be positive.

Any positive quaternion-Kähler manifold has a unique spin structure if m iseven, whereas the only positive quaternion-Kähler spin manifold with m odd is thequaternionic projective space [Sal82].

The condition for a quaternion-Kähler manifold to be positive is rather restric-tive: C. LeBrun and S. Salamon proved that, for a given integer m � 2, up to ahomothetic change of the metric, there are only �nitely many positive quaternion-Kähler manifolds of dimension 4m; see [LS94]. The only known examples are

7.2. Quaternion-Kähler spinor bundle decomposition 179

Wolf’s symmetric spaces (cf. [Wol65] or Theorem 14.51 in [Bes87]). It is conjec-tured that there are no other examples [LS94].

Other (important) properties of quaternion-Kähler manifolds are brie�y men-tioned in Section 10.1.

7.2 Quaternion-Kähler spinor bundle decomposition

From now on .M 4m; g/ is a positive quaternion-Kähler spin manifold (with4m � 8). A key ingredient in the proof of the Dirac eigenvalue estimate is thedecomposition of the spinor bundle †M into parallel subbundles corresponding tothe irreducible parts of the decomposition of the spin representation.

We �rst consider the following parameterization of the irreducible representa-tions of the group Sp1 � Spm. Since this group can be identi�ed with a subgroup ofSpmC1 (cf. Section 13.3), and since the standard maximal torus T of SpmC1 is alsoa maximal torus of Sp1 � Spm, irreducible representations can be parametrized bytheir dominant weight relatively to this maximal torus T . Such a dominant weighthas the form

.r; `1; : : : ; `m/;

where r and the ì are integers verifying the conditions r � 0, and `1 � `2 � � � � �`m � 0, (cf. Section 15.4).

Indeed, any such .mC1/-tuple may be viewed as the dominant weight of a tensorproduct of the form �˝ �, where � (resp. �) is an irreducible representation of Sp1(resp. Spm), relative to the product of the standard tori of Sp1 and Spm, respectively.

In the following we often identify an irreducible representation of Sp1 � Spmwith the dominant weight .r; `1; : : : ; `m/ with such a tensor product �˝ �.

More precisely, let H ´ C2 ' H and E ´ C

2m ' Hm be the standard

complex representations of the groups Sp1 and Spm, respectively. Then � is thenatural representation in the space SymrH of symmetric tensors of order r overH (see Theorem 12.41), and � an irreducible representation of Spm with dominantweight .`1; : : : ; `m/ (cf. Section 12.4.3).

The irreducible representations of Spm which are often considered are theso-called fundamental ones (cf. Section 12.4.3), whose dominant weights are givenby

.1; : : : ; 1; 0; : : : ; 0„ ƒ‚ …`

/;

where 0 � ` � m � 1. They appear as the standard representations on the spacesƒm�`

ı E, 0 � ` � m � 1, where ƒm�`ı E denotes the orthogonal complement of

the space ! ^ .ƒm�`�2E/ in the spaceƒm�È. Here, ! is the standard symplectic2-form on E.


Example 7.2. One can easily verify that the representation of Sp1 � Spm in thespace QC ´ Q ˝R C is irreducible with dominant weight .2; 0; : : : ; 0/. HenceQC ' Sym2H and the bundle QCM ´ QM ˝R C is isomorphic to the bundleP �Sp1 � Spm

Sym2H .

Since Sp1 � Spm is simply connected, the homomorphism

˛W Sp1 � Spm �! Sp1 � Spm

lifts to a unique homomorphism

Q W Sp1 � Spm �! Spin4m

(see Examples 14.13). The group Sp1 � Spm acts on the spinor space †4m by therepresentation e�4m D �4m ı Q . The decomposition of this representation may beobtained by determining itsmaximal vectors (see Section 12.3 and [BS83], [Wan89],and [HM95a]),

†4m DmM

rD0†4m;r ; (7.6)

where †4m;r is the space of the irreducible representation with dominant weight

.r; 1; : : : ; 1; 0; : : : ; 0„ ƒ‚ …r

/:

Thus

†4m;r ' SymrH ˝ƒm�rı E:

This algebraic decomposition induces the following decomposition of the spinorbundle †M :

†M DmM

rD0†rM; (7.7)

where †rM is a globally parallel subbundle of rank .r C 1/��2mm�r

��

2mm�r�2

��.

Indeed, if m is even, all the Sp1 � Spm-representations in the spaces †4m;rdescend to Sp1 � Spm. In this case †M can be identi�ed with the well-de�ned

bundleP �Sp1 �Spm†4m, and one gets the decomposition by considering the bundles

†rM ´ P �Sp1 �Spm†4m;r . If m is odd, then the spin condition implies the exis-

tence of a lifting of P to a Sp1 � Spm-principal bundle zP ; see [Sal82]. In this case,

the spinor bundle is isomorphic to zP �Sp1 �Spm†4m and one gets the decomposition

by considering the bundles †rM ´ zP �Sp1 � Spm†4m;r .

7.2. Quaternion-Kähler spinor bundle decomposition 181

Viewed as an endomorphism of the spinor bundle, the fundamental form � hasthe following local expression (cf. [HM95b]):

� D3X

˛D1�˛ ��˛ C 6mId: (7.8)

Note that, as endomorphisms of the spinor bundle, the local 2-forms�˛ are de�nedby

�˛ D 1

2

4mX

iD1ei � J˛ei : (7.9)

Moreover, it is easy to see that, for all ˛; ˇ D 1; 2; 3, they satisfy the relations

Œ�˛; �ˇ � D 4"123˛ˇ � ;

4�˛ DX

ˇ;

"123˛ˇ �ˇ �� ; (7.10)

and

Œ�;�˛� D 0: (7.11)

Since � is .Sp1 � Spm/-invariant, the restriction of its action on each subbundle†rM is a scalar multiple of the identity, and one has [HM95a]

�j†rM D .6m � 4r.r C 2// Id: (7.12)

A second step in the proof is the decomposition of the “twistor” bundle TMC˝†Minto parallel subbundles. By the above result, this amounts to the decompositionof the bundles TMC ˝ †rM , which are induced by the decompositions of thecorresponding Sp1� Spm-representations in the spaces C

4m ˝†4m;r .The tensor product C

4m ˝†4m;r splits as (cf. [KSW99] or [Mil99])

C4m ˝†4m;r D Wr�1; Nr ˚WrC1; Nr ˚Wr�1;rC1 ˚WrC1;r�1

˚Wr�1;r�1 ˚WrC1;rC1;(7.13)

denoting byWr;s the space of the irreducible representation of Sp1 � Spm with dom-inant weight

.r; 1; : : : ; 1; 0; : : : ; 0„ ƒ‚ …s

/;

and byWr;Ns, the space of the irreducible representation of Sp1 � Spm with dominantweight

.r; 2; 1; : : : ; 1; 0; : : : ; 0„ ƒ‚ …s

/;


setting Wr;s D f0g if r < 0 or s > m, and Wr;Ns D f0g if r < 0 or s > m � 1.Note that the last two terms in (7.13) are respectively isomorphic to †4m;r�1 and†4m;rC1.

Exactly as for the spinor bundle, this decomposition induces the decompositionof the twistor bundle into globally parallel subbundles:

TMC ˝†rM D Wr�1; NrM ˚WrC1; NrM ˚Wr�1;rC1M ˚WrC1;r�1M

˚Wr�1;r�1M ˚WrC1;rC1M I(7.14)

the two last bundles in the sum are respectively isomorphic to†r�1M and†rC1M .

7.3 The main estimate

Let ‰ 2 �.†rM/. According to the decomposition (7.14), r‰ splits into

r‰ D .r‰/r�1; Nr C .r‰/rC1; Nr C .r‰/r�1;rC1 C .r‰/rC1;r�1C .r‰/r�1;r�1 C .r‰/rC1;rC1:

(7.15)

Since the direct sum in (7.14) is orthogonal, we get

jr‰j2 D j.r‰/r�1; Nr j2 C j.r‰/rC1; Nr j2 C j.r‰/r�1;rC1j2 C j.r‰/rC1;r�1j2

C j.r‰/r�1;r�1j2 C j.r‰/rC1;rC1j2:(7.16)

Asmentioned before, the last two terms in (7.15) are sections of bundles respectivelyisomorphic to†r�1M and†rC1M . Using elementary arguments in representationtheory, we �rst note that the last two terms in (7.16) may be expressed in terms ofthe square norms jD�‰j2 and jDC‰j2, where D�‰ and DC‰ are the only twocomponents of D‰ given by (7.7).

Lemma 7.3. Let r be the restriction of the Clifford multiplication to

C4m ˝†4m;r :

One has

Ker r D Wr�1; Nr ˚WrC1; Nr ˚Wr�1;rC1 ˚WrC1;r�1:

The restriction of r toWr�1;r�1 (resp.WrC1;rC1) is an isomorphism onto†4m;r�1(resp. †4m;rC1).

7.3. The main estimate 183

Furthermore,

j .w/j2 D 2.r C 1/.m � r C 1/

rjwj2; w 2 Wr�1;r�1; (7.17)

and

j .w/j2 D 2.r C 1/.mC r C 3/

r C 2jwj2; w 2 WrC1;rC1: (7.18)

(The Hermitian product h�; �i on C4m ˝ †4m is the natural one, constructed from

the Hermitian products of C4m and †4m, respectively).

Proof. The �rst part is a simple consequence of the Schur lemma. For the proofof (7.17), for simplicity, denote by � the restriction of r to Wr�1;r�1. This is an(Sp1 � Spm)-equivariant isomorphism onto †4m;r�1.

Now let .�; �/ be the Hermitian scalar product on Wr�1;r�1 de�ned by

.w; w0/ D h �.w/; �.w0/i;

where h�; �i is the usual Hermitian scalar product on†4m. This is an (Sp1 � Spm)-in-variant scalar product on Wr�1;r�1 (since the spin representation is unitary for thescalar product h�; �i).

By the Schur lemma, the two .Sp1 � Spm/-invariant scalar products .�; �/ andh�; �i on Wr�1;r�1 have to be proportional: .�; �/ D ch�; �i. The value of the constantc is obtained from the computation of .w�; w�/ and hw�; w�i respectively, wherew� is (for instance) the maximal vector ofWr�1;r�1. The proof of (7.18) is similarto (7.17). �

Considering now the restriction of the Dirac operatorD to the space �.†rM/ ofsections of the globally parallel subbundle †rM , we may conclude from the abovelemma that

DW�.†rM/r�! �.TM ˝†rM/

�! �.†r�1M ˚†rC1M/:

Hence, for any ‰ 2 �.†rM/, the spinor �eld D‰ splits as

D‰ D D�‰ C DC‰;

where

D�‰ ´ .D‰/r�1;r�1 and DC‰ ´ .D‰/rC1;rC1:

Moreover, by (7.17) and (7.18), we also have


Lemma 7.4. For any ‰ 2 �.†rM/, the following relations hold:

j.r‰/r�1;r�1j2 D r

2.r C 1/.m� r C 1/jD�‰j2

and

j.r‰/rC1;rC1j2 D r C 2

2.r C 1/.mC r C 3/jDC‰j2:

We now consider the endomorphism

QW C4m ˝†4m �! C

4m ˝†4m;

de�ned on decomposable tensors x ˝ , x 2 C4m, 2 †4m, by

Q.x ˝ / D Ix ˝ .�I � /C Jx ˝ .�J � /CKx ˝ .�K � /;

where .I; J;K/ is the canonical basis ofQ (cf. (7.3)), and where �I , �J , �K arethe 2-forms associated with the three endomorphisms I; J;K. More precisely,

�I .x; y/ D hIx; yi; �J .x; y/ D hJx; yi; �K.x; y/ D hKx; yi:

Note that, by the SO3-invariance of the expression I˝�I CJ˝�J CK˝�K,we can consider any standard basis ofQ in the de�nition of Q.

This indeed implies the .Sp1 � Spm/-equivariance of the endomorphismQ, whichnaturally extends to a bundle endomorphism, also denoted byQ:

QW�.TM ˝†M/ �! �.TM ˝†M/;

de�ned locally, for all X 2 �.TM/ and ‰ 2 �.†M/, by

Q.X ˝‰/ D3X

˛D1J˛X ˝ .�˛ �‰/;

where .J˛/˛D1;2;3 is a standard local frame of the bundleQM .

Lemma 7.5. The restriction ofQ to sections of TM ˝†0M vanishes identically.

Proof. Let J˛ be a standard local frame of QM . For any ‰ 2 �.TM ˝ †0M/,one has X

˛

j�˛ �‰j2 D �DX

˛

�˛ ��˛ �‰;‰E

D 0;

since the restriction ofP˛�˛ ��˛ to †0M vanishes identically. �


Now, the endomorphism Q being .Sp1 � Spm/-equivariant, its restriction to anyirreducible component in decomposition (7.13) is a scalar multiple of the identity.This implies that the bundle endomorphism Q acts as a scalar multiple of the iden-tity on sections of the subbundles occurring in decomposition (7.14). Computing(for instance) the image of a maximal vector in each component in decomposi-tion (7.13), one gets the following lemma.

Lemma 7.6. The restriction of the bundle endomorphism Q to any subbundle indecomposition (7.14) of TM ˝†M is given by

QjWr�1; NrM D 2.r C 2/Id; QjWrC1; NrM D �2rId;

QjWr�1;rC1M D 2.r C 2/Id; QjWrC1;r�1M D �2rId;

QjWr�1;r�1M D 2.r C 2/Id; QjWrC1;rC1M D �2rId:

Now let ‰ 2 �.†rM/. From (7.15) we obtain

Q.r‰/ D 2.r C 2/.r‰/r�1; Nr � 2r.r‰/rC1; Nr C 2.r C 2/.r‰/r�1;rC1� 2r.r‰/rC1;r�1 C 2.r C 2/.r‰/r�1;r�1 � 2r.r‰/rC1;rC1:

(7.19)

Hence,

jQ.r‰/j2 D 4.r C 2/2j.r‰/r�1; Nr j2 C 4r2j.r‰/rC1; Nr j2

C 4.r C 2/2j.r‰/r�1;rC1j2 C 4r2j.r‰/rC1;r�1j2

C 4.r C 2/2j.r‰/r�1;r�1j2 C 4r2j.r‰/rC1;rC1j2:

This further yields

j.r‰/r�1; Nr j2 C j.r‰/r�1;rC1j2 C j.r‰/r�1;r�1j2

D 1

4.r C 2/2jQ.r‰/j2

� r2

.r C 2/2.j.r‰/rC1; Nr j2 C j.r‰/rC1;r�1j2 C j.r‰/rC1;rC1j2/:

Hence, we may write equation (7.16) as

jr‰j2

D 1

4.r C 2/2jQ.r‰/j2

C 4.r C 1/

.r C 2/2.j.r‰/rC1; Nr j2 C j.r‰/rC1;r�1j2 C j.r‰/rC1;rC1j2/:

(7.20)


Note that, by Lemma 7.5, (7.20) reduces to (7.16) for ‰ 2 �.†0/.

On the other hand, the operator Q satis�es the following property.

Lemma 7.7. For any ‰ 2 �.†rM/, one has

Z

M

jQ.r‰/j2�g D 4r.r C 2/

Z

M

�jr‰j2 C Scal

2.mC 2/j‰j2

��g :

Proof. Let p be an arbitrary point in M . On a neighborhood U of p, let fe1; : : : ;e4mg be a local orthonormal frame of TM , such that

reiej .p/ D 0;

and let .J˛/ be a standard local frame of QM . For any ‰ 2 �.†r/, using (7.8),(7.12), and (7.10), at the point p, one gets

jQ.r‰/j2 DX

˛;ˇ;i;j

hJ˛ei ˝�˛ � rei‰; Jˇej ˝�ˇ � rej‰i

DX

˛;ˇ;i;j

hJ˛ei ; Jˇej ih�˛ � rei‰;�ˇ � rej‰i

DX

˛;ˇ;i;j

hei ; J˛Jˇ ej ihrei‰;�˛ ��ˇ � rej‰i

D �X

˛;i;j

ıij hrei‰;�˛ ��˛ � rej‰i

CX

˛;ˇ; ;i;j

hei ; J ej ihrei‰; "˛ˇ �˛ ��ˇ � rej‰i

D 4r.r C 2/jr‰j2 C 4X

i

Drei

‰;X

;j

hei ; J ej i� � rej‰E:

(7.21)

Now, since the expressionP˛ J˛.�/ ��˛ is parallel (it comes from an algebraic

(Sp1 � Spm)-invariant object), we may write the second term in the right-hand sideof (7.21) as

4X

i

ei

�D‰;X

;j

hei ; J ej i� �rej‰E�

�4D‰;X

;i;j

hei ; J ej i� �reirej‰

E: (7.22)

The �rst term in (7.22) being a divergence term, it gives zero by integration overM ,


so we only need to consider the second term. We have

� 4X

;i;j

hei ; J ej i� � reirej‰

D �2X

;i;j

hei ; J ej i� � reirej‰ � 2

X

;j;i

hej ; J ei i� � rej rei‰

D �2X

;i;j

hei ; J ej i� � .reirej � rej rei

/‰

D �2X

;i;j

hei ; J ej i� � Rei ;ej‰

D �2X

;j

� � RJ ej ;ej‰

D 1

2

X

;j;k;l

R.ej ; J ej ; ek; el/� � ek � el �‰:

Since the last expression is tensorial, we may now consider an adapted local frameon U , that is an orthonormal frame of the form

.e1; J1e1; J2e1; J3e1; : : : ; em; J1em; J2em; J3em/:

Since any ej has the form ˙J ei , the Bianchi identity gives

X

j;k

R.ej ; J ej /ek D 2X

i;k

R.ei ; ek/J ei :

Therefore,

�4X

;i;j

hei ; J ej i� � reirej‰ D

X

;i;k;l

R.ei ; ek; J ei ; el/� � ek � el � ‰:

Now, for any quaternion-Kähler manifold, the following basic equality holds;cf. for instance [Ish74], [Bes87], or, [HM95b]:2

ŒR.X; Y /; J˛� D Scal

4m.mC 2/

X

ˇ;

"123˛ˇ �ˇ .X; Y /J : (7.23)

2With a change of sign depending on the de�nition of the curvature.


From this equality, we deduce that

X

;i;k;l

R.ei ; ek; J ei ; el/� � ek � el �‰

DX

;i;k;l

hJ .R.ei ; ek/ei/; eli� � ek � el � ‰

C Scal

4m.mC 2/

X

;˛;ˇi;k;l

"123 ˛ˇ�˛.ei ; ek/hJˇei ; eli� � ek � el �‰

D �X

;i;k;l

hR.ei ; ek/ei ; J eli� � ek � el �‰

C Scal

4m.mC 2/

X

;˛;ˇi;k;l

"123 ˛ˇ hJ˛ei ; ekihJˇ ei ; eli� � ek � el � ‰

DX

;i;k;l

R.ei ; ek; J el ; ei/� � ek � el �‰

C Scal

4m.mC 2/

X

;˛;ˇ;i

"123 ˛ˇ� � J˛ei � Jˇei �‰

DX

;k;l

Ric.ek ; J el /� � ek � el �‰

C Scal

4m.mC 2/

X

;˛;ˇ;i

"123 ˛ˇ� � J˛ei � Jˇei �‰:

Using that the manifold is Einstein, and applying once again the fact that any eihas the form ˙J˛ej , we obtainX

;i;k;l

R.ei ; ek; J ei ; el/� � ek � el � ‰

DX

;k;l

Scal

4mhek ; J eli� � ek � el �‰

� Scal

4m.mC 2/

X

;˛;ˇ;i

"123 ˛ˇ� � ei � JˇJ˛ei �‰

DX

;l

Scal

4m� � J el � el �‰ C Scal

4m.mC 2/

X

;˛;ˇ;�;i

"123˛ˇ "123˛ˇ�� ei � J�ei � ‰:


Hence by (7.9), it follows thatX

;i;k;l

R.ei ; ek; J ei ; el/� � ek � el �‰

D �Scal

2m

X

� �� ‰ C Scal

m.mC 2/

X

� �� ‰

D � Scal

2.mC 2/

X

� �� ‰

D 4r.r C 2/Scal

2.mC 2/‰;

which yields the result. �

Plugging this last result into the integrated version of equation (7.20), we obtainZ

M

jr‰j2�g

D rScal

4.mC 2/

Z

M

j‰j2�g

C 2.r C 1/

r C 2

Z

M

.j.r‰/rC1; Nr j2 C j.r‰/rC1;r�1j2 C j.r‰/rC1;rC1j2/�g :(7.24)

This yields the following inequality:Z

M

jr‰j2�g � r Scal

4.mC 2/

Z

M

j‰j2�g � 2.r C 1/

r C 2

Z

M

j.r‰/rC1;rC1j2�g � 0:

But by Lemma 7.4, this inequality can be written asZ

M

jr‰j2�g � r Scal

4.mC 2/

Z

M

j‰j2�g � 1

mC r C 3

Z

M

jDC‰j2�g � 0:

The �nal “trick” consists in applying the equality above to a convenient spinor�eld ‰ in �.†rM/.

Let � be the �rst eigenvalue of the Dirac operator. Consider the space

E�M ´ f‰ 2 �.†M/WD2‰ D �2‰g:Since the action of the 4-form � on �.†M/ commutes with D2 (this results fromthe Schrödinger–Lichnerowicz formula), this space splits into the orthogonal de-composition

E�M DmM

rD0E�rM; E�rM ´ E�M \ �.†rM/:


Lemma 7.8. Let rmin be the smallest integer r for which E�rM ¤ f0g. For anynon-trivial ‰ 2 E�rmin

M , the following holds.

(i) D�‰ D 0, thus D‰ D DC‰.

(ii) ˆ ´ D‰ is a non-trivial spinor �eld in E�rminC1M , such that DCˆ D 0.Thus 0 � rmin � m � 1.

Proof. (i) If D�‰ ¤ 0, then it should be a non-trivial spinor in E�rmin�1M , contra-dicting the de�nition of rmin.

(ii) By the Lichnerowicz theorem, since the scalar curvature is assumed to bepositive, there is no harmonic spinor on M , thus ˆ is non-trivial. Furthermore,since D‰ D DC‰, ˆ 2 E�rminC1M , and since Dˆ D D2‰ D �2‰ 2 E�rmin

M ,one has DCˆ D 0. �

With the notations of Lemma 7.8, let ‰ be a non-trivial spinor �eld in E�rminM .

Since D�‰ D 0 and DC‰ D D‰, the above inequality givesZ

M

�mC rmin C 2

mC rmin C 3�2 � mC rmin C 2

mC 2

Scal

4

�j‰j2�g � 0:

Hence we obtain the inequality

�2 � mC rmin C 3

mC 2

Scal

4: (7.25)

Since 0 � rmin � m � 1, we conclude that

�2 � mC 3

mC 2

Scal

4:

7.4 The limiting case

Assume that the �rst eigenvalue � of the Dirac operator satis�es

�2 D mC 3

mC 2

Scal

4:

Then (7.25) implies that necessarily rmin D 0. By Lemma 7.8, there exists a spinor�eld ‰0 such that

‰0 2 �.†0M/ and D2‰0 D mC 3

mC 2

Scal

4‰0:


Moreover, the spinor �eld‰1 ´ D‰0

satis�es

‰1 2 �.†1M/ and D2‰1 D mC 3

mC 2

Scal

4‰1:

Now, the covariant derivatives of ‰0 and ‰1 can be obtained from the proof of thelower bound, and this is a �rst step in the study of the limiting case.

Applying equation (7.24) to the spinor �eld ‰0 givesZ

M

jr‰0j2�g DZ

M

.j.r‰0/1; N0j2 C j.r‰0/1;1j2/�g :

By Lemma 7.4, we get

j.r‰0/1;1j2 D 1

mC 3jD‰0j2;

soZ

M

j.r‰0/1;1j2�g D 1

mC 3

Z

M

jD‰0j2�g

D �2

mC 3

Z

M

j‰0j2�g

D 1

mC 2

Scal

4

Z

M

j‰0j2�g :

On the other hand, by the Schrödinger–Lichnerowicz formula,Z

M

jr‰0j2�g D��2 � Scal

4

� Z

M

j‰0j2�g

D 1

mC 2

Scal

4

Z

M

j‰0j2�g :

Therefore, Z

M

j.r‰0/1; N0j2�g D 0;

and so.r‰0/1; N0 D 0:

Hence, the splitting of r‰0 under decomposition (7.14) has only one component:

r‰0 D .r‰0/1;1:

Since the restriction of the Clifford multiplication to W1;1M gives an isomor-phism onto †1M , this suggests that one may express r‰0 as the inverse image of .r‰0/ D D‰0 D ‰1.


Lemma 7.9. For any vector �eldX , denoting by .X �‰1/0 the component ofX �‰1belonging to �.†0M/, one gets

rX‰0 D � 1

mC 3.X �‰1/0

D � 1

4.mC 3/

�X �‰1 C 1

2

X

˛

J˛X ��˛ �‰1�:

Proof. Let fe1; : : : ; e4mg, be a local orthonormal frame of TM . One has

X

i

ˇˇrei

‰0 C 1

mC 3.ei �‰1/0

ˇˇ2

DX

i

jrei‰0j2 C 2

mC 3ReX

i

hrei‰0; ei �‰1i

C 1

.mC 3/2

X

i

hei �‰1; .ei � ‰1/0i

D jr‰0j2 � 2

mC 3RehD‰0; ‰1i � 1

.mC 3/2

D‰1;

X

i

ei � .ei �‰1/0E

D jr‰0j2 � 2

mC 3j‰1j2 � 1

.mC 3/2

D‰1;

X

i

ei � .ei �‰1/0E:

Now, denoting .ei �‰1/2 the component of ei �‰1 belonging to �.†2M/, one getsby (7.12)

� � ei �‰1 D 6m.ei �‰1/0 C .6m � 32/.ei �‰1/2:

Hence

.ei �‰1/0 D 1

32..32� 6m/ei �‰1 C� � ei �‰1/:

Using the fact thatX

i

ei �� ei D .8 � 4m/�;

this impliesX

i

ei � .ei �‰1/0 D �.mC 3/‰1:


Hence, since

jr‰0j2 D j.r‰0/1;1j2 D 1

mC 3j‰1j2;

one obtainsX

i

ˇˇrei

‰0 C 1

mC 3.ei �‰1/0

ˇˇ2

D 0;

hence the �rst result. To conclude the second one, it is easy to verify that

.X �‰1/0 D 1

32..32 � 6m/X � ‰1 C� � X � ‰1/

D 1

32.20X �‰1 C Œ�;X� � ‰1/

D 1

4

�X �‰1 C 1

2

X

˛

J˛X ��˛ �‰1�: �

We now attempt to apply the above arguments to the spinor �eld ‰1. Usingequation (7.24) for the spinor �eld ‰1 yields

Z

M

jr‰1j2�g

D Scal

4.mC 2/

Z

M

j‰1j2�g

C 4

3

Z

M

.j.r‰1/0; N1j2 C j.r‰1/2;0j2 C j.r‰1/2;2j2/�g :

Since the left-hand side of the above equation is equal to

Scal

4.mC 2/

Z

M

j‰1j2�g ;

this implies that

.r‰1/0; N1 D .r‰1/2;0 D .r‰1/2;2 D 0;

which together with (7.19) gives

Q.r‰1/ D 6r‰1;

hence implying the following result.


Lemma 7.10. On the domain of de�nition of a standard local frame .J˛/ of QM ,let D˛ , ˛ D 1; 2; 3; be the local operators de�ned by

D˛ D4mX

iD1J˛ei � rei

; (7.26)

where .ei / is a local orthonormal frame of the bundle TM . Then

X

˛

�˛ � D˛‰1 D 0 (7.27)

and

8D˛‰1 DX

ˇ;

"123˛ˇ �ˇ � D ‰1: (7.28)

Proof. Since .Q.r‰1// D 6 .r‰1/ D 6D‰1, we get

X

˛;i

J˛ei ��˛ � rei‰1 D 6D‰1:

For any spinor �eld ‰, any tangent vector �eld X , and for ˛ D 1; 2; 3, it is easy tosee that

�˛ �X �‰ D X ��˛ �‰ C 2J˛X � ‰: (7.29)

This yields X�˛ � D˛‰1 C 6D‰1 D 6D‰1;

hence (7.27).To prove (7.28), we �rst express locally the equality Q.r‰1/ D 6r‰1:

X

˛

X

i

J˛ei ˝�˛ � rei‰1 D 6

X

i

ei ˝ rei‰1:

Since for any ˛ D 1; 2; 3, the sumPi J˛ei ˝�˛ � rei

‰1 is a tensorial expressionin the ei , we may assume that the local frame .ei/ is adapted to the correspondingalmost structure J˛: each ej is up to a sign equal to some J˛ei . We get

X

i

J˛ei ˝�˛ � rei‰1 D �

X

i

ei ˝�˛ � rJ˛ei‰1:


Hence X

i

ei ˝�

�X

˛

�˛ � rJ˛ei‰1 � 6rei

‰1

�D 0;

so

rei‰1 D �1

6

X

˛

�˛ � rJ˛ei‰1:

We then obtain

6D˛‰1 D 6X

˛;i

J˛ei � rei‰1

D �X

˛;ˇ;i

J˛ei ��ˇ � rJˇei‰1

DX

˛;ˇ;i

J˛Jˇei ��ˇ � rei‰1

D �X

˛;i

ei ��˛ � rei‰1 C

X

˛;ˇ;i

"123˛ˇ J ei ��ˇ � rei‰1

D �X

˛

�˛ � D‰1 C 2X

˛

D˛‰1 CX

˛;ˇ;

"123˛ˇ �ˇ � D ‰1

� 2X

˛;ˇ;

"123˛ˇ JˇJ ei � rei‰1:

SinceD‰1 2 �.†0M/, we haveP˛�˛ �D‰1 D 0, by Lemma 7.5. Furthermore,

�2X

˛;ˇ;

"123˛ˇ JˇJ ei � rei‰1 D �2

X

˛;ˇ;

."123˛ˇ /2D˛‰1 D �4D˛‰1;

hence the result. �

Thus, the proof of the estimate provides the decomposition

r‰1 D .r‰1/2; N1 C .r‰1/0;2 C .r‰1/0;0;

where only the last term in the right-hand side may be expressed in terms of ‰0 byLemma 7.4.

To overcome this dif�culty, we consider the twisted bundle QM ˝ †M , andintroduce the operator

{W�.†M/ �! �.QM ˝†M/;

locally de�ned on the domain of a standard local frame .J˛/˛D1;2;3 ofQM by

‰ 7�!X

˛

J˛ ˝�˛ �‰:


Since this operator is induced by an .Sp1 � Spm/-equivariant algebraic operatorbetween the corresponding �bres, it preserves the decomposition of

†M DmM

rD0†rM I

i.e., we have

{W�.†rM/ �! �.QM ˝†rM/:

(this can also be obtained by using the commutation relations (7.11)).In particular, by Lemma 7.5,

{j�.†0M/ � 0:

Proposition 7.11. Let D be the twisted Dirac operator acting on sections of thetwisted bundleQM ˝†M . For any spinor �eld ‰ 2 �.†M/,

D.{‰/ D {.D‰/ � 2X

˛

J˛ ˝ D˛‰;

where the D˛ are the .local/ operators introduced in (7.26).

Proof. Being induced by an Sp1 � Spm-equivariant operator, the operator { is suchthat

r.{‰/ D {.r‰/; ‰ 2 �.†M/: (7.30)

Let fe1; : : : ; e4mg be a local orthonormal frame of TM , and .J˛/˛D1;2;3, be a stan-dard local frame of QM , both de�ned on the same domain. Using (7.29), for anyspinor �eld ‰ 2 �.†M/, one has

D.{‰/ DX

i

ei � rei.{‰/

DX

i

ei � {.rei‰/

DX

i

ei ��X

˛

J˛ ˝�˛ � rei‰�

DX

˛

J˛ ˝�X

i

ei ��˛ � rei‰�

D {.D‰/ � 2X

˛

J˛ ˝ D˛‰: �


This and the fact that D‰1 2 �.†0M/ yield

D.{‰1/ D �2X

˛

J˛ ˝ D˛‰1: (7.31)

Now let be the Clifford multiplication acting on sections ofQM ˝†M .By the properties of the Clifford multiplication acting on sections of †rM , the

restriction of to TM ˝ .QM ˝†1M/ takes values in QM ˝ .†0M ˚†2M/.Using the Clebsch–Gordan formulas, this last bundle has a �ber isomorphic to

Sym2H ˝ .ƒmı ˚ .Sym2H ˝ƒm�2ı //

' .Sym2H ˝ƒmı /˚ .Sym4H ˝ƒm�2ı /˚ .Sym2H ˝ƒm�2

ı /˚ƒm�2ı :

Being induced by an Sp1 � Spm-equivariant operator, the operator ı .Id ˝ {/WTM ˝†1M �! QM ˝ .†0M ˚†2M/;

preserves the decomposition of r‰1 as r‰1 D .r‰1/2; N1C.r‰1/0;2C.r‰1/0;0,hence r‰1 and its image by ı .Id ˝ {/ must be sections of isomorphic bundles.Considering the decomposition above, this implies that

ı .Id ˝ {/ � .r‰1/0; N1 D 0;

ı .Id ˝ {/ � .r‰1/0;0 D 0;

andD.{‰1/ D ı .Id ˝ {/ � r‰1 D ı .Id ˝ {/ � .r‰1/0;2:

Therefore, D.{‰1/ is a section of a bundle with �ber isomorphic to ƒm�2ı .

Now one can prove with the same arguments as in Lemma 7.4 the followingresult.

Lemma 7.12. For any X ˝‰ 2 �.TM ˝†1M/, one hasˇ ı .Id ˝ {/ � .X ˝‰/

ˇ2 D 16.mC 4/jX ˝‰j2:This implies the relation

jD.{‰1/j2 D 4X

˛

jD˛‰1j2 D 16.mC 4/j.r‰1/0;2j2: (7.32)

Lemma 7.13. We haveZ

M

jD.{‰1/j2�g D 4

Z

M

X

˛

jD˛‰1j2�g

D��2 � Scal

m.mC 2/

� Z

M

j{‰1j2�g

D 12.mC 4/.m � 1/m.mC 2/

Scal

4

Z

M

j‰1j2�g :


Proof. The �rst equality follows from (7.31). To prove the second one, one observesthat Z

M

jD.{‰1/j2�g DZ

M

hD2.{‰1/; {‰1i�g :

But by the Schrödinger–Lichnerowicz formula for twisted bundles,

D2.{‰1/ D rr�.{‰1/C Scal

4{‰1 C RQM � {‰1;

whereRQM denotes the action of the curvatureRQM of the bundleQM on sectionsof the twisted bundle QM ˝†M (cf. Theorem 8.17 in [LM89]).

By (7.30),

rr�.{‰1/ D {.rr�‰1/;

hence

rr�.{‰1/C Scal

4{‰1 D �2{‰1:

On the other hand,

RQM � {‰1 D 1

2

X

˛

X

i;j

.RQM .ei ; ej /J˛/˝ ei � ej ��˛ �‰1:

But since QM is a subbundle of the bundle End.TM/, its curvature RQM is givenby

RQM .X; Y /J D ŒR.X; Y /; J �; X; Y 2 �.TM/; J 2 �.QM/:

Hence, using (7.23), one gets

1

2

X

˛

X

i;j

.RQM .ei ; ej /J˛/˝ ei � ej ��˛ � ‰1

D Scal

4m.mC 2/

X

˛;ˇ;

"123˛ˇ J ˝�ˇ ��˛‰1

D � Scal

m.mC 2/

X

J ˝� ‰1 (using (7.10))

D � Scal

m.mC 2/{‰1:


Therefore,

D2.{‰1/ D��2 � Scal

m.mC 2/

�{‰1: (7.33)

Moreover,

j{‰1j2 DX

˛

h�˛ �‰1; �˛ �‰1i D �DX

˛

�˛ ��˛‰1; ‰1E

D 12j‰1j2;

hence the result. �

From (7.32) we then haveZ

M

j.r‰1/0;2j2�g D 3.m � 1/

4m.mC 2/

Scal

4

Z

M

j‰1j2�g :

On the other hand, using Lemma 7.4 and the fact that D�‰1 D D‰1, we also haveZ

M

j.r‰1/0;0j2�g D mC 3

4m.mC 2/

Scal

4

Z

M

j‰1j2�g :

This impliesZ

M

�j.r‰1/0;2j2 C j.r‰1/0;0j2

��g

D 1

mC 2

Scal

4

Z

M

j‰1j2�g

DZ

M

jr‰1j2�g

DZ

M

.j.r‰1/2; N1j2 C j.r‰1/0;2j2 C j.r‰1/0;0j2/�g :

Consequently, Z

M

j.r‰1/2; N1j2�g D 0;

and so.r‰1/2; N1 D 0:

The splitting of r‰1 under the decomposition (7.14) is therefore given by

r‰1 D .r‰1/0;2 C .r‰1/0;0:

This suggests that the covariant derivative of ‰1 may be expressed in terms of thespinor �elds D‰1 D �2‰0 and D.{‰1/ D

P˛ J˛ ˝ D˛‰1.


Lemma 7.14. For any vector �eld X ,

rX‰1 D � �2

4mX �‰0 � 1

4.mC 4/

X

˛

J˛X � D˛‰1:

Proof. Let fe1; : : : ; e4mg be a local orthonormal frame of TM , and .J˛/˛D1;2;3, bea standard local frame of QM , both de�ned on the same domain. The proof relieson the computation of

X

i

ˇˇˇrei

‰1 C �2

4mei �‰0 C 1

4.mC 4/

X

˛

J˛ei � D˛‰1

ˇˇˇ

2

:

One has

2ReX

i

hrei‰1; ei �‰0i D �2RehD‰1; ‰0i D �2�2j‰0j2;

and

2ReX

i

Drei

‰1;X

˛

J˛ei � D˛‰1E

D �2X

˛

jD˛‰1j2:

By (7.27) in Lemma 7.10, one gets

2ReX

i

Dei � ‰0;

X

˛

J˛ei � D˛‰1E

D �4D‰0;

X

˛

�˛ � D˛‰1E

D 0:

Furthermore, using (7.28) in Lemma 7.10, it follows that

X

i

DX

˛

J˛ei � D˛‰1;X

ˇ

Jˇei � Dˇ‰1E

D �X

˛

DD˛‰1;

X

ˇ;i

J˛ei � Jˇei � Dˇ‰1E

DX

˛

DD˛‰1;

X

ˇ;i

ei � JˇJ˛ei � Dˇ‰1E

D 4mX

˛

jD˛‰1j2 C 2X

˛

DD˛‰1;

X

ˇ;

"123˛ˇ �ˇ � D ‰1E

D 4.mC 4/X

˛

jD˛‰1j2:


Finally,

X

i

ˇˇˇrei

‰1 C �2

4mei �‰0 C 1

4.mC 4/

X

˛

J˛ei � D˛‰1

ˇˇˇ

2

D j.r‰1/0;2j2 C j.r‰1/0;0j2 � �4

4mj‰0j2 � 1

4.mC 4/

X

˛

jD˛‰1j2

D 0;

by (7.32) and Lemma 7.4. �

Remark 7.15. Since ‰0 2 �.†0M/ and {j�.†0M/ � 0, one has D.{‰0/ D 0.On the domain of any standard frame .J˛/˛D1;2;3, by Proposition 7.11, this impliesthat

�˛ �‰1 D 2D˛‰0:

Hence, for any vector �eld X , the covariant derivative of the spinor �elds ‰0 and‰1 may be written as

rX‰0 D � 1

4.mC 3/X � D‰0 � 1

4.mC 3/

X

˛

J˛X � D˛‰0;

rX‰1 D � 1

4mX � D‰1 � 1

4.mC 4/

X

˛

J˛X � D˛‰1:

Note that those equations may be seen as quaternion-Kähler analogues of theKählerian twistor-spinors equations.

So the limiting case of the lower bound provides two spinor �elds‰0 2 �.†0M/

and ‰1 2 �.†1M/ such that

‰ ´ �‰0 C‰1

is an eigenspinor of the Dirac operator for the lowest eigenvalue

� DrmC 3

mC 2

Scal

4;

together with the two sections of the twisted bundleQM˝†M , {.‰1/ andD.{‰1/,such that

z‰ ´ �{.‰1/C D.{‰1/

is an eigenspinor of the twisted Dirac operator for the eigenvalue

� Ds.mC 4/.m� 1/m.mC 2/

Scal

4:


Wewill see in Chapter 11 that‰0,‰1, {.‰1/, andD.{‰1/may be identi�ed withspinors on the canonical 3-Sasakian SO3-principal bundle associated with QM ,and that a convenient combination of these spinor �elds will give rise to a non-trivial Killing spinor on that bundle. This will imply that the only limiting positivequaternion-Kähler manifolds are the quaternionic projective spaces.

7.5 A systematic approach

The general approach presented below is a powerful tool that allows one to pro-vide not only simple proofs of eigenvalue estimates for the Dirac and the Laplaceoperators, but also results on harmonic forms giving vanishing theorems for Bettinumbers, both for positive and negative scalar curvature. It is based on the followingformulation of the Weitzenböck decompositions of Laplace operators (cf. Section I,Chapter 1 in [Bes87]).

7.5.1 General Weitzenböck formulas

Let .M n; g/ be an orientable n-dimensional Riemannian manifold. It is well knownthat the representation

AdW SOn �! GL.son/

can be identi�ed with the 2-wedge product of the standard representation

ƒ2�stdW SOn �! GL.ƒ2.Rn//;

(the vector spaceƒ2.Rn/ being endowed with its natural scalar product h; i), by theSOn-invariant isomorphism

ƒ2.Rn/ �! son;

v ^w 7�! .x 7! hx; viw � hx; wiv/:

The above identi�cation allows to associate to any representation .�; V / of SOnthe SOn-equivariant map

c�Wƒ2Rn ' son��! End.V /:

This induces a morphism of vector bundles, also denoted by c�,

c�Wƒ2M �! End.E�/;

where E� is the associated vector bundle corresponding to the representation �.

7.5. A systematic approach 203

Now let c2� be the SOn-equivariant map de�ned by the following composition

ƒ2Rn ˝ƒ2Rnc�˝c��! End.V /˝ End.V /

ı7�! End.V /;

the secondmap being the composition of endomorphisms. This induces a morphismof vector bundles, also denoted by c2� ,

c2� Wƒ2M ˝ƒ2M �! End.E�/:

Considering the curvature tensor R as a section of the bundle ƒ2M ˝ ƒ2M , onethus gets a section c2�.R/ of the bundle End.E�/.

If .ei / is a local orthonormal frame, then c2�.R/ is locally de�ned by the formula

c2�.R/ D 1

4

X

i;j;k;l

R.ei ; ej ; ek; el/c�.ei ^ ej / ı c�.ek ^ el/: (7.34)

Note that, using the isomorphism son ' spinn, the same construction holds, atleast locally (globally if the manifold is spin), for any representation of Spinn.If for instance, one considers the spin representation �†, then the Schrödinger–Lichnerowicz formula can be written as

D2 D r�r C 2c2�†.R/: (7.35)

Consider now the twisted Dirac operator acting on sections of the vector bundle†M ˝ E�, where E� is now the associated vector bundle (may be locally de-�ned as well as the bundle †M ) corresponding to a representation � of Spinn.The Schrödinger–Lichnerowicz formula is then given by

D2 D r�r C Scal

4Id†M ˝ IdE� C RE� ;

where the operator RE� acts, via the curvature RE� of the bundle E�, on decom-posable tensors of the form ' ˝ s by

RE� .' ˝ s/ ´ 1

2

X

i;j

ei � ej � ' ˝RE�.ei ; ej /s;

(cf. Theorem 8.17 in [LM89]). It is easy to see that

RE� D c2�†˝�.R/� c2�†.R/˝ IdE� � Id†M ˝ c2�.R/;

hence using

c2�†.R/ D Scal

8Id†M ;


one gets

D2 D r�r C c2�†˝�.R/C Scal

8Id†M ˝ IdE� � Id†M ˝ c2� .R/: (7.36)

In particular, choose for � the spin representation �†. Recall the equivalence of thefollowing SOn-representations:

�† ˝ �† ' �ƒ ´

8ˆˆ<ˆˆ:

nM

pD0ƒp�C

std; for n even,

Œn2 �M

pD0ƒ2p�C

std 'Œn

2 �M

pD0ƒ2pC1�C

std; for n odd,

cf. for instance Corollary 5.19 in [LM89].Since D2 is the Hodge Laplacian � acting on

nM

pD0ƒpT�

CM for n even;

and

Œn2 �M

pD0ƒ2pT�

CM 'Œn

2 �M

pD0ƒ2pC1T�

CM for n odd,

cf. [ABS64], [Hit74], and [LM89], one deduces immediately from (7.36) and

c2�†.R/ D Scal

8Id†M ;

that� D r�r C c2�ƒ

.R/: (7.37)

Of course, this formula can also be stated directly; it is a basic example in [SW01]and [SW02].

We now assume that the holonomy group of .M n; g/ is contained in a closedsubgroup H of SOn.

This implies the existence of a reductionP of the bundlePSOnM to the groupH ,compatible with the Levi-Civita connection (that is, such that the Levi-Civitaconnection reduces to a connection on P ).

Under the isomorphism SOn ' ƒ2Rn above, the adjoint representation of Hon its Lie algebra h � son can be identi�ed with a representation on a subspaceƒ2hR

n of ƒ2Rn. Let ƒ2hM be the vector bundle associated with P by this repre-sentation. It is well known that the holonomy condition implies that the curvature


tensor R, viewed as an operator ƒ2M ! ƒ2M , takes values in ƒ2hM , and sinceR is a symmetric map, it can in fact be viewed as an operator ƒ2hM ! ƒ2hM .Consequently, one may de�ne, as above, for any representation .�; V / of H ,a section c2�.R/ of the bundle End.E�/, where E� is the vector bundle associatedwith P by the representation �. Here again, the same construction holds, at leastlocally, if � is a representation of some �nite covering ofH .

Now, the decomposition of the representation �ƒ (resp. the spin representa-tion �†) into irreducible components under the action of H (or possibly some�nite covering ofH ) induces a global decomposition of the vector bundleƒ�T�

CM

(resp. †M ) into parallel subbundles.By the property of the curvature relative to the holonomy condition, it is clear

from theWeitzenböck decompositions (7.37) and (7.35) that the Hodge Laplacian�(resp. the square of the Dirac operator D2) preserves this decomposition.

The main ingredient of the approach consists in a comparison between therestriction of � (resp. D2) to these parallel subbundles and the following second-order differential operators.

Definition 7.16. Let .�; V / be a representation of H and let E� be the associatedvector bundle. The (elliptic) second-order differential operator �� de�ned on thespace �.E�/ of sections of E� by

�� ´ r�r C c2�.R/; (7.38)

is called the canonical Laplacian induced by the representation.

Remark 7.17. Indeed there is a geometric reason to consider this operator insteadof the rough Laplacian: for a symmetric space G=H , differential forms or spinor�elds may be identi�ed with particular functions on the group G, and under thisidenti�cation, the operator �� corresponds to a Casimir operator on G (see Chap-ter 15).

Now, let .�; V / be an irreducible complex representation ofH and letE� be theassociated vector bundle.

The representation � is “contained” in the representation �ƒ (with the terminol-ogy of representation theory) if and only if the space HomH .V;ƒ�

Cn/ ofH -equiv-

ariant C-linear homomorphisms V ! ƒ�Cn is non-trivial. Each non-trivial vec-tor … 2 HomH .V;ƒ�

Cn/ induces a parallel embedding E� ,! ƒ�T�

CM , also

denoted by….Considering the canonical Laplacian ��, one gets easily from the Weitzenböck

decomposition (7.37) that

� ı… D … ı��: (7.39)


Analogously, let � be an irreducible complex representation of H (or eventuallysome �nite covering ofH ) contained in the spin representation �†. This induces aparallel embedding…WE� ,! †M . It is immediate from the Weitzenböck decom-position (7.35) that

D2 ı… D … ı�� C Scal

8…: (7.40)

Note that equation (7.39) (resp. (7.40)) implies not only that the restriction of �(resp. D2) to any of the parallel subbundles considered above may be de�ned byconsidering some parallel embedding of the form …WE� ,! ƒ�T�

CM (resp. of the

form…WE� ,! †M ), but also that this restriction does not depend on this speci�cembedding.

Consider now the twisted Dirac operatorD acting on sections of the vector bun-dle†M˝E�, whereE� is the associated vector bundle corresponding to a complexrepresentation (not necessarily irreducible) � of H (or possibly a �nite coveringofH ).

Because of the last term Id†M˝c2�.R/ in theWeitzenböck decomposition (7.36),the operator D2 does not in general preserve the decomposition of †M ˝E� intothe sum of parallel subbundles corresponding to the decomposition of the represen-tation �† ˝ � into irreducible components under the action ofH (or possibly some�nite covering of H ). However, if this term acts by scalar multiplication, then D2

preserves this decomposition and satis�es the same properties relative to parallelembeddings as the Hodge Laplacian or the square of the untwisted Dirac opera-tor. This situation occurs in the quaternion-Kähler setting for “natural” twists of thebundle †M , providing the key formula of this approach.

7.5.2 Application to quaternion-Kähler manifolds

Let P be a reduction of the bundle PSOnM to the group Sp1 � Spm, compatible withthe Levi-Civita connection. A vector bundle associated with P via a (complex)representation �W Sp1 � Spm ! GLC.V / is denoted by PV .

Let � W Sp1 ! GLC.H/ and �W Spm ! GLC.E/ be the standard complex rep-resentations. Any irreducible representation of the group Sp1� Spm is containedin a tensor product of the form .˝pH/ ˝ .˝qE/ (cf. Proposition 12.9 and Theo-rem 12.15), and then factors through the group Sp1 � Spm D Sp1 �Z2

Spm if andonly if p C q is even. Moreover, the representation inherits a real structure in thiscase, induced by the quaternionic structures of � and �.

With the notations of Section 7.2, we consider the (irreducible) representations

�kW Sp1� Spm �! Sp1 �! GLC.SymkH/

and

�`W Sp1� Spm �! Spm �! GLC.ƒ`ıE/;


and denote their tensor product by

�k;` ´ �k ˝ �`W Sp1 � Spm �! SymkH ˝ƒ`ıE:

Note that the associated vector bundle PSymkH˝ƒ`ıE

is well de�ned if k C `

is even. However, even this is not the case, its tensor product with another not well-de�ned bundle, may produce a well-de�ned bundle, hence we do not pay attentionto this condition in what follows. Note moreover that, as noted in Section 7.2, thespin condition ensures that one can lift P to a Sp1 � Spm-principal bundle zP , hencein any case, we get well-de�ned bundles under that condition.

Lemma 7.18 ([SW01] and [SW02]). For each representation of the form �k;`,the endomorphism c2�k;`

.R/ acts by scalar multiplication on the vector bundlePSymkH˝ƒ`

ıE. More precisely

c2�k;`.R/ D Scal

8m.mC 2/.k.k C 2/C `.2m � `C 2//IdP

Symk H˝ƒ`ıE:

Proof. We only sketch the proof (see [SW01] and [SW02] for details). It is basedon the decomposition of the curvature tensor R into irreducible components underthe action of the group Sp1 � Spm; see [Ale68] and [Sal82].

First, the two complex irreducible representations AdCW Spm ! GLC.spm;C/

and Spm ! GLC.Sym2E/ are equivalent. To see this, recall that the complexi�edLie algebra spm;C is the Lie algebra of the group Sp2m;C, (cf. (12.28)), so it can beidenti�ed with the space Sym2E by the map

Sym2E ,�! E ˝EŠ!EE ˝E� D End.E/;

x _ y 7�! x ˝ y C y ˝ x Š !E .x; �/y C !E .y; �/x;(7.41)

the space E being identi�ed with its dual E� by means of the standard symplecticform !E . In the same way, the complexi�ed Lie algebra sp1;C is identi�ed withSym2H .

Hence the curvature tensor R may be seen as a section of the bundle

Sym2.PSym2H ˚ PSym2E /;

de�ned by the representation of Sp1� Spm in the space

Sym2.Sym2H ˚ Sym2E/:

But this space splits into

Sym2.Sym2H ˚ Sym2E/

D Sym2.Sym2H/˚ Sym2H _ Sym2E ˚ Sym2.Sym2E/;


where _ is the symmetrized product. Note that this is not a decomposition into irre-ducible parts. It is easy to see that the irreducible decomposition of Sym2.Sym2H/has a unique one-dimensional factor. Since the C-linear extension BC

sp1of the

Killing form of the Lie algebra sp1 is Sp1-invariant, this factor is actually CBCsp1

.Now the symplectic form !H induces an Sp1-invariant symmetric form BH onSym2H , using Gram’s determinant. Thus BC

sp1is a scalar multiple of BH . Choos-

ing a particular vector in Sym2H , it follows that BCsp1

D 8BH .In the same way, it can be shown that the Spm-invariant symmetric form BE

on Sym2E, induced by the symplectic form !E using Gram’s determinant, is ascalar multiple of the C-linear extension BC

spmof the Killing form of spm and that

BCspm

D 4.mC 1/BE .Now the invariant symmetric bilinear formBH , (resp.BE ) induces a sectionRH

(resp. BE ) of the bundle PSym2.Sym2H/, (resp. PSym2.Sym2E/), obtained by paralleltransport along the �bers.

It then turns out thatR admits the following decomposition (see [Ale68], [Sal82],[KSW99], [SW01], and [SW02]):

R D � Scal

8m.mC 2/.RH CRE /CRhyper;

where Rhyper is a section of the bundle PSym2.Sym2E/. (The notation “Rhyper” comesfrom the fact that this part of the curvature tensor behaves like the curvature tensorof a Riemannian manifold with holonomy contained in Spm, that is a hyper-Kählermanifold).

Since, by (7.34), the expression c2�k;`.R/ depends linearly on R, one gets

c2�k;`.R/ D � Scal

8m.mC 2/.c2�k;`

.RH /C c2�k;`.RE //C c2�k;`

.Rhyper/:

Now because BCsp1

D 8BH , it is easy to see that c2�k;`.BH / is nothing else but

�8Cassp1.�k/, where Cassp1

.�k/ is the Casimir operator of the representation �kof Sp1. It is known that Cassp1

.�k/ acts by scalar multiplication on the irreducibleSp1-space Sym

kH . Furthermore, this Casimir eigenvalue can be computed in termsof the dominant weight of the representation by means of the Freudenthal formula(cf. the comments following Theorem 15.10 for details). It is easy to verify that

Cassp1.�k/jSymkH D 1

8k.k C 2/IdSymkH :

Hence,

c2�k;`.RH / D �k.k C 2/IdP

SymkH˝ƒ`ıE:


Using analogous arguments, one can show that

c2�k;`.BE / D �4.mC 1/Casspm

.�`/

and

Casspm.�`/jƒ`

ıED 1

4.mC 1/`.2m� `C 2/Idƒ`

ıE;

giving

c2�k;`.RE / D �`.2m � `C 2/IdP

Symk H˝ƒ`ıE:

Finally, we claim that the last component c2�k;`.Rhyper/ acts trivially on sections

of PSymkH˝ƒ`ıE

. To see this, it is suf�cient to show that c2�`.w/ acts trivially on

ƒ`ıE, for anyw 2 Sym2.Sym2E/. Since anyw 2 Sym2.Sym2E/ can be expressedas a sum of terms of the form e4, e 2 E, it is indeed suf�cient to check that c2�`

.e4/

acts trivially on ƒ`ıE, for any e 2 E. Now, using the isomorphism Sym2E 'spm;C, given in (7.41), the action of c2�`

.e4/ on ƒ`ıE is, up to a scalar,

c2�`.e4/ D .e ^ e]³/ ı .e ^ e]³/;

where ] is the isomorphism E ! E� induced by the symplectic form !E . So

c2�`.e4/ D �.e ^ e^/ ı .e] ³ e]³/ D 0;

which completes the proof. �

Thus, the following result can be deduced from (7.36).

Proposition 7.19 ([SW01] and [SW02]). Let Dk;` be the twisted Dirac operatoracting on sections of the bundle †M ˝ PSymkH˝ƒ`

ıE.

The square D2k;`

of this operator preserves the decomposition of the bundle†M ˝ PSymkH˝ƒ`

ıEinto parallel subbundles corresponding to the decomposi-

tion of the representation �† ˝ �k;`, into irreducible components under the actionof Sp1� Spm.

Moreover, if PE% is the vector bundle de�ned by an irreducible representation %contained in �† ˝ �k;`, the restriction D2

k;`j% of D2

k;`to sections of the subbundle

given by the parallel embedding PE% ,! †M ˝PSymkH˝ƒ`ıE

does not depend onthis speci�c embedding and satis�es (up to the obvious identi�cation of the spaceof sections of PE% with the space of sections of the embedding)

�% D D2k;`j% C Scal

8m.mC 2/.k C ` �m/.k � `CmC 2/Id; (7.42)

where �% is the canonical Laplacian induced by the representation %; see (7.38).


7.5.3 Proof of the estimate

With the notations of Section 7.2, for r D 0; : : : ; m, let

†rM ' PSymrH ˝ Pƒm�rı E

be one of the parallel subbundles of †M ; see (7.7). Suppose for the moment thatthere exists a parallel embedding†rM ,! †M˝PSymkH˝ƒ`

ıE. Then, combining

formulas (7.40) and (7.42), one gets

D2j†r D D2Rk;` j†r C Scal

8m.mC 2/.k2 C 2k � `2 C 2`.mC 1//: (7.43)

Thus if ‰ 2 �.†r/ is an eigenspinor of D2 with eigenvalue �2, then

�2 � Scal

8m.mC 2/.k2 C 2k � `2 C 2`.mC 1//:

To obtain the best estimate in this way, one has to determine the twistsPSymkH˝ƒ`ıE

such that there is a parallel embedding †rM ,! †M ˝ PSymkH˝ƒ`ıE

, (the ad-missible twists, in the terminology of [SW01] and [SW02]), and among those par-ticular twists, the twist(s) for which the last term in the right-hand side of equa-tion (7.43) is maximal (themaximal twist(s) for†rM , in the terminology of [SW01]and [SW02]).

This is a purely algebraic problem. As a �rst step, one has to determine the(algebraic) twists SymkH ˝ ƒ`ıE such that the irreducible decomposition of thespace†4m ˝ .SymkH ˝ƒ`ıE/ under the group Sp1 � Spm contains the irreduciblemodule SymrH ˝ ƒm�r

ı E. Then one has to determine among these twists, thosefor which the expression k2 C 2k � `2 C 2`.mC 1/ is maximal.

In [SW01] and [SW02], U. Semmelmann and G. Weingart give a characteriza-tion of the admissible twists of the irreducible representations SymkH ˝ ƒ

a;b> E,

whereƒa;b> E is the irreducible representation of highest weight in the tensor prod-uct3 ƒaıE ˝ƒbıE, and classify the maximal twists of those irreducible representa-tions. We will not give (even a sketch of) the proof here, since it is rather long andrequires some non-trivial knowledge on the representation theory of the group Spm.

It turns out that the maximal twist for †rM D PSymrH ˝ Pƒm�rı E is

PSymmCrH˝ƒm�rı E :

In this case equation (7.43) reads

D2j†r D D2mCr;m�r j†r C mC r C 2

mC 2

Scal

4:

3This irreducible representation is often called the Cartan summand of the tensor product.


Now let � be the �rst eigenvalue of the Dirac operator. With the notations ofLemma 7.8, let ‰ be a non-trivial spinor �eld in E�rmin

M . Since

ˆ ´ D‰ 2 �.†rminC1/

is a non-trivial eigenspinor ofD2 for the eigenvalue �2, one deduces from the aboveequation that

�2 � mC rmin C 3

mC 2

Scal

4:

Since rmin 2 f0; : : : ; m� 1g, it follows that

�2 � mC 3

mC 2

Scal

4:

Part III

Special spinor fields

and geometries

Chapter 8

Special spinors on Riemannian manifolds

8.1 Parallel spinors on spin and Spinc manifolds

In this section we study spin and Spinc manifolds carrying parallel spinors. Thelocal structure of these manifolds is now completely understood (see [Wan89]and [Mor97b]), but the global situation is, to our knowledge, far from being solved.In the locally irreducible case, however, the complete list of Riemannian mani-folds admitting a spin structure with parallel spinors is available in terms of theirholonomy groups (see [Wan95] and [MS00]). The similar problem in the conformalsetting will be studied in Section 9.2 below.

8.1.1 Parallel spinors on spin manifolds

Let .M n; g/ be a spin manifold with (complex) spinor bundle†M , and letr denotethe Levi-Civita covariant derivative on TM , as well as that on†M . Parallel spinorsare sections ‰ of†M satisfying the differential equation r‰ � 0. They obviouslycorrespond to �xed points (in †n) of the restriction of the spin representation �n tothe spin holonomy group eHol.M/ � Spinn.

We consider �rst the local problem. A spin manifoldM is said to admit a localparallel spinor if its universal cover has a (globally de�ned) parallel spinor. Usingthe Berger–Simons Holonomy theorem, Hitchin [Hit74] and later onWang [Wan89]obtained the following result.

Theorem 8.1. Let .M n; g/ be a locally irreducible spin manifold of dimensionn � 2. ThenM carries a local parallel spinor if and only if its local holonomy groupHol0.M; g/ is one of the following: G2 .n D 7/, Spin7 .n D 8/, SUm .n D 2m/,Spk .n D 4k/.

Proof. IfM carries a local parallel spinor, it cannot be locally symmetric. Indeed,M is Ricci-�at by Corollary 2.8, and Ricci-�at locally symmetric manifolds are �at.This would contradict the irreducibility hypothesis. One can thus use the Berger–Simons theorem, which states that the local holonomy group of M belongs to thefollowing list: G2 .n D 7/, Spin7 .n D 8/, SUm .n D 2m/, Spk .n D 4k/,Um .n D 2m/, Sp1 � Spk .n D 4k/, SOn.

216 8. Special spinors on Riemannian manifolds

On the other hand,M carries a local parallel spinor if and only if there exists a�xed point in †n of the restricted spin holonomy group eHol0.M/, which is equiv-alent to the existence of a vector 2 †n on which the Lie algebra fhol.M/ Dhol.M/ of eHol0.M/ acts trivially. It is an easy exercise of representation theory tocheck that this holds exactly for the �rst four groups from the Berger–Simons list(cf. [Wan89]). �

We now study the problem of the existence of global parallel spinors on a locallyirreducible spin manifold M . If M is simply connected, the answer is of coursegiven by the previous theorem. As we will need this result later on, we restate it,for the reader’s convenience.

Theorem 8.2. Let .M n; g/ be a simply connected irreducible spin manifold ofdimension n � 2. ThenM carries a parallel spinor if and only if its holonomy groupHol.M; g/ is one of the following: G2 .n D 7/, Spin7 .n D 8/, SUm .n D 2m/,Spk .n D 4k/.

In the non-simply connected case, the �rst step is to obtain some algebraic re-strictions on the holonomy group of M . In [Wan95], Wang obtains the list of allpossible holonomy groups of irreducible Ricci-�at manifolds (which can be consid-erably reduced ifM is compact; see [McI91]). Then, the following simple observa-tion ofWang (whichwe state from a slightly different point of view, more convenientfor our purposes), gives a criterion for a subgroup of SOn to be the holonomy groupof an n-dimensional manifold with parallel spinors:

Lemma 8.3. Let .M n; g/ be a spin manifold admitting a parallel spinor. Then thereexists an embedding

Q�WHol.M/ �! Spinn

such that

� ı Q� D IdHol.M/;

where

�W Spinn �! SOn :

Moreover, the restriction of the spin representation to Q�.Hol.M// has a �xed pointon †n.

Using this criterion, a case by case analysis yields

Theorem 8.4 ([Wan95]). Let .M n; g/ be an irreducible Riemannian spin manifoldwhich is not simply connected. If M admits a non-trivial parallel spinor, then thefull holonomy group Hol.M/ belongs to those in Table 3.

8.1. Parallel spinors on spin and Spinc manifolds 217

Table 3

Hol0.M/ dim.M/ Hol.M/ N conditions

SUm 2m SUm 2

SUm ÌZ=2Z 1 m � 0.4/

Spm mC 1

Spm 4m Spm �Z=dZ .mC 1/=d d > 1; dodd; ddividesmC 1

Spm �� see [Wan95] m � 0.2/

Spin7 8 Spin7 1

G2 7 G2 1

Here � is either Z=2dZ .d > 1/, or an in�nite subgroup of U1 Ì Z=2Z, or abinary dihedral, tetrahedral, octahedral, or icosahedral group, and N denotes thedimension of the space of parallel spinors. If, moreover, M is compact, then onlythe possibilities described in Table 4 may occur.

Table 4

Hol0.M/ dim.M/ Hol.M/ N conditions

SUm 2m SUm 2 m odd

SUm ÌZ=2Z 1 m � 0.4/

Spm 4m Spm �Z=dZ .mC 1/=d d > 1; dodd; ddividesmC 1

G2 7 G2 1

Let us point out that the algebraic restrictions on the holonomy group given byWang theorem are actually suf�cient for the existence of a spin structure carryingparallel spinors. The main tool is the following converse of Lemma 8.3.

Lemma 8.5. Let .M n; g/ be a Riemannian manifold and suppose that there existsan embedding Q�WHol.M/ ! Spinn which makes the diagram

Spinn

�

��Hol.M/

Q�;;✈✈✈✈✈✈✈✈✈✈✈✈✈

�// SOn

commutative. Then M carries a spin structure whose holonomy group is exactlyQ�.Hol.M//, hence isomorphic to Hol.M/.


Proof. Let � be the inclusion of Hol.M/ into SOn and Q�WHol.M/ ! Spinn besuch that � ı Q� D �. We �x a frame u 2 PSOn

M and let P � PSOnM de-

note the holonomy bundle of M through u, which is a Hol.M/ principal bundle(see [KN63], Chapter 2). Then there is a canonical bundle isomorphism P��SOn 'PSOn

M and it is clear that P �Q� Spinn together with the canonical projection ontoP �� SOn, de�nes a spin structure on M . The spin connection comes of coursefrom the restriction to P of the Levi-Civita connection of M and hence the spinholonomy group is just Q�.Hol.M//, as claimed. �

Now recall that Table 3 was obtained in the following way: among all possibleholonomy groups of non-simply connected irreducible Ricci-�at Riemannian man-ifolds, one selects those whose holonomy group lifts isomorphically to Spinn andsuch that the spin representation has �xed points when restricted to this lift. UsingLemma 8.5 we then deduce the converse of Theorem 8.4.

Theorem 8.6. An oriented non-simply connected irreducible Riemannian mani-fold has a spin structure carrying parallel spinors if and only if its Riemannianholonomy group appears in Table 3 (or, equivalently, if it satis�es the conditions inLemma 8.3).

There is still an important point to be clari�ed here. Let G D Hol.M/ be theholonomy group of a manifold such that G belongs to Table 3. Suppose that thereare two lifts Q�i WG ! Spinn, i D 1; 2, of the inclusion G ! SOn. By Lemma 8.5,each of these lifts gives rise to a spin structure on M carrying parallel spinors,and one may legitimately ask whether these spin structures are equivalent or not.The answer to this question is given by the following (more general) result.

Theorem 8.7. Let G � SOn and let P be a G-structure onM which is connectedas a topological space. Then the extensions to Spinn of P using two different liftsof G to Spinn are not equivalent as spin structures.

Proof. Recall that two spin structures Q and Q0 are said to be equivalent if thereexists a bundle isomorphism F WQ ! Q0 such that the diagram

QF //

��✼✼✼

✼✼✼✼

✼✼✼✼

✼✼✼

Q0

��✞✞✞✞✞✞✞✞✞✞✞✞✞✞

PSOnM

commutes. Let Q�i WG ! Spinn .i D 1; 2/ be two different lifts of G and supposethat P �Q�i Spinn are equivalent spin structures on M . Assume that there exists a


bundle map F which makes the diagram

P �Q�1 SpinnF //

!!❇❇❇

❇❇❇❇

❇❇❇❇

❇❇❇❇

❇❇P �Q�2 Spinn

}}⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤⑤

P �� SOn

commutative. This easily implies the existence of a smooth mapping

f WP � Spinn �! Z=2Z

such thatF.u �Q�1 a/ D u �Q�2 f .u; a/a; u 2 P; a 2 Spinn: (8.1)

As P and Spinn are connected, we deduce that f is constant, say f � ".Then (8.1) immediately implies Q�1 D "Q�2, hence " D 1, since Q�i are grouphomomorphisms (and both map the identity element of G to the identity elementof Spinn), so Q�1 D Q�2, which contradicts the hypothesis. �

Using these results, one can construct examples of compact Riemannian mani-folds with several spin structures carrying parallel spinors (see [MS00] for details).

8.1.2 Parallel spinors on Spinc manifolds

In this section we classify all simply connected complete Spinc manifolds .M; g;S;L; A/ admitting parallel spinors. As before, a parallel spinor corresponds to a �xedpoint of the holonomy group of the Spinc connection, but in contrast to the spin case,there is no analogue of the Berger–Simons theorem in this situation. The main ideahere is to use a special complex distribution on M de�ned by the parallel spinor;see (8.6).

The curvature form dA of A can be viewed as an imaginary-valued 2-form onM , and the corresponding real 2-form will be denoted by � ´ �idA.Remark 8.8. If we �x .M; g;S; L/ and the curvature form i�, there is a canonicalfree action of C1.M/ on the set of pairs .A;‰/, whereA is a connection onLwithcurvature i� and ‰ is a non-zero parallel spinor with respect to this connection.This action is given by

f:.A;‰/ D .A � 2idf; eif‰/:

When seeking for Spinc structures with parallel spinors, we will thus most of thetime implicitly divide by this free action.


Lemma 8.9. Suppose there exists a parallel spinor ‰ onM n, i.e.,

rAX‰ D 0; X 2 �.TM/: (8.2)

ThenRic.X/ � ‰ D iX ³� �‰; X 2 �.TM/: (8.3)

Proof. Let fe1; : : : ; eng be a local orthonormal tangent frame. From (8.2) we easilyobtain

RAX;Y‰ D 0; X; Y 2 �.TM/: (8.4)

A local computation shows that the curvature operator on the spinor bundle is givenby the formula

RA D R C i

2�; (8.5)

where

RX;Y D 1

2

X

j<k

R.X; Y; ej ; ek/ej � ek �

Using (8.4) and (8.5) together with the Ricci identity (2.3), we get

0 DX

j

ej � RAej ;X

‰

DX

j

ej � .Rej ;X‰ C i

2�.ej ; X/‰/

D 1

2Ric.X/ � ‰ � i

2X ³� �‰: �

We consider Ric as a (1,1) tensor onM and for every x 2 M denote by K.x/the image of Ric, i.e.,

K.x/ D fRic.X/I X 2 TxM g

and by L.x/ the orthogonal complement of K.x/ in TxM , which by (8.3) can bewritten as

L.x/ D fX 2 TxM I Ric.X/ D 0g D fX 2 TxM I X ³� D 0g:

Since ‰ is parallel, TM � ‰ and iTM � ‰ are two parallel sub-bundles of †M .This shows that their intersection is also a parallel sub-bundle of†M . LetE be theinverse image of TM �‰ \ iTM � ‰ by the isomorphism

F WTM �! TM �‰


given byF.X/ D X � ‰:

The �ber at some x 2 M of E can be described as

Ex D fX 2 TxM I there exists Y 2 TxM such that X �‰ D iY � ‰g: (8.6)

By the discussion above, E and E? are well-de�ned parallel distributions of M .Moreover, (8.3) shows thatK.x/ � Ex for all x, soE?

x � L.x/, i.e., the restrictionof � to E? vanishes.

Assuming now thatM is complete and simply connected, we use the de Rhamdecomposition theorem and obtain that M is isometric to a Riemannian productM D M1 �M2, whereM1 andM2 are arbitrary integral manifolds of E and E?

respectively.The Spinc structure ofM induces Spinc structures onM1 andM2 with canon-

ical line bundles whose curvature is given by the restriction of � to E and E?,and (using the correspondence between parallel spinors and �xed points of the spinholonomy representation) the parallel spinor ‰ induces parallel spinors ‰1 and ‰2onM1 andM2, which satisfy (8.3).

It is clear that (since the restriction of � to E? vanishes) the canonical linebundle of the Spinc structure on M2 is trivial and has vanishing curvature, so byLemma 2.12, ‰2 is actually a parallel spinor of a spin structure on M2 (see Re-mark 8.8 above).

On the other hand, by the very de�nition of E, one easily obtains that the con-dition

X �‰ D iJX �‰; (8.7)

de�nes an almost complex structure J onM1.

Lemma 8.10. The almost complex structure J de�ned by equation (8.7) is parallel,so .M1; J / is a Kähler manifold.

Proof. Taking the covariant derivative (onM1) of (8.7) in an arbitrary direction Yand using (8.2) gives

rYX �‰ D irY .JX/ �‰: (8.8)

On the other hand, replacing X by rYX in (8.7) and subtracting from (8.8) yields

rY .JX/ �‰ D JrYX �‰;

so ..rY J /.X// � ‰ D 0, and �nally .rY J /.X/ D 0, since ‰ never vanishes onM1. As X and Y were arbitrary vector �elds, we conclude that rJ D 0. �

We �nally point out that the restriction of the Spinc structure ofM toM1 is justthe canonical Spinc structure ofM1, since (8.7) and (8.3) show that the restrictionof � toM1 is the Ricci form ofM1.


Remark 8.11. Of course, replacing J by �J just means switching from the canon-ical to the anti-canonical Spinc structure of M1, but we solve this ambiguity by“�xing” the sign of J with the help of (8.7).

Conversely, the Riemannian product of two Spinc manifolds carrying parallelspinors is again a Spincmanifold carrying parallel spinors, and as we already pointedout in the �rst section, the canonical Spinc structure of everyKähler manifold carriesparallel spinors. Consequently, we have proved the following theorem.

Theorem 8.12. A complete, simply connected Spinc manifoldM carries a parallelspinor if and only if it is isometric to the Riemannian product M1 � M2 of a sim-ply connected Kähler manifold and a simply connected spin manifold carrying aparallel spinor. Moreover, the Spinc structure ofM is the product of the canonicalSpinc structure ofM1 and the spin structure ofM2 (modulo the action of the gaugegroup C1.M/).

There are two natural questions that one may ask.

Question 1. What is the dimension of the space of parallel spinors onM ?

Question 2. How many Spinc structures onM carry parallel spinors?

We can assume thatM is irreducible, since otherwise we decomposeM , endoweach component with the induced Spinc structure, and apply the argument belowfor each component separately.

Using Theorem 8.12, we can thus always suppose thatM is either an irreduciblespin manifold carrying parallel spinors, or an irreducible Kähler manifold (not Ricci�at, since these ones are already contained in the �rst class), endowed with thecanonical Spinc structure. We call such manifolds of type S (spin) and of type K(Kähler) respectively. Then the answers to the above questions are the followingones.

1. For manifolds of type S, the answer is given by M. Wang’s classi�cation;see [Wan89]. For manifolds of type K, we will show that the dimension of thespace of parallel spinors is 1.

Suppose we have two parallel spinors ‰1 and ‰2 on M . Correspondingly wehave two Kähler structures J1 and J2. Moreover, the Ricci forms of these Kählerstructures are both equal to�, so J1 D J2 when restricted to the image of the Riccitensor. On the other hand, the vectors X for which J1X D J2X form a paralleldistribution onM , which, by irreducibility, is either the whole of TM or empty.

In the �rst case we have J1 D J2, so both ‰1 and ‰2 are parallel sections of†0M whose complex dimension is 1. The second case is impossible, since it wouldimply that the Ricci tensor vanishes.

8.2. Special holonomies and relations to warped products 223

2. This question has ameaning only for manifolds of type K. A slight modi�cationin the above argument shows that on a manifold of type K there are no other Kählerstructures except for J and �J .

On the other hand, we have seen that a parallel spinor with respect to some Spinc

structure induces on such a manifold M a Kähler structure whose canonical Spinc

structure is just the given one. It is now clear that we can have at most two differentSpinc structures with parallel spinors: the canonical Spinc structures induced by Jand �J . These are just the canonical and anti-canonical Spinc structures on .M; J /,and they both carry parallel spinors.

Summarizing, we have the following result.

Proposition 8.13. The only Spinc structures on an irreducible not Ricci-�at Käh-ler manifold which carry real Killing spinors are the canonical and anti-canonicalones. The dimension of the space of parallel spinors is in both cases equal to 1.

8.2 Special holonomies and relations to warped products

This section is intended to provide a dictionary between the different geometriesof the base and the total space of a special class of warped products, namely theRiemannian cones. They will be needed in the next section for the classi�cation ofcomplete spin or Spinc manifolds with Killing spinors (cf. Theorem 2.41).

8.2.1 Sasakian structures

Let .M; gM / be a Riemannian manifold with Levi-Civita covariant derivative r.

Definition 8.14. A Sasakian structure on a Riemannian manifold .M; gM / is aKilling vector �eld � of unit norm with the property

r2X;Y � D �gM .X; Y /� C �.Y /X; X; Y 2 TM; (8.9)

where � ´ gM .� �; �/ is the 1-form corresponding to �.

Taking the scalar product with � in (8.9) shows that the (1,1)-tensor ' ´ �r�vanishes on � and de�nes an orthogonal complex structure on �?. Since � is aKilling vector �eld, its covariant derivative is related to the exterior differential ofthe associated 1-form � by

gM .rX�; Y / D 1

2d�.X; Y /; X; Y 2 TM: (8.10)


Definition 8.15. The Riemannian cone over .M; gM / is the manifold

xM D M � RC;

endowed with the Riemannian metric

g xM ´ r2gM C dr2:

The covariant derivative of the Levi-Civita connection of g xM will be denotedby xr. It satis�es the warped product relations (2.32)–(2.34), where X and Y arevector �elds onM , identi�ed with their canonical extensions to xM .

Theorem 8.16. The Sasakian structures on M are in one-to-one correspondencewith the Kähler structures on the Riemannian cone over M preserved by the �owof the radial vector �eld r@r of xM .

Proof. Suppose �rst that (�; �; ') is a Sasakian structure onM . We de�ne a tensor Jof type (1,1) on xM by

J.r@r/ D �; J � D �r@r ; JX D �'.X/; for X ? @r ; X ? �:

The remark above shows that J is an orthogonal almost complex structure on xM .By (8.10), the associated 2-form �.�; �/ ´ gM .J �; �/ is given by

� D r2

2d�C rdr ^ �;

so d� D 0 on xM .Let N J denote the Nijenhuis tensor of J , de�ned by

N J .X; Y / ´ ŒX; Y �C J ŒX; J Y �C J ŒJX; Y �� ŒJX; J Y �

D J.LXJ /Y � .LJXJ /Y:

We claim thatLr@rJ D 0 andL�J D 0. The �rst relation follows from the fact that

the �ow of r@r is given by 't .r; x/ D .etr; x/ and clearly preserves J . The vector�eld � is Killing for gM , hence also for g xM . Moreover, L�� D 0, thus proving thesecond relation.

Let now X and Y denote vector �elds on xM induced by arbitrary vector �eldsonM perpendicular to �. Using (2.33) and (2.34), we get

J.xrXY / D J.rXY / � rJ.gM .X; Y /@r /D J.gM .rXY; �/�/C J.rXY � gM .rXY; �//� gM .X; Y /�D � rgM .�;rXY /@r � '.rXY / � gM .X; Y /�;


and

xrX.J Y / D �xrX.'.Y // D �rX.'.Y // � rgM .X;rY �/@r :

Since gM .�;rXY / D �gM .Y;rX�/ D gM .X;rY �/, this shows that

.xrXJ /.Y / D J.xrXY / � xrX.J Y / D .rX'/.Y / � gM .X; Y /�D gM .X; Y /� � gM .�; Y /X � gM .X; Y /� D 0;

so in particular N J .X; Y / D 0. Combining these results we get N J D 0, so xM isKähler by [KN69], Proposition 4.2, Chapter 9.

Conversely, suppose that J is a Kähler structure on xM preserved by the �owof r@r . We identify M with f1g � M � xM and denote by � the restriction to Mof J@r . From (2.33), rX� equals �JX when X 2 TM and X ? �, and

r�� D xr�� C @r D J.xr�@r /C @r D J � C @r D 0;

so � is a unit Killing vector �eld onM .To check (8.9), we use (2.33) and (2.34):

�r2V;W � D �rVrW � C rrVW �

D �xrVrW � � gM .rW �; V /@r C xrrVW � C gM .rVW; �/@rD �xrV xrW � � xrV .gM .W; �/@r/ � gM .J.rW @r/; V /@r

C xrxrVW� C xrgM .V;W /@r

� C gM .xrVW; �/@rD �J xrVW � gM .xrVW; �/@r � gM .W; xrV �/@r � gM .W; �/xrV @r

� gM .JW; V /@r C J.xrVW /C gM .V;W /� C gM .xrVW; �/@rD �gM .W; JV /@r � gM .W; �/V � gM .JW; V /@r C gM .V;W /�

D gM .V;W /� � gM .W; �/V:

The two constructions above are clearly inverse to each other. �

8.2.2 3-Sasakian structures

Definition 8.17. A 3-Sasakian structure on a Riemannian manifold .M; g/ is atriple .�i ; �i ; 'i/ ; i 2 f1; 2; 3g, of Sasakian structures such that the following rela-tions hold:

Œ�1; �2� D 2�3; Œ�2; �3� D 2�1; Œ�3; �1� D 2�2I (8.11)

'3'2 D �'1 C �2 ˝ �3; '2'3 D '1 C �3 ˝ �2I (8.12)

'1'3 D �'2 C �3 ˝ �1; '3'1 D '2 C �1 ˝ �3I'2'1 D � '3 C �1 ˝ �2; '1'2 D '3 C �2 ˝ �1:


Remark 8.18. It is easy to see that .8.11/ is equivalent to

r�i�j D "ijk�k ; (8.13)

where "ijk denotes the signature of the permutation fijkg if i; j; k are all different,and 0 otherwise.

Definition 8.19. A hyper-Kähler structure on a Riemannian manifold .M; g/ is atriple fI; J;Kg of Kähler structures satisfying IJ D �JI D �K:Theorem 8.20. The 3-Sasakian structures onM are in one-to-one correspondencewith the hyper-Kähler structures on the Riemannian cone overM preserved by the�ow of the radial vector �eld r@r of xM .

Proof. Let .�i ; �i ; 'i / ; i 2 f1; 2; 3g be a 3-Sasakian structure on M . By Theo-rem 8.16, we obtain three Kähler structures I , J and K on xM . Let us check, forinstance, that IJ D � K:

From (8.13) we have '1.�2/ D �3; '2.�3/ D �1; etc., so

IJ.r@r/ D I.�2/ D �'1.�2/ D ��3 D �K.r@r/;

IJ �1 D I.�'2.�1// D I.�3/ D �'1.�3/ D '3.�1/ D �K.�1/;

IJ �2 D I.�r@r/ D ��1 D '3.�2/ D �K.�2/;

IJ �3 D I.�'2.�3// D I.��1/ D r@r D �K.�3/;

IJV D '1.'2.V // D '3.V /C �2.V /�1 D '3.V / D �K.V /;

for all vectors V orthogonal to �1; �2; �3:Conversely, a hyper-Kähler structure on xM preserved by the �ow of r@r

induces three Sasakian structures .�i ; �i ; 'i /iD1;2;3 on M , as in Theorem 8.16.To check (8.13), we compute

r�1�2 D rI.@r /J.@r/

D xrI.@r /J.@r/ � g xM .I.@r /; J.@r//@r

D JI.@r/C g xM .JI.@r/; @r/@r

D �3 C g xM .K.@r/; @r/@r

D �3:

Finally, we check, for instance, the �rst equation of (8.12):

'3.'2.�1// D �'3.JI@r/ D �'3.�3/ D 0 D .�'1 C �2 ˝ �3/.�1/; (8.14)

etc. �


8.2.3 The exceptional group G2

Let V be the 7-dimensional Euclidean space with canonical base fe1; : : : ; e7g andscalar product h�; �i and denote by f!1; : : : ; !7g the canonical base of V �. We usethe notation !ijk for !i ^ !j ^ !k . Following Bryant (see [Bry87], p. 539), wede�ne G2 as the subgroup of GL.V / preserving a suitable 3-form of ƒ3V .

Definition 8.21. We set G2 ´ fg 2 GL.V / g�.�/ D �g, where

� ´ !123 C !145 C !167 C !246 C !257 C !347 C !356: (8.15)

We refer to [Bry87] for the proof of the following classical result.

Theorem 8.22 ([Bry87]). The groupG2 � GL.V / is a compact, connected, simple,simply connected 14-dimensional subgroup of SO.V /. Moreover, it acts irreduciblyand transitively on V and transitively on the Grassmannian of 2-planes of V .

We de�ne the bilinear map

P WV � V �! V

associated with � via h�; �i as

hP.x; y/; zi D �.x; y; z/; x; y; z 2 V I

P satis�esP.x; y/ D � P.y; x/; P.e1; e2/ D e3;

and moreover

hP.x; y/; P.x; y/i D hx; xihy; yi � hx; yi2 (8.16)

and

P.x; P.x; y// D �hx; xiy C hx; yix: (8.17)

To check (8.16) and (8.17) we note that G2 preserves � by de�nition and the scalarproduct h�; �i by Theorem 8.22, so P is G2-invariant. Consequently, using the lastpart of Theorem 8.22, we see that it is enough to prove (8.16) and (8.17) for x D˛e1 and y D ˇe1 C e2, which is obvious.

Lemma 8.23. Let 2 ƒ3V . The following assertions are equivalent.(i) For every unit vector X 2 V , the restriction of X ³ to X? de�nes an

orthogonal complex structure on X? and

X� ^X ³ ^ X ³ ^X ³ D 6!1234567:

(ii) There exists g 2 SO.V / such that D g��:


Proof. (i) H) (ii). We have to show that one can �nd an orthonormal basis of Vwith the property that has in this basis the same form as � in the basis fe1; : : : ; e7g.

For any unit vector X of V we denote by JX the complex structure de�nedby X ³ on X?. We now choose X D e1. One can �nd an orthonormal basisfv2; : : : ; v7g of e?

1 such that

Je1v2 D v3; Je1

v4 D v5; Je1v6 D v7: (8.18)

Let X D v2. One sees immediately that Jv2e1 D �v3, therefore Jv2

preserves thesubspace spanned by v4; v5; v6; v7: By making a rotation in the plane (v4; v5), onecan assume that Jv2

v4 ? v7. Then Jv2v4 ? v5 too: if not, contains the terms

e1 ^ v4 ^ v5 and ˛v2 ^ v4 ^ v5, ˛ ¤ 0 (one identi�es the vectors with the dualforms), therefore in any case the “coef�cient” of v4 ^ v5 in would be a vector Yof norm r D kY k > 1. By taking X D Y=r in (i), we conclude that JX maps v4 to.1=r/v5, which is impossible.

We can thus suppose Jv2v4 D v6. Then

Jv2v5 D av4 C bv6 C cv7; a2 C b2 C c2 D 1;

so�v5 D .J 2v2

/v5 D Jv2.av4 C bv6 C cv7/ D av6 � bv4 C cJv2

v7;

which implies c2 D 1C a2 C b2. Thus a D b D 0, so we obtained

Jv2e1 D �v3; Jv2

v4 D v6; Jv2v5 D ˙v6: (8.19)

From (8.18), (8.19) and (i) applied to X D e3 and X D e5 we have

Jv4v3 D ˙v7; Jv5

v3 D ˙v6: (8.20)

Using (8.18), (8.19), and (8.20) we get

D e1 ^ v2 ^ v3 C e1 ^ v4 ^ v5 C e1 ^ v6 ^ v7 C v2 ^ v4 ^ v6˙ v2 ^ v5 ^ v7 ˙ v3 ^ v4 ^ v7 ˙ v3 ^ v5 ^ v7 C � � � : (8.21)

Now, each pair of subscripts between 1 and 7 is contained in exactly one of thetriples of subscripts appearing in (8.21). Thus, the same argument as before showsthat there is no other term in the expression of given by (8.21):

D e1 ^ v2 ^ v3 C e1 ^ v4 ^ v5 C e1 ^ v6 ^ v7 C v2 ^ v4 ^ v6˙ v2 ^ v5 ^ v7 ˙ v3 ^ v4 ^ v7 ˙ v3 ^ v5 ^ v7:

Finally, to prove that the ambiguous signs are all “�”, we just use the second con-dition of (i) with X D v2 ; X D v4 and X D v6:


(ii) H) (i) Observe �rst that the conditions in (i) remain true if we replace by g� ; g 2 SO.V /. It is thus enough to check (i) for �. We use Theorem 8.22:(i) holds for X D e1, and if X is arbitrary, we choose g 2 G2 � SO7 such thatX D g.e1/. Then,

X ³ � D .g�/�1.e1 ³ �/ D g.e1 ³ �/ ; X� D .g�/�1.e�1 / D g.e�

1 /

and it is clear that X ³� D g.e1 ³�/ de�nes a complex structure on X? D g.e?1 /,

and that

X� ^X ³ ^X ³ ^ X ³ D g.e�1 ^ e1³ ^ e1 ³ ^ e1 ³ /

D g.6!1234567/

D 6!1234567: �

The 3-forms satisfying the equivalent conditions of the above lemma are calledgeneric 3-forms. For later use, we introduce the following notion

Definition 8.24. A weak G2-structure on a 7-dimensional manifoldM is given bya generic 3-form ' satisfying r' D �', with respect to the covariant derivative andthe duality operator given by the metric induced by '.

8.2.4 Nearly Kähler manifolds

Definition 8.25. A nearly Kähler structure on a Riemannian manifold .M; g/ is analmost Hermitian structure J such that

.rXJ /.X/ D 0; X 2 TM: (8.22)

A nearly Kähler structure is called of constant type ˇ 2 R if

j.rXJ /.Y /j2 D ˇ.jX j2jY j2 � hX; Y i2 � hJX; Y i2/; X; Y 2 TM: (8.23)

Note that (8.23) implies (8.22) and that a nearly Kähler structure of constanttype ˇ is Kähler if and only if ˇ D 0.

From (8.22) we obtain immediately

.rXJ /Y C .rY J /X D 0 and .rXJ /.JX/ D 0; (8.24)

.rXJ /J Y D .rJXJ /Y D � J..rXJ /Y /;and

h.rXJ /Y;Zi D � h.rXJ /Z; Y i: (8.25)

The relations above show thatrJ is a skew-symmetric tensor of type (3,0)+(0,3).In particular, a 4-dimensional nearly Kähler manifold is automatically Kähler.


Proposition 8.26. Every nearly Kähler 6-dimensional manifold M is of constanttype.

Proof. We may suppose that M is not Kähler. Let e1; Je1; e2; Je2 be mutuallyorthogonal unit vector �elds on an open set U ofM and de�ne a positive function˛ and a unit vector �eld on U by .re1

J /e2 D ˛e3. As before, it is easy to checkthat .e1; Je1; e2; Je2; e3; Je3/ is an orthonormal frame on U . Moreover, we have.re2

J /e3 ? .e2; Je2; e3; Je3/,

h.re2J /e3; e1i D h.re1

J /e2; e3i D ˛;

and

h.re2J /e3; Je1i D h.re1

J /e2; Je3i D 0;

thus showing that.re2

J /e3 D ˛e1:

Similarly,.re3

J /e1 D ˛e2:

From these formulas, (8.23), with ˇ D ˛2, follows directly. Note also that bypolarization (8.23) is equivalent to

h.rV J /X; .rY J /Zi D ˇfhV; Y ihX;Zi � hV;ZihX; Y i� hV; J Y ihX; JZi C hV; JZihX; J Y ig (8.26)


We now come to the central result of this section.

Theorem 8.27. The nearly Kähler structures of constant type 1 on a 6-dimensionalRiemannian manifold M are in one-to-one correspondence with parallel generic3-forms on the Riemannian cone overM .

Proof. Let ' be a parallel generic 3-form on xM . As before, we identify M withM � f1g � xM and using Lemma 8.23, we de�ne an almost Hermitian structure JonM by

hX; J Y i D '.@r ; X; Y /:

Then

h.rXJ /Z; Y i D hrX .JZ/; Y i � hJrXZ; Y iD XhY; JZi � hrXY; JZi � hJrXZ; Y iD X:'.@r ; Y; Z/� '.@r ;rXY;Z/� '.@r ; Y;rXZ/D X:'.@r ; Y; Z/� '.@r ; xrXY;Z/� '.@r ; Y; xrXZ/D '.xrX@r ; Y; Z/C .xrX'/.@r ; Y; Z/D '.X; Y; Z/;

which shows that J is nearly Kähler of constant type 1.


Conversely, let J be a nearly Kähler structure of constant type 1 on M . Forevery x 2 M; r 2 R

C; and X; Y;Z 2 TxM � T.x;r/ xM; @r 2 TrRC � T.x;r/ xM ,we de�ne

'.@r ; X; Y / D � '.X; @r ; Y / D '.X; Y; @r/ ´ r2hX; J Y iM ;

'.X; Y; Z/ ´ r3hY; .rXJ /.Z/iM :The fact that ' is skew-symmetric follows from (8.24)–(8.25). To show that '

is generic, let fe1; Je1; e2; Je2; e3; Je3g be a local orthonormal frame on M suchthat e3 D .re1

J /e2. By taking v1 D @r , v2 D e1=r , v3 D Je1=r , v4 D e2=r ,v5 D Je2=r , v6 D e3=r , and v7 D Je3=r , we obtain a local orthonormal frame ofxM around .x; r/. Relations (8.24)–(8.25) then show that the expression of ' in thisbasis is just (8.15).

Finally, we check that ' is parallel (we continue to use the notation X; Y; : : : forvector �elds onM , identi�ed with their canonical extension to xM ):

(1) we have

.xr@r'/.X; Y; Z/ D @r.'.X; Y; Z//� '.xr@r

X; Y;Z/

� '.X; xr@rY;Z/� '.X; Y; xr@r

Z/

D 3r2hY; .rXJ /Zi � 3r2hY; .rXJ /ZiD 0I

(2) we have

.xr@r'/.@r ; X; Y / D @r .'.@r ; X; Y // � '.@r ; xr@r

X; Y /

� '.@r ; X; xr@rY /

D 2rhX; J Y iM � 2rhX; J Y iMD 0I

(3) we have

.xrX'/.@r ; Y; Z/ D X.'.@r ; Y; Z//� '.xrX.@r/; Y; Z/� '.@r ; xrXY;Z/� '.@r ; Y; xrXZ/D r2XhY; JZiM � r2hY; .rXJ /ZiM

� r2hrXY; JZiM � r2hY; JrXZiMD 0: �

To show that .xrX'/.Y; Z;W / D 0, we use (8.26) and the following result ofGray (see [Gra76]).


Proposition 8.28. For every vector �elds X; Y;Z;W on a nearly Kähler mani-foldM ,

2h.r2X;Y J /W;Zi D S

Y;W;Zh.rXJ /Y; J.rW J /Zi:

We then conclude (using h�; �i instead of h�; �iM for simplicity) that

h.r2X;Y J /W;Zi D �1

2S

Y;W;Zh.rXJ /Y; .rW J /JZi

D �12

SY;W;Z

fhX;W ihY; JZi � hX; JZihY;W i

C hX; JW ihY;Zi � hX;ZihY; JW ig;

and after some calculations

h.r2X;Y J /W;Zi D hX;ZihY; JW i C hX; Y ihW; JZi C hX;W ihZ; J Y i:

Consequently

0 D h.r2X;Y J /W;Zi � fhX;ZihY; JW i C hX; Y ihW; JZi C hX;W ihZ; J Y ig

D hrX ..rY J /W / � .rrXY J /W � .rY J /.rXW /;Zi� fhX;ZihY; JW i C hX; Y ihW; JZi C hX;W ihZ; J Y ig;

and �nally

.xrX'/.Y; Z;W /D X.'.Y;Z;W // � '.rXY � rhX; Y i@r ; Z;W /

� '.Y;rXZ � rhX;Zi@r ; W / � '.Y;Z;rXW � rhX;W i@r /

D r3fXhZ; .rY J /W i � .hZ; .rrXY J /W i � hX; Y ihZ; JW i/� .hrXZ; .rY J /W i C hX;ZihY; JW i/� .hZ; .rY J /.rXW /i � hX;W ihY; JZi/g

D r3fhrX..rY J /W / � .rrXY J /W � .rY J /.rXW /;Zi� fhX;ZihY; JW i C hX; Y ihW; JZi C hX;W ihZ; J Y igg

D 0:


8.2.5 The group Spin7

In this section we introduce the group Spin7 as the stabilizer of a suitable 4-formin R

8, and describe its relationship with the exceptional group G2.We view the Euclidean space R

8 as the direct sum

VC ´ Re0 ˚ V;

where V is the 7-dimensional Euclidean space appearing in the construction of G2.Let fe0; : : : ; e7g be the standard basis of VC, let f!0; : : : ; !7g be the dual basisof V �

C, and denote by h�; �i the standard Euclidean scalar product. We de�ne the4-form ˆ 2 ƒ4.V �

C/ by

ˆ D !0 ^ � C ��;

where � is the standard generic 3-form on V de�ned by (8.15). One easily computes

ˆ ^ˆ D 14!01234567;

and

ˆ D 1

2˛ ^ ˛ C Re.ˇ/;

where

˛ D !01 C !23 C !45 C !67;

ˇ D .!0 C i!1/ ^ .!2 C i!3/ ^ .!4 C i!5/ ^ .!6 C i!7/:

We de�ne

G D fg 2 GL.VC/I g�.ˆ/ D ˆg

and quote the following result.

Theorem 8.29 ([Bry87], p. 545). The groupG is a compact, connected, simply con-nected 21-dimensional Lie group preserving the Euclidean product h�; �i.It acts irreducibly on VC and transitively on each Grassmannian of k-planes of VC.k � 8/. Its center is Z=2Z D f˙IdVCg and G=.Z=2Z/ ' SO7, so G is isomor-phic to Spin7.

For the proof, we refer to [Bry87], but we point out that the correct de�nition ofthe subgroupH introduced there is

H D fh 2 SO7I h�.˛/ D ˛; h�.ˇ/ D ˇg:


Lemma 8.30. Let ‰ 2 ƒ4.V �C/. The following assertions are equivalent.

(i) For every unit vector X 2 VC, the restriction of X ³ ‰ to X? is a generic3-form ' on X?, and the restriction of ‰ itself to X? equals �7', where�7 denotes the Hodge duality operator on X? with respect to the inducedEuclidean product.

(ii) The assertion (i) holds for X D e0.

(iii) The form‰ can be written as‰ D !0^'C�7', where ' is a generic 3-formon e?

0 .

Proof. The only non-trivial point is to prove (iii) H) (i). Let ‰ D !0 ^ ' C �7':The fact that (i) holds for X D e0 is tautological. For X 2 VC arbitrary, we useTheorem 8.29 to �nd g 2 Spin7 � SO8 with X D g.e0/. Then, on the one hand,we obtain that

X ³‰ D .g�/�1.e0 ³‰/ D g.'/

is a generic 3-form on X? D g.e?0 /, and on the other hand

‰jX? D g.‰/jg.e?0 /

D g.‰/je?0

D g.�7'/ D �7g.'/: �

Definition 8.31. A 4-form onR8 satisfying the equivalent conditions of Lemma 8.30

will be called a generic 4-form.

Theorem 8.32. There is a one-to-one correspondence between the weak G2-struc-tures on a 7-dimensional manifold M and the parallel generic 4-forms on theRiemannian cone xM ofM .

Proof. Let ˆ be a parallel generic 4-form on xM . We de�ne a generic 3-form '

onM by'.X; Y; Z/ D ˆ.@r ; X; Y; Z/:

It is clear that ˆ D dr ^ ' C �', so

.rW '/.X; Y; Z/ D W.ˆ.@r ; X; Y; Z//�ˆ.@r ;rWX; Y;Z/�ˆ.@r ; X;rW Y;Z/�ˆ.@r ; X; Y;rWZ/

D Wˆ.@r ; X; Y; Z/�ˆ.@r ; xrWX; Y;Z/�ˆ.@r ; X; xrW Y;Z/�ˆ.@r ; X; Y; xrWZ/

D .xrWˆ/.@r ; X; Y; Z/Cˆ.xrW @r ; X; Y; Z/D ˆ.W;X; Y; Z/

D .�'/.W;X; Y; Z/;

thus proving that ' de�nes a weak G2-structure onM .


Conversely, let ' be a generic 3-form onM satisfying r' D �':We de�ne a4-form ˆ on xM by

ˆ.@r ; X; Y; Z/D r3'.X; Y; Z/; (8.27)

ˆ.W;X; Y; Z/ D r4.rW '/.X; Y; Z/ D r4.�'/.W;X; Y; Z/: (8.28)

To avoid any possible confusion, we denote by “�” the duality operator onM andby “� xM ” the duality operator on the sub-bundle E � T� xM of forms on xM whichdo not contain dr . Consider the 3-form Q' on xM given by

Q'.x;r/.@r ; X; Y / D 0;

Q'.x;r/.X; Y; Z/ D 'x.rX; rY; rZ/:

Since the transformation TxM ! T.x;r/ xM given by Xx 7! rX.x;r/ is an isometry,we obtain

.� xM Q'/.W;X; Y; Z/ D .�'/.rW; rX; rY; rZ/D r4.�'/.W;X; Y; Z/;

so (8.27) and (8.28) imply that

ˆ D dr ^ Q' C .� xM Q'/;

i.e., ˆ is a generic 4-form on xM .It remains to check that ˆ is parallel:

.xr@rˆ/.@r ; X; Y; Z/ D @r.r

3'.X; Y; Z//� 3r2'.X; Y; Z/ D 0;

.xr@rˆ/.X; Y; Z; T / D @r.r

4.�'/.X; Y; Z; T // � 4r3.�'/.X; Y; Z; T / D 0;

and

.xrWˆ/.@r ; X; Y; Z/ D W.r3'.X; Y; Z//�ˆ�Wr;X; Y; Z

�� r3'.rWX; Y;Z/

� r3'.X;rW Y;Z/� r3'.X; Y;rWZ/

D r3.rW '/.X; Y; Z/�ˆ�Wr;X; Y; Z

�

D 0:

In order to show that .xrWˆ/.X; Y; Z; T / D 0 for all vectors W;X; Y;Z; Tof xM orthogonal to @r , we observe that


• the volume form ofM is parallel, so rW .�'/ D �.rW '/ and• ifW � denotes the dual 1-form of W with respect to the metric h�; �i, then

�.W ³ .�'// D ' ^W �

(by linearity, it suf�ces to check this for ' D dx1^dx2^dx3 andW D @=@x3orW D @=@x4/:

Consequently,

rW .�'/ D �.rW '.X; Y; Z; T // D �.W ³ .�'// D ' ^W �;

which yields

.xrWˆ/.X; Y; Z; T /

D r4.rW .�'//C r3'.Y;Z; T / � rhW;Xi � r3'.X;Z; T / � rhW; Y iC r3'.X; Y; T / � rhW;Zi � r3'.X; Y; Z/ � rhW; T i

D 0: �

The results of this section are due both to Bär [Bär93] and Bryant [Bry87], aswell as Gray [Gra76].

8.3 Classi�cation of manifolds admitting real Killing spinors

Real Killing spinors appeared naturally in Chapter 5, where it was shown that theycharacterize the limiting case of Friedrich’s inequality between the �rst eigenvalueof the Dirac operator and the scalar curvature of a compact spin manifold. In thissection we give a complete geometrical description of all compact simply connectedspin and Spinc manifolds carrying real Killing spinors. By Lemma 2.12, it suf�cesto treat the Spinc case.

All over this section, .M n; g;S; L; A/ will be a simply connected Spinc mani-fold carrying a real Killing spinor ‰, i.e., satisfying the equation

rAX‰ D �X �‰; X 2 TM;

for some �xed real number � ¤ 0. By rescaling the metric if necessary, we canassume, without loss of generality, that � D ˙1

2. Moreover, for n even we can

assume that � D 12, by taking the conjugate of ‰ if necessary.

Consider the Riemannian cone . xM; Ng/ overM and, as before, denote by @r theradial vector �eld. Using the formulas for warped products (2.32)–(2.34), one easilycomputes the curvature tensor xR of xM (cf. [O’N66], p. 210):

xR.X; @r/@r D xR.X; Y /@r D xR.X; @r/Y D 0; (8.29)

xR.X; Y /Z D R.X; Y /Z C g.X;Z/Y � g.Y;Z/X: (8.30)

8.3. Classi�cation of manifolds admitting real Killing spinors 237

In particular, we have the following result.

Lemma 8.33. The Riemannian cone xM is �at if and only if M is a space form ofconstant sectional curvature equal to 1.

We now quote an important result concerning the holonomy of Riemannian conemetrics, originally due to S. Gallot (see [Gal79], Proposition 3.1).

Lemma 8.34. IfM is complete, then xM is irreducible or �at.

Note that the completeness hypothesis is essential here. Indeed, it is straight-forward to see that the Riemannian product of two Riemannian cones is again aRiemannian cone over a non-complete manifold (cf. [MO08], for instance). Thismeans that there are lots of examples of non-�at Riemannian cones with reducibleholonomy over non-complete manifolds.

The use of the Riemannian cone over M is the key idea for the classi�cationof manifolds with real Killing spinors. From Lemma 2.40 and Theorem 2.41, weobtain at once the following corollary.

Corollary 8.35. The natural Spinc structure of the Riemannian cone over a Spinc

manifoldM n carrying real Killing spinors with Killing constant˙12admits parallel

spinors. Moreover, for n odd, a Killing spinor with Killing constant 12generates a

positive parallel half-spinor and aKilling spinor with Killing constant�12generates

a negative parallel half-spinor. The converse is also true.

As in the case of parallel spinors, the gauge group C1.M/ acts freely on the pairs.A;‰/ of connections A on L with a given curvature form and Killing spinors ‰.

Using the corollary above together with Lemma 8.34 and Theorem 8.12, we ob-tain that the Riemannian cone xM is either an irreducible spin manifold with parallelspinors, or a Kähler manifold with the canonical Spinc structure, or a �at manifold.In the last case, the Spinc structure of xM is also �at, because of (8.3).

Thus, modulo the action of the gauge group we have the following result.

Theorem 8.36. A simply connected complete Spinc manifold M carries a realKilling spinor with Killing constant 1

2if and only if the associated Riemannian cone

satis�es one of the two conditions below:

(1) xM is a simply connected spin manifold carrying parallel spinors;

(2) xM is a Kähler manifold.

Wenow study these two situations in more detail. The �rst case is directly settledby Lemma 8.34, Lemma 8.33, and the results of the previous sections (Theorem 8.1,Theorem 8.16, Theorem 8.20, Theorem 8.27, and Theorem 8.32).


Theorem 8.37 ([Bär93]). A simply connected complete spin manifoldM n carriesa real Killing spinor if and only if it is homothetic to one of the following manifolds:

(1) the sphere Sn,

(2) an Einstein–Sasakian manifold,

(3) a 3-Sasakian manifold,

(4) a nearly Kähler manifold of constant type 1 .n D 6/,

(5) a weak G2-manifold .n D 7/.

The second case of Theorem 8.36 is equivalent to M being Sasakian (by The-orem 8.16). The following result provides an analogue in the Sasakian setting forthe canonical Spinc structure on Kähler manifolds.

Proposition 8.38. Every Sasakian manifold .M 2kC1; g; �/ has a canonical Spinc

structure. IfM is Einstein, then the auxiliary bundle of the canonical Spinc structureis �at, so if in additionM is simply connected, then it is spin.

Proof. The �rst statement follows directly from the fact that the Riemannian coneoverM is Kähler, and thus carries a canonical Spinc structure, whose restriction toM is the desired canonical Spinc structure.

IfM is Einstein, then its Einstein constant is 2k (see [BFGK91], p. 78), so xM isRicci �at by (8.29) and (8.30). The auxiliary bundle of the canonical Spinc structureof xM , which is just the canonical bundle K D ƒkC1;0 xM , is thus �at, and the sameis true for its restriction toM . The last statement follows from Lemma 2.12. �

One can construct the Spinc structure more directly as follows. The frame bundleof every Sasakian manifold restricts to Uk , by considering only adapted frames,i.e., orthonormal frames of the form f�; e1; '.e1/; : : : ; ek; '.ek/g. Then extend thisbundle of adapted frames to a Spinc2k-principal bundle using the canonical inclusionUk ! Spinc2k (cf. [LM89], p. 392).

Just as in the case of almost Hermitian manifolds, one can de�ne an anti-canon-ical Spinc structure for Sasakian manifolds, which has the same properties as thecanonical one.

We recall that the parallel spinor of the canonical Spinc structure of a Kählermanifold M 2kC2 lies in †0M , so is always a positive half-spinor, and the parallelspinor of the anti-canonical Spinc structure lies in †kC1M , so it is positive (nega-tive) for k odd (resp. even). Collecting these remarks together with Corollary 8.35we obtain the following result.

8.4. Detecting model spaces by Killing spinors 239

Corollary 8.39. The non-Einstein Sasakian manifoldsM 2kC1 endowed with theircanonical or anti-canonical Spinc structure are the only simply connected Spinc

manifolds admitting real Killing spinors other than the spin manifolds. For thecanonical Spinc structure, the dimension of the space of Killing spinors, for theKilling constant 1

2, is always equal to 1, and there is no Killing spinor for the con-

stant �12. For the anti-canonical Spinc structure, the dimensions of the spaces of

Killing spinors, for the Killing constants 12and�1

2, are 1 and 0 .0 and 1/ for k odd

.resp. even/.

The results in this section are due to Ch. Bär in the spin case [Bär93] and toA. Moroianu in the Spinc case [Mor97b].

8.4 Detecting model spaces by Killing spinors

In this section we characterize the round sphere as being the only complete con-nected spin manifold of dimension 3; 4; 7 or 8 admitting non-trivial Killing spinorsassociated with the positive and the negative �rst eigenvalues of the Dirac opera-tor. Note that in dimension 4 and 8 the spectrum of the Dirac operator is symmet-ric around zero. The result is due to Th. Friedrich [Fri81] in dimension 4 and toNieuwenhuizen and Warner [vNW84] in dimension 7. Here we present a uniformsimple proof (see [Hij86a]) based on the use of real spinor �elds and the Obata–Lichnerowicz theorem for the scalar Laplacian.

For this, recall the following de�nition.

Definition 8.40. A real Killing spinor‰ is a spinor �eld satisfying, for any tangentvector �eld X , the differential equation

rX‰ C �0

nX �‰ D 0; (8.31)

where

�0 D ˙r

n

4.n� 1/Scal0;

is the smallest eigenvalue of the Dirac operator. The space of such spinors is denotedby KS˙.

Theorem 8.41. Any complete connected Riemannian spin manifold of dimension3; 4; 7 or 8 for which dimKSC > 0 and dimKS� > 0 is isometric to the roundsphere.


Proof. To show this it is suf�cient to prove that the conditions of the Obata–Lich-nerowicz theorem are satis�ed, i.e., there exist a positive constant c and a non-trivialreal function f onM n such that

Ric � c Id and �f D n

n � 1cf;

where � is the scalar Laplacian.Consider ‰ 2 KSC and x‰ 2 KS� and let f ´ Reh‰; x‰i. With the help

of (8.31) it is straightforward to see that

�f D Scal

n � 1f:

Hence, since the manifold is Einstein, it is suf�cient to take c D Scaln

and to provethat f 6� 0.

For each integer n, denote

dn ´ dimR†;

where † is an irreducible R-module of Cln and

dCn ´ dimC†

C;

where †C is an irreducible C-module of Cln (hence for Cln ´ Cln ˝RC). FromChapter 1, see [LM89], we have Table 5.

Table 5

n 1 2 3 4

Cln C H H ˚ H H.2/

dn 2 4 4 8

Cln C ˚ C C.2/ C.2/˚ C.2/ C.4/

dCn 1 2 2 4

n 5 6 7 8

Cln C.4/ R.8/ R.8/˚ R.8/ R.16/

dn 8 8 8 16

Cln C.4/˚ C.4/ C.8/ C.8/˚ C.8/ C.16/

dCn 4 8 8 16

8.5. Generalized Killing spinors 241

From this table we can read that for n D 1; 2; 3; 4; 5, the bundle of real spinorsis the same as the bundle of complex spinors (considered as real vector bundles) andfor n D 6; 7; 8, the bundle of complex spinors is the complexi�cation of the bundleof real spinors. In both cases, the real and the complex Dirac operators have thesame spectrum. We denote by †M the real spinor bundle.

We �rst consider the case n D 3 or 7. For ‰ 2 KSC de�ne the linear injectivemap

‰ W TM �! †M;

X 7�! X �‰;which is an isomorphism onto its image ‰.TM/, a subspace in †M of codimen-sion 1, for n D 3 or 7.

Since ‰ is orthogonal to ‰.TM/, the function f ´ Reh‰; x‰i has no zeroonM .

For n D 4 or 8, we consider the decomposition of the real spinor bundle †Minto positive and negative spinors, i.e.,†M D †CM ˚†�M and‰ D ‰C C‰�.We note that the linear injective map

‰CW TM �! †�M;

X 7�! X �‰C;

is an isomorphism since in these dimensions, dimR TM D dimR†M . Note that inthis case, if ‰ 2 KSC, then x‰ ´ ‰C � ‰� 2 KS�. We then conclude by notingthat, for any tangent vector �eld X ,

X.f / D �4�0n.X �‰C; ‰�/ 6� 0: �

8.5 Generalized Killing spinors

If Z is a Riemannian spin manifold, any oriented hypersurface M � Z inheritsa spin structure, and by Lemma 2.38, the restriction to M of the complex spinorbundle †Z if n is even (resp. †CZ if n is odd) is canonically isomorphic to thecomplex spinor bundle †M . Moreover, Theorem 2.39 shows that ifW denotes theWeingarten tensor ofM , the spin covariant derivatives rZ on †Z and r on †Mare related by

.rZX‰/jM D rX.‰jM / � 1

2W.X/ � .‰jM /; X 2 �.TM/;

for all spinors (resp. half-spinors for n odd) ‰ on Z . This motivates the followingde�nition.


Definition 8.42. A generalized Killing spinor on a spin manifold .M; g/ is a non-zero spinor 2 �.†M/ satisfying for all vector �elds X onM the equation

rX D A.X/ � ; (8.32)

where A 2 �.EndC.TM// is some �eld of symmetric endomorphisms called thestress-energy (or energy-momentum) tensor of .

We thus see that if ‰ is a parallel spinor on Z , its restriction to any hyper-surface M is a generalized Killing spinor on M . It is natural to ask whether theconverse holds.

Question. If is a generalized Killing spinor onM n, does there exist an isometricembedding of M into a spin manifold .ZnC1; gZ/ carrying a parallel spinor ‰whose restriction toM is ?

This question is the Cauchy problem for metrics with parallel spinors askedin [BGM05].

The answer is known to be positive in several special cases: if the stress-energytensor A of is the identity [Bär93], if A is parallel [Mor03], and if A isa Codazzi tensor [BGM05]. Even earlier, Friedrich [Fri98] had worked out thecase n D 2, which is also covered by [BGM05], Theorem 8.1, since on surfaces thestress-energy tensor of a generalized Killing spinor is automatically a Codazzi ten-sor. Some related embedding results were also obtained by Kim [Kim02], Lawn andRoth [LR10], and Morel [Mor05]. The common feature of these cases is that onecan actually construct in an explicit way the “ambient” metric gZ as a generalizedcylinder metric on the product .�"; "/�M (see also [BG92]).

Our aim in this section is to show that the same holds more generally, under thesole additional assumption that .M; g/ and A are analytic.

Theorem 8.43 ([AMM13]). Let be a spinor �eld on an analytic spin manifold.M n; g/, andA an analytic �eld of symmetric endomorphisms of TM . Assume that is a generalized Killing spinor with respect toA, i.e., it satis�es (8.32). Then thereexists a unique metric gZ of the form gZ D dt2 C gt with g0 D g on a suf�cientlysmall neighborhood Z of f0g �M inside R �M , such that .Z; gZ/, endowed withthe spin structure induced fromM , carries a parallel spinor ‰ whose restriction toM is .

In particular, the solution gZ must be Ricci-�at. Einstein manifolds are analytic,but of course hypersurfaces can lose this structure, so our hypothesis is restrictive.Einstein metrics with smooth initial data can be constructed for small time asmetricsof constant sectional curvature when the second fundamental form is a Codazzitensor; see [BGM05], Theorem 8.1. In particular, in dimensions n D 1 and n D 2

8.6. The Cauchy problem for Einstein metrics 243

Theorem 8.43 remains valid in the smooth category since the tensor A associatedto a generalized Killing spinor is automatically a Codazzi tensor.

Note that several particular instances of Theorem 8.43 have been proved in re-cent years, based on the characterization of generalized Killing spinors in termsof exterior forms in low dimensions. Indeed, in dimensions 5, 6 and 7, general-ized Killing spinors are equivalent to so-called hypo, half-�at and co-calibrated G2structures, respectively. Hitchin proved that the cases n D 6 and n D 7 can besolved up to the local existence of a certain gradient �ow [Hit01]. Later on, Contiand Salamon [CS06] and [CS07] treated the cases n D 5, n D 6, and n D 7 in theanalytical setting, cf. also [Con11] for further developments.

8.6 The Cauchy problem for Einstein metrics

Let .Z; gZ/ be an oriented Riemannian manifold of dimension n C 1, and M anoriented hypersurface with induced Riemannian metric

g ´ gZ jM :

We start by �xing some notations. Denote by rZ and r the Levi-Civita covariantderivatives on .Z; gZ/ and .M; g/, by � the unit normal vector �eld alongM com-patible with the orientations, and by W 2 �.EndC.TM// the Weingarten tensor,de�ned by

rZX � D �W.X/; X 2 TM: (8.33)

Using the normal geodesics issued fromM , the metric on Z can be expressed in aneighborhoodZ0 ofM as gZ D dt2Cgt , where t is the distance function toM andgt is a family of Riemannian metrics onM with g0 D g (cf. [BGM05]). The vector�eld � extends toZ0 as � D @=@t and (8.33) de�nes a symmetric endomorphism onZ0 which can be viewed as a family Wt of endomorphisms ofM , symmetric withrespect to gt , and satisfying (cf. [BGM05], equation (4.1))

gt .Wt .X/; Y / D �12

Pgt .X; Y /; X; Y 2 TM:

By [BGM05], equations (4.5)–(4.8), the Ricci tensor and the scalar curvature of Zsatisfy for every vectors X; Y 2 TM the relations

RicZ.�; �/ D tr.W 2t / � 1

2trgt . Rgt /; (8.34)

RicZ.�; X/ D d tr.Wt /.X/C ıgt .W /.X/; (8.35)

RicZ.X; Y / D Ricgt .X; Y /C 2gt .WtX;WtY /

C 1

2tr.Wt / Pgt .X; Y / � 1

2Rgt .X; Y /;

(8.36)


andScalZ D Scalgt C 3 tr.W 2

t / � tr2.Wt / � trgt . Rgt /; (8.37)

where in (8.35) the divergence operator

ıg WEnd.TM/ �! T�M

is de�ned in a local g-orthonormal basis feig of TM by

ıg.A/.X/ D �nX

iD1g..rei

A/.ei/; X/:

Using (8.34) and (8.37) we get

� 2RicZ.�; �/C ScalZ D Scalgt C tr.W 2t / � tr2.Wt /: (8.38)

Assume now that the metric gZ is Einstein with scalar curvature .nC 1/�, i.e.,RicZ D �gZ . Evaluating (8.35) and (8.38) at t D 0 yields

d tr.W /C ıgW D 0; (8.39)

and

Scalg C tr.W 2/ � tr2.W / D .n � 1/�: (8.40)

Ifgt WEnd.TM/ �! T�M ˝ T�M

is the isomorphism de�ned by

gt .A/.X; Y / ´ gt .A.X/; Y /

andg�1t WT�M ˝ T�M �! End.TM/

denotes its inverse, then taking (8.34) into account, (8.36) reads

Rgt D 2Ricgt C Pgt .g�1t . Pgt /�; �/� tr.g�1

t . Pgt // Pgt � 2�gt :

Using the Cauchy–Kowalewskaya theorem (see, e.g., [Die75]), Koiso provedthe following existence and unique continuation result for Einstein metrics startingfrom an analytic metric and an analytic stress-energy tensor satisfying the aboveconstraints.

Theorem 8.44 ([Koi81]). Let .M n; g/ be an analytic Riemannian manifold andlet W be an analytic �eld of symmetric endomorphisms on M satisfying (8.39)and (8.40). Then for " > 0, there exists a unique analytic germ near f0g � M

of an Einstein metric gZ with scalar curvature .nC1/� of the form gZ D dt2Cgton Z ´ R �M with g0 D g, whose Weingarten tensor at t D 0 is W .


Corollary 8.45. Assume that .M n; g/ is an analytic spin manifold carrying a non-trivial generalized Killing spinor with analytic stress-energy tensor A. Then ina neighborhood of f0g �M in Z ´ R �M there exists a unique Ricci-�at metricgZ of the form gZ D dt2 C gt whose Weingarten tensor at t D 0 isW D 2A.

Proof. We just need to check that the constraints (8.39) and (8.40) are a consequenceof (8.32). In order to simplify the computations, we will drop the reference to themetric g and denote respectively by r, R, Ric and Scal the Levi-Civita covariantderivative, curvature tensor, Ricci tensor and scalar curvature of .M; g/. As usual,feig will denote a local g-orthonormal basis of TM .

We will use the following classical formula in Clifford calculus which says thatthe Clifford contraction of a symmetric tensor A depends only on its trace:

nX

iD1ei � A.ei/ D � tr.A/: (8.41)

Now, let be a non-trivial generalized Killing spinor satisfying (8.32) anddenote W WD 2A. Being parallel with respect to a modi�ed connection on †M , is nowhere vanishing (and actually of constant norm).

Taking a further covariant derivative in (8.32) and skew-symmetrizing yields

RX;Y D 1

4.W.Y / �W.X/ �W.X/ �W.Y // �

C 1

2..rXW /.Y / � .rYW /.X// �

for all X; Y 2 TM . In this formula we set Y D ei , take the Clifford product withei and sum over i . From (8.41) and the classical formula for the Ricci tensor (2.3)we get

Ric.X/ � D �12

nX

iD1ei � .W.ei/ �W.X/ �W.X/ �W.ei // �

�nX

iD1ei � ..rXW /.ei/ � .rei

W /.X// �

D 1

2tr.W /W.X/ �

C 1

2

nX

iD1.�W.X/ � ei � 2g.W.X/; ei// �W.ei/ �

CX.tr.W // CnX

iD1ei � .rei

W /.X/ � ;


whence

Ric.X/ � D tr.W /W.X/ � �W 2.X/ � C X.tr.W //

CnX

iD1ei � .rei

W /.X/ � :(8.42)

We set X D ej in (8.42), take the Clifford product with ej , and sum over j .Using (8.41) again we obtain

�Scal D � tr2.W / C tr.W 2/ C r.tr.W // �

CnX

i;jD1ej � ei � .rei

W /.ej / �

D � tr2.W / C tr.W 2/ C d tr.W / �

CnX

i;jD1.�ei � ej � 2ıij / � .rei

W /.ej / �

D � tr2.W / C tr.W 2/ C 2d tr.W / � C 2ıW � ;

which implies simultaneously (8.39) and (8.40) (indeed, if f D X � for somereal f and vector X , then �jX j2 D X � X � D X � .f / D f 2 , so both fand X vanish). �

Theorem 8.46. Let .Z; gZ/ be a Ricci-�at spin manifold with Levi-Civita connec-tion rZ and letM � Z be any oriented analytic hypersurface. Assume there existssome spinor 2 �.†ZjM / which is parallel alongM :

rZX D 0; X 2 TM � TZ : (8.43)

Assume moreover that the map �1.M/ ! �1.Z/ induced by the inclusion is onto.Then there exists a parallel spinor ‰ 2 �.†Z/ such that ‰jM D .

Proof. Any Ricci-�at manifold is analytic, cf. [Bes87], p. 145, thus the analyticityofM makes sense.

Let � denote the unit normal vector �eld along M . Every x 2 M has anopen neighborhood V in M such that the exponential map .�"; "/ � V ! Z ,.t; y/ 7! expy.t�/, is well de�ned for some " > 0. Its differential at .0; x/ beingthe identity, one can assume, by shrinking V and choosing a smaller " if necessary,that it maps .�"; "/ � V diffeomorphically onto some open neighborhood U of xin Z . We extend the spinor to a spinor ‰ on U by parallel transport along thenormal geodesics expy.t�/ for every �xed y. It remains to prove that ‰ is parallelon U in horizontal directions.


Let feig be a local orthonormal basis along M . We extend it on U by paralleltransport along the normal geodesics, and note that fei ; �g is a local orthonormalbasis on U . More generally, every vector �eld X along V gives rise to a uniquehorizontal vector �eld on U also denoted X , satisfying r�X D 0. For every suchvector �eld we get

rZ� .rZ

X‰/ D RZ.�; X/‰ C rZ

Œ�;X�‰ D RZ.�; X/‰ C rZ

W.X/‰; (8.44)

where in this section RZ.�; X/‰ stands for RZ�;X‰.

Since Z is Ricci-�at, (2.3) applied to the local orthonormal basis fei ; �g of Zyields

0 D 12RicZ.X/ �‰ D

nX

iD1ei � RZ.ei ; X/‰C � � RZ.�; X/‰: (8.45)

We take the Clifford product by � in this relation, differentiate again with respect to�, and use the second Bianchi identity to obtain

rZ� .R

Z.�; X/‰/ D rZ�

��

nX

iD1ei � RZ.ei ; X/‰

�

D � �nX

iD1ei � .rZ

� RZ/.ei ; X/‰

D � �nX

iD1ei � ..rZ

eiRZ/.�; X/‰ C .rZ

XRZ/.ei ; �/‰/;

whence

rZ� .R

Z.�; X/‰/

D � �nX

iD1ei � .rZ

ei.RZ.�; X/‰/C RZ.W.ei/; X/‰ � RZ.�;rZ

eiX/‰

� RZ.�; X/rZei‰ C rZ

X .RZ.ei ; �/‰/ � RZ.rZ

X ei ; �/‰

C RZ.ei ; W.X//‰ � RZ.ei ; �/rZX‰/:

(8.46)

Let �? denote the distribution orthogonal to � on U and consider the sections

B;C 2 �..�?/� ˝†U/ and D 2 �.ƒ2.�?/� ˝†U/


de�ned for all X; Y 2 �? by

B.X/ ´ rZX‰;

C.X/ ´ RZ.�; X/‰;

D.X; Y / ´ RZ.X; Y /‰:

We have noted that the metric gZ is analytic since it is Ricci-�at. From the assump-tions thatM is analytic and that is parallel along M it follows that ‰, and thusthe tensors B , C , andD, are analytic.

Equations (8.44) and (8.46) read in our new notation

.rZ� B/.X/ D C.X/C B.W.X//;

and

.rZ� C/.X/ D � �

nX

iD1ei �..rZ

eiC/.X/CD.W.ei/; X/ � RZ.�; X/B.ei/

� .rZXC/.ei/CD.ei ; W.X//� RZ.ei ; �/B.X//:

(8.47)

Moreover, the second Bianchi identity yields

.rZ� D/.X; Y / D .rZ

� RZ/.X; Y /‰

D .rZXR

Z/.�; Y /‰ C .rZYR

Z/.X; �/‰

D rZX .R

Z.�; Y /‰/ � RZ.rZX �; Y /‰

� RZ.�;rZX Y /‰ � RZ.�; Y /rZ

X‰

� rZY .R

Z.�; X/‰/C RZ.rZY �; X/‰

C RZ.�;rZY X/‰ C RZ.�; X/rZ

Y ‰

D .rZXC/.Y /CD.W.X/; Y / � RZ.�; Y /rZ

X‰

� .rZY C/.X/CD.X;W.Y //C RZ.�; X/rZ

Y ‰;

thus showing that

.rZ� D/.X; Y / D.rZ

XC/.Y /CD.W.X/; Y / � RZ.�; Y /.B.X//

� .rZY C/.X/CD.X;W.Y //C RZ.�; X/.B.Y //:

(8.48)


Hypothesis (8.43) is equivalent to B D 0 for t D 0. Differentiating thisagain in the direction of M and skew-symmetrizing yields D D 0 for t D 0.Finally, (8.45) shows that C D 0 for t D 0. We thus see that the sectionS ´ .B; C;D/ vanishes along the hypersurface f0g � V of U .

The system (8.47)–(8.48) is a linear PDE for S and the hypersurfaces t Dconstant are clearly non-characteristic. The Cauchy–Kowalewskaya theorem showsthat S vanishes everywhere on U . In particular, B D 0 on U , thus proving ourclaim.

Nowweprove that there exists a parallel spinor‰ 2 �.†Z/ such that‰jM D .Take any x 2 M and an open neighborhood U like in Theorem 8.46 on whicha parallel spinor ‰ extending is de�ned. The spin holonomy group eHol.U; x/thus preserves ‰x . Since any Ricci-�at metric is analytic, the restricted spin holon-omy group eHol0.Z; x/ is equal to eHol0.U; x/ for every x 2 Z and for every openneighborhood U of x. By the local extension result proved above, eHol0.U; x/ actstrivially on ‰x , thus showing that ‰x can be extended (by parallel transport along

every curve in zZ starting from x) to a parallel spinor z‰ on the universal cover zZof Z . The group of deck transformations acts trivially on z‰ since every element in�1.Z; x/ can be represented by a curve inM (here we use the surjectivity hypothe-sis) and‰ was assumed to be parallel alongM . Thus z‰ descends to Z as a parallelspinor. This completes the proof of Theorem 8.46. �

Theorem 8.43 is now a direct consequence of Corollary 8.45 and Theorem 8.46.

Chapter 9

Special spinors on conformal manifolds

In this chapter we continue the study of conformal spin manifolds initiated in Chap-ter 2. The �rst section is devoted to the conformal analogue of the Schrödinger–Lichnerowicz formula, which is derived in the framework of conformal geometry(i.e., without making use of Riemannian metrics). We then give in the next sectionan application of this formula to the problem of the existence of parallel spinors onconformal manifolds. Finally, the Hijazi inequality (see Chapter 5) is proved in thelast section by using methods of conformal spin geometry. The notations are thoseof Section 2.2.3.

9.1 The conformal Schrödinger–Lichnerowicz formula

Let .M n; c/ be a conformal spin manifold and let D be a Weyl structure on M .Recall that the Clifford product (denoted by ) of 1-forms on weighted spinorslowers the conformal weight by one. We de�ne the conformal Dirac operatorD.k/

(or simply D when there is no risk of confusion), mapping sections of †.k/M tosections of †.k�1/M , as the composition D.k/ WD ıD,

�.†.k/M/D�! �.T�M ˝†.k/M/

�! �.†.k�1/M/:

The conformal invariance of theDirac operator, �rst observed byHitchin [Hit74],can be stated in the following way:

Theorem 9.1. The Dirac operator D.k/ is independent of D for k D 1�n2.

In particular, for every Riemannian metric g in the conformal class c, it coin-cides with the usual Dirac operator after the identi�cation, de�ned in Section 2.2.3,of the weighted spinor bundles with the g-spinor bundle.

Proof. Let g be a (local) metric in the conformal class c. It is suf�cient to observethat, according to Theorem 2.21, the conformal Dirac operator D.k/ written in thismetric satis�es

D.k/ˆ D DˆC�k C n� 1

2

�� ˆ: �

We now prove the following conformal analogue of Corollary 2.8.

252 9. Special spinors on conformal manifolds

Proposition 9.2. Let ˆ be a spinor of weight k on a conformal manifoldM . Then,for every local conformal frame feig and vector X onM ,

nX

iD1e�i �RD

X;eiˆ D �1

2RicDs .X/ �ˆ� 1

2X �F �ˆC

�kC n � 4

4

�.X ³F / �ˆ; (9.1)

where RicDs denotes the symmetric part of the Ricci tensor and F is the Faradayform of D.

Proof. We �x a local conformal basis feig, i.e., satisfying c.ei ; ej / D l2ıij forsome section l of L. Then its dual basis can be written as e�

i D ei l�2 (via the

identi�cation by c of T�M with TM ˝ L�2).We start by noting that (with the notations of Section 2.2.3)

RD;.k/X;Y ˆ D 1

2RDa .X; Y / �ˆC kF.X; Y /ˆ; (9.2)

the proof being similar to that of Theorem 2.7. Then, using the “Bianchi-type”relation (2.10), Lemma 2.16, and the Clifford relations

e�j � e�

k � e�i D ei � e�

j � e�k � 2l�2e�

j ıik C 2l�2e�kıij

similarly to (2.4), we get

X

i

e�i �RDa .X; ei / �ˆ

D 1

2

X

i;j;k

c.RDa .X; ei/ej ; ek/e�i � e�

j � e�k �ˆ

D 1

6Si;j;k

X

i;j;k

c.RDa .X; ei /ej ; ek/e�i � e�

j � e�k �ˆ �

X

i

RicD.X; ei/e�i �ˆ

D 1

6Si;j;k

X

i;j;k

F.ej ; ei/c.X; ek/e�i � e�

j � e�k �ˆ

� RicDs .X/ˆC n � 22

.X ³ F / �ˆ

D �X � F �ˆ� RicDs .X/ˆC n� 4

2.X ³ F / �ˆ:

This relation, together with (9.2), proves the desired formula. �

9.1. The conformal Schrödinger–Lichnerowicz formula 253

Wenow introduce the conformal spin Laplacian�D mapping sections of†.k/Mto sections of †.k�2/M , as the composition �D WD �.c ˝ Id/ ıD ıD,

�.†.k/M/D�! �.T�M ˝†.k/M/

D�!�.T�M ˝ T�M ˝†.k/M/

�c˝Id��! �.†.k�2/M/:

We end this section by proving the following theorem.

Theorem 9.3 (Conformal Schrödinger–Lichnerowicz formula). The square of theconformal Dirac operator and the conformal spin Laplacian, both acting on thespinor bundle of weight k, are related by

D.k�1/ ı D.k/ˆ D �DˆC ScalD

4ˆC

�k C n � 2

2

�F �ˆ;

where ScalD is the (L.�2/-valued) conformal scalar curvature ofD.

Proof. Consider a local conformal frame feig as before. For a local section ˆ of†.k/M we have

�Dˆ D �X

i;j

c ˝ Id.e�i ˝Dei

.e�j ˝Dejˆ//

D �X

i;j

c ˝ Id.e�i ˝ e�

j ˝DeiDejˆC e�

i ˝Deie�j ˝Dejˆ/

D l�2X

i

.DDeieiˆ �Dei

Deiˆ/:

On the other hand, using the Clifford relations we compute

D.k�1/ ı D.k/ˆ DX

i;j

e�i �Dei

.e�j �Dejˆ/

DX

i;j

.e�i �Dei

e�j �DejˆC e�

i � e�j �Dei

Dejˆ/

D �X

i;j

e�i � e�

j �DDeiejˆ � l�2

X

i

DeiDei

ˆ

C 1

2

X

i;j

e�i � e�

j � .DeiDejˆ �DejDei

ˆ/

D �DˆC 1

2

X

i;j

e�i � e�

j � RD;.k/ei ;ej

ˆ;


where RD;.k/ denotes the curvature tensor of the connection D acting on spinorsof weight k. We thus have

D.k�1/ ı D.k/ˆ D �DˆC 1

2

X

i;j

e�i � e�

j � RD;.k/ei ;ej

ˆ;

which together with (9.1) yields the desired result. �

The conformal Schrödinger–Lichnerowicz formula is a result due to V. Buch-holz [Buc98], P. Gauduchon [Gau95a], and A. Moroianu [Mor98a].

9.2 Parallel spinors with respect to Weyl structures

Consider the following question.

Problem 9.4. Classify the triples ..M; c/;D; k/, whereM is an n-dimensional spinmanifold with conformal structure c,D is a Weyl structure onM and k 2 R, whichadmit D-parallel spinors of weight k.

We will give a partial answer to this question in the non-compact case, and solveit completely forM compact.

The �rst remark is that, if .M; g/ is a Riemannian spin manifold with parallelspinors, then ..M; Œg�/;rg ; k/ is a solution to our problem for every k, by The-orem 2.21. Such solutions are called Riemannian. The Weyl structure of everyRiemannian solution is exact and conversely, a solution with exact Weyl structure isRiemannian. Indeed, ifD is exact, we can choose a metric g in the conformal classc such that D D rg , and the D-parallel spinor can be identi�ed with a parallelspinor on .M; g/, again by Theorem 2.21.

The second remark is that every solution ..M; c/;D; k/ of our problem withk ¤ 0 is Riemannian. Indeed, if ˆ is a D-parallel spinor of weight k ¤ 0, thenkˆk 1

k is aD-parallel section of L, so, by de�nition, D is exact. Here k � k denotesthe norm associated with the L2k-valued Hermitian product on †.k/M .

Accordingly, from now on we will only considerD-parallel spinors of weight 0.In particular, every such spinor has constant (scalar-valued) norm.

9.2.1 Parallel conformal spinors on Riemann surfaces

It happens quite often in conformal geometry that the 2-dimensional case has tobe treated separately. In this section we consider our problem on spin Riemannsurfaces .M; c/.

9.2. Parallel spinors with respect to Weyl structures 255

Recall that a conformal structure in dimension 2 is equivalent to a complex struc-ture. Up to a change of orientation, we may suppose that the parallel spinor lies in†CM . Let D be a Weyl structure on M , let g be a metric in the conformal classc and let � be the Lee form of D with respect to g (identi�ed by g with a vector�eld).

Lemma 9.5. A spinor �eld ‰ 2 �.†CM/ isD-parallel if and only if it satis�es

rgX‰ D i

2g.X; J�/‰; X 2 �.TM/:

Proof. For every ‰ 2 �.†CM/ we have JX � X �‰ D i jX j2‰, so

ŒX � � C �.X/� �‰ Dhg�X;

�

j� j� �

j� j � � C g�X;J�

j� j�J�

j� j � � C g.X; �/i

�‰

D ig.X; J�/‰:

The statement now follows immediately from Theorem 2.21. �

This lemma implies that every spinor ‰ of constant norm on a Riemann surfaceis parallel with respect to some Weyl structure. Indeed, we can write

rgX‰ D ˛.X/‰

for some purely imaginary-valued 1-form ˛ and consider the Weyl structure Dwhose Lee form � satis�es

i

2J� D ˛:

However, this does not answer our problem, since we need to determine whethera given Weyl structure D admits D-parallel spinors. We have already seen that anecessary condition is the triviality of the spinor bundle †CM . Let ˆ be a unitsection of †CM and de�ne a 1-form ˛ by

rgXˆ D i˛.X/ˆ:

Here g is a �xed metric onM and we denote by � the Lee form ofD with respect tog. Every spinor of constant norm can be written as f ˆ for some map f WM ! S

1.By Lemma 9.5, it is clear that f ˆ is D-parallel if and only if

1

2J� � ˛ D �idf

f:

Then the answer to our problem is given by the following lemma.


Lemma 9.6. A 1-form on a manifold M can be written as idff

for some mapf WM ! U1, if and only if it is closed and has integral cohomology class.

Proof. Let A be the sheaf of germs of U1-valued functions and let F be the sheafof germs of closed 1-forms onM . The exact sequence of sheaves

0 �! U1 �! Af 7! idf

f��! F �! 0

induces a long exact sequence of cohomology groups

� � � �! H 0.M;A/ �! H 0.M;F/ı�! H 1.M;U1/ �! � � � :

This exact sequence shows that a closed 1-form ! can be written as idff

for somef WM ! U1 if and only if ı.!/ D 0. On the other hand, we also have a long exactsequence

� � � �! H 1.M;Z/ �! H 1.M;R/exp�! H 1.M;U1/ �! � � �

and, moreover, if Œ � � denotes the natural mapping H 0.M;F/ ! H 1.M;R/, thenexp ıŒ � � D ı (by naturality). Thus ı.!/ D 0 if and only if exp.Œ!�/ D 0, which, bythe last exact sequence, is equivalent to Œ!� 2 H 1.M;Z/. �

We thus, have proved the following theorem.

Theorem 9.7. Let .M; c/ be a spin Riemann surface with a Weyl structureD, let gbe a metric in the conformal class c, and let � be the Lee form ofD with respect to g.Then †CM carries aD-parallel spinor �eld if and only if it is trivial .as complexline bundle/ and 1

2J��˛ is a closed integral form, where i˛ denotes the connection

form of the Levi-Civita spin connection written in some arbitrary gauge of †CM .

9.2.2 The non-compact case

Consider now the n-dimensional case, with n > 2.

Theorem 9.8. Let .M n; c/ .n > 2/ be a conformal spin manifold with Weyl struc-ture D. IfM carries aD-parallel spinor �eld ‰ and n ¤ 4, thenD is closed.

Proof. The conformal Schrödinger–Lichnerowicz formula implies that

ScalD‰ D �2.n� 2/F � ‰: (9.3)

By taking here the scalar product with ‰, we obtain

ScalDjj‰jj2 D �2.n� 2/hF �‰;‰i:


As the left term is real and the right term is purely imaginary, they must vanish, soin particular ScalD D 0. As n > 2, (9.3) shows that

F �‰ D 0: (9.4)

This equation, together with (9.1) gives

2RicDs D .n � 4/F; (9.5)

so, for n ¤ 4, the Faraday form must vanish. �

This theorem says that – in every dimension other than 2 and 4 – the solutions toProblem 9.4 are locally Riemannian (i.e., everyD-parallel spinor is locally rg -par-allel for some local metric g). In the previous section we have seen that this theoremdoes not hold in dimension 2, and the example below shows that it also fails forn D 4. We �rst emphasize the special relations between forms and spinors whichhold on 4-dimensional manifolds.

Recall that for n even, every weighted spinor bundle decomposes under theaction of the complex volume element !C 2 ƒnM˝Ln as†M D †CM˚†�M ,where

†˙M D f‰I !C �‰ D ˙‰g: (9.6)

This decomposition is preserved by every Weyl structure D since the volume ele-ment isD-parallel. As usual, the projections of a spinor‰ on†˙M are denoted by‰˙. Then (9.6) shows that changing the orientation ofM is equivalent to switch-ing from †CM to †�M . Thus, if ‰ is a D-parallel spinor, we can assume that‰ 2 �.†CM/.

Now letM be 4-dimensional. In this case, the conformal Hodge operator � actsinvolutively onƒ2M ˝L2. On the other hand,ƒ2M ˝L2 is canonically isomor-phic to the bundle A.M/ of skew-symmetric endomorphisms of the tangent bundle.Hence we obtain an orthogonal decomposition A.M/ D AC.M/˚A�.M/, whereAC.M/ (resp. A�.M/) denotes the eigenspace of � corresponding to the eigen-value 1 (resp. �1). The elements of AC.M/ (resp. A�.M/) are called self-dual(resp. anti-self-dual) endomorphisms. Using the isomorphism between A.M/ andƒ2M˝L2, we see that the skew-symmetric endomorphisms act by Clifford producton spinors and preserve their weight.

Lemma 9.9. (i) If A 2 A�.M/ is a �xed anti-self-dual endomorphism, then therestriction to †CM of the Clifford product by A is the zero endomorphism.

(ii) The endomorphism uWAC.M/ ! †CM de�ned by A 7! A � ‰ is injectivefor every �xed spinor ‰ 2 †CM .


Proof. (i) The Clifford product by A is a skew-Hermitian endomorphism of†CM .By (1.15), the weightless volume form vol satis�es vol D �!C. If A is anti-self-dual we thus have

A � A �‰ D .A ^ A/ � ‰ � hA;Ai‰D �.A ^ �A/ �‰ � hA;Ai‰D �hA;Aivol �‰ � hA;Ai‰D 0;

for all ‰ 2 †CM . The statement follows from the fact that a skew-Hermitianendomorphism whose square vanishes must itself vanish.

(ii) If A 2 Ker.u/, then

0 D A � A �‰D .A ^ A/ � ‰ � hA;Ai‰D .A ^ �A/ �‰ � hA;Ai‰D hA;Aivol �‰ � hA;Ai‰D �2hA;Ai‰;

so A D 0. �

Now, by Theorem 2.6, the spinor bundle of a 4-dimensional manifold carries aquaternionic structure, i.e., a parallel complex anti-linear endomorphism j commut-ing with the Clifford product by real vectors and satisfying j2 D �1. In particular,j preserves the decomposition †M D †CM ˚†�M .

Moreover, the uniqueness of the Hermitian product on spinors shows that thereexists some positive real constant ˛ such that

hj‰; jî D ˛h‰;î; ‰;ˆ 2 �.†M/:

Applying this twice, we get ˛2 D 1. Thus j satis�es

hj‰; jî D h‰;î: (9.7)

Lemma 9.10. There is a canonical isomorphism between the complexi�cation ofthe bundle of self-dual endomorphisms and the second symmetric power of †CM .


Proof. Indeed, let Ai , i D 1; 2; 3 be a local orthonormal basis of AC.M/. Theendomorphism

F W†CM ˝†CM �! AC.M/˝ C

de�ned byF.‰ ˝ˆ/ D

X

i

AihAi �‰; jî

is symmetric by (9.7), so it descends to an endomorphism

FsW Sym2.†CM/ 7�! AC.M/˝ C:

Using Lemma 9.9 and a local basis of Sym2.†CM/ we easily obtain that Fs isinjective, thus an isomorphism (for dimensional reasons). �

Definition 9.11. Let J be an almost complex structure on a conformal manifold.M; c/ with Weyl structure D. The 1-form �DJ given by

�DJ .X/ D trfV 7! J.DV J /Xg

is called the Lee form ofD with respect to J .

Lemma 9.12. On a conformal manifold with almost complex structure J , thereexists exactly one Weyl structure D whose Lee form with respect to J vanishes.

Proof. Choose a metric g in the conformal class c, and let D be an arbitrary Weylstructure whose Lee form with respect to g is � . An easy calculation using for-mula (2.8) yields

�DJ D �J ıg�C .n � 2/J�;where �.X; Y / ´ g.JX; Y /. Thus �DJ D 0 if and only if � D 1

n�2 ıg�. �

The connection D is sometimes called the canonical Weyl structure associatedto .M; c; J /. Note that if J isD-parallel for some Weyl structure D, thenD has tobe the canonical Weyl structure and moreover J has to be integrable, being parallelwith respect to a torsion-free connection. Conversely, in dimension 4, we have thefollowing

Proposition 9.13. A Hermitian structure on a conformal 4-dimensional manifoldis parallel with respect to its canonical Weyl structure.

Proof. Let D be the canonical Weyl structure of J and let �.X; Y / ´ c.JX; Y /

be the associated (L2-valued) Kähler form. It is straightforward to check that indimension 4 one has dD� D ıD� ^�. In our case, this implies that dD� D 0.The fact that J isD-parallel is then a direct consequence of the conformal analogueof [KN69], Proposition 4.2, Chapter IX. �


Definition 9.14. A hyper-Hermitian structure on a conformal manifold .M; c/ isgiven by three complex structures J1; J2; J3 which are compatible with c and satisfythe quaternionic relations.

Lemma 9.15. If J1; J2; J3 de�ne a hyper-Hermitian structure, then for each Weylstructure D, the Lee forms �DJi

are all equal.

Proof. It is well known that an almost complex structure J is integrable if and onlyif JDXJ D DJXJ , no matter what Weyl structure D is used. We then compute,using a local conformal basis feig

nX

iD1.Dei

J1/J2ei DnX

iD1.DJ1ei

J1/J2J1ei

D �nX

iD1.J1Dei

J1/J1J2ei

D �nX

iD1.Dei

J1/J2ei ;

sonX

iD1.Dei

J1/J2ei D 0:

This yields

�DJ3D

nX

iD1J3.Dei

J3/ei DnX

iD1J3f.Dei

J1/J2ei C J1.DeiJ2/eig

DnX

iD1J3J1.Dei

J2/ei

D �DJ2: �

From Proposition 9.13 and Lemma 9.15 we obtain the following corollary.

Corollary 9.16. On a 4-dimensional hyper-Hermitian conformal manifoldM thereexists a unique Weyl structure with respect to which the three Hermitian structuresare parallel.

Theorem 9.17. LetM be a 4-dimensional conformal spin manifold. IfM carries aWeyl structureD and aD-parallel spinor, then it is hyper-Hermitian. The converseis also true ifM is simply connected.


Proof. Let‰ be aD-parallel spinor, and suppose as before that it lies in �.†CM/.Then f‰; j‰g de�nes a parallelization of †CM , so, by Lemma 9.10, one can �ndthreeD-parallel orthonormal sections J1; J2; J3 of AC.M/. It is easy to check thatJi are almost Hermitian structures satisfying the quaternion relations. Their inte-grability is obvious, since they are parallel with respect to a torsion-free connection.

Conversely, suppose thatM has a hyper-Hermitian structure J1; J2; J3. Corol-lary 9.16 implies that Ji are all D-parallel for some Weyl structure D onM . Thisshows that AC.M/ is D-�at, thus Sym2.†CM/ is D-�at, too. Note that forevery non-zero spinor ‰, the symmetric product with ‰ maps †CM injectivelyinto Sym2.†CM/. Let ‰ ˇ‰ be a section of Sym2.†CM/. Then

0 D RDX;Y .‰ ˇ ‰/ D 2.RD

X;Y‰/ˇ‰;

so by the above remark, †CM is itself D-�at. IfM is simply connected, one canthus �nd aD-parallel spinor. �

Example 9.18 ([Apo96]). Let uWU � C2 ! C

� be a smooth function and considerthe Riemannian metric on U given by

gu D dz1d Nz1 C juj2dz2d Nz2:

The relations�1 C i�2 D Nudz1 ^ d Nz2

and�3 D i.dz1 ^ d Nz1 C juj2dz2 ^ d Nz2/

de�ne the Kähler forms of three almost Hermitian structures J1; J2; J3 on .U; gu/.A straightforward computation shows that the Lee forms of J1 and J2 are both

equal to

� D 1

u

@u

@z1dz1 C 1

Nu@ Nu@ Nz1

d Nz1 C 1

Nu@ Nu@z2

dz2 C 1

u

@u

@ Nz2d Nz2;

while the Lee form of J3 is

� D 2@

@z1ln jujdz1 C 2

@

@ Nz1ln jujd Nz1:

Moreover, J1; J2; J3 are integrable if and only if � D � , which is equivalent to ubeing holomorphic.

Thus, if u is holomorphic, fJ1; J2; J3g de�ne a hyper-Hermitian structure on.U; gu/. In that case, the Weyl structure de�ned by each of fJ1; J2; J3g is closed ifand only if d� D 0, which is equivalent to

u2@2u

@z1@z2D @u

@z1

@u

@z2: (9.8)


By Theorem 9.17, we deduce that if U is simply connected and u is a non-zero holomorphic function on U which does not satisfy (9.8), then .U; gu/ carriesparallel spinors with respect to the non-closed Weyl structure de�ned above.

9.2.3 The compact case

In this section wewill provide a complete solution to Problem 9.4 under the assump-tion that M is compact. The �rst step is to note that, in the compact case, Theo-rem 9.8 also holds for n D 4. Indeed, if we express equation (9.4) in a �xed metric,Lemma 9.9 implies that the self-dual part of the Faraday form F of D vanishes.Since F is exact, the compactness ofM easily implies F D 0. This proves

Corollary 9.19. Let .M n; c/, n > 2, be a compact conformal spin manifold withWeyl structure D, carrying a D-parallel spinor ‰. ThenD is closed.

Actually, much more can be said on such manifolds.

Theorem 9.20. A conformal manifold .M; c/ with Weyl structure D carries aD-parallel spinor if and only if there exists a metric g in the conformal class c.which turns out to be unique up to a constant rescaling/ such that one of the twostatements below holds:

• D D rg and .M; g/ is a spin manifold with parallel spinors;

• the universal cover of .M; g/ is isometric toN �R, whereN is a Riemannianmanifold with Killing spinors, and D is the Weyl structure on M whose Leeform with respect to g is dt .

Proof. The main ingredient of the proof is the use of the standard or Gauduchonmetric associated with every Weyl structure (in dimension greater than 2). Recallthe following theorem.

Theorem 9.21 ([Gau84]). Let .M; c/ be a compact conformal manifold of dimen-sion n > 2 and let D be a Weyl structure onM . Then c contains a unique metricg0 .up to homothety/ such that the Lee form ofD with respect to g0 is g0-coclosed.This metric is called the standard metric associated toD.

Now, Corollary 9.19 together with (9.5) show that D is a Weyl–Einstein struc-ture (see De�nition 2.17). Let g be the standard metric of D, and denote by � theLee form of D with respect to g. A fundamental observation by P. Tod is that theLee form of a closedWeyl–Einstein structure on a compact manifold is parallel withrespect to the standard metric (cf. [Gau95b]). We distinguish 2 cases.

If the Lee form � vanishes, thenD D rg and theD-parallel spinor of†.0/.M; c/induces a parallel spinor on †.M; g/.

9.3. A conformal proof of the Hijazi inequality 263

The second case is when � is a non-zero parallel form on .M; g/. By rescalingthe metric, we can suppose that � has unit norm. Let zM be the universal coverofM , which inherits by pull-back the Weyl structure D, the D-parallel spinor andthe metric g fromM . By the de Rham decomposition theorem, . zM;g/ is isometrictoN�R for some Riemannian (spin) manifoldN . With respect to this identi�cation,we have of course � D dt . Let ‰ be the D-parallel spinor �eld, expressed in themetric g. By Theorem 2.21, it satis�es

rgX‰ D 1

2X � � �‰ C 1

2�.X/‰:

As before, by changing the orientation ofM if necessary, we can suppose that, for neven,‰ is a non-zero section of†CM . Then Theorem 2.38 shows that (for every n),by restriction ‰ induces a well-de�ned spinor onN , which, by (2.27) and (2.28), isa Killing spinor with Killing constant 1

2. Conversely, the same equations show that

a manifold satisfying one of the two cases above has a Weyl structure with parallelspinors. This proves the theorem. �

9.3 A conformal proof of the Hijazi inequality

In this section we give a proof of the Hijazi inequality which makes essential use ofconformal geometry.

Theorem 9.22. Let .M n; g/ be a compact Riemannian spin manifold of dimensionn � 3. Then, the �rst eigenvalue �1 of the Dirac operator on the spinor bundlesatis�es

�21 � n

4.n� 1/�1;

where �1 is the �rst eigenvalue of the scalar curvature operator Lg de�ned by

Lg D 4n � 1

n � 2�g C Scalg :

Proof. Consider an arbitrary conformal change of the metric given by Ng D f �2g.We have de�ned in Section 2.2.3 a family of isomorphisms

ˆ.k/W†gM �! † NgM;

which satisfy (2.13).Every choice of a Weyl structure on .M; Œg�/ induces Dirac and Penrose oper-

ators, Dg and Pg , acting on †gM , as well as weighted Dirac and Penrose opera-tors,D andP , which act on the weighted spinor bundles and lower the weights by 1.


Moreover, for k D �.n� 1/=2 (resp. k D 1=2) the weighted Dirac operator (resp.Penrose operator) is conformally invariant (i.e., does not depend on the choice ofthe Weyl structure), by Theorem 9.1. As a consequence, this yields

Dg D .ˆ.�12 .1Cn///�1 ı D Ng ıˆ. 1

2 .1�n//;

and

.Pg/X D .ˆ.12 //�1 ı .P Ng /X ıˆ. 1

2 /:

The Schrödinger–Lichnerowicz formula relating the Dirac and Penrose opera-tors associated to a metric g can be written as

n� 1

n.Dg/�.Dg/ � 1

4Scalg D .Pg/�.Pg/ :

By integration overM with respect to the volume element �g we getZ

M

�n� 1

njDg j2 � 1

4Scalg j j2

��g D

Z

M

jPg . /j2g�g :

This formula also holds for the metric Ng D f �2g and the spinor x D ˆ. /,where ˆ is the isomorphism ˆ.

1�n2 / above. Hence

Z

M

�n � 1

njD Ng x j2 � 1

4Scal Ng j x j2

�� Ng D

Z

M

jP Ng . x /j2Ng� Ng ; (9.9)

On the other hand, we obviously have

� Ng D f �n�g ; jD Ng x j2Ng D f nC1jDg j2g ; jP Ng x j2Ng D f jPg .f n2 /j2g ;

and moreover, by 5.19,

Scal Ng D h� nC2n�2 Lgh;

where h D f � n�22 . Plugging these formulas into (9.9) yieldsZ

M

�n � 1n

jDg j2 � 1

4h�1.Lgh/j j2

�h� 2

n�2 �g

DZ

M

f 1�njPg .f n2 /j2g�g :

(9.10)

Let us now choose a spinor and a function h such that

Dg D �1 and Lgh D �1h:

By the maximum principle, h is everywhere strictly positive, so (9.10) yields

�21 � n

4.n � 1/ �1

as required. �

Chapter 10

Special spinors on Kähler manifolds

It was pointed out in Section 5.6 that Friedrich’s inequality (see [Fri80]) is not sharpon Kähler manifolds, for which better estimates hold (see Kirchberg’s inequali-ties, Theorem 6.1). In this chapter we show that Kirchberg’s inequalities are sharp,we characterize their limiting cases and we give an application in complex contactgeometry.

The �rst section is devoted to twistor theory, which will play a central role inthe sequel. The next section treats the limiting case of Kirchberg’s inequality inodd complex dimensions, following closely [Mor95]. An important application ofthis result – the characterization of positive Kähler–Einstein contact manifolds – isgiven in Section 10.3, making use of Spinc geometry.

The last section in this chapter deals with the limiting case of Kirchberg’sinequality in even complex dimensions. This problem was �rst studied by A. Lich-nerowicz [Lic90] and the complete solution was obtained in [Mor97a] and [Mor99].

10.1 An introduction to the twistor correspondence

The twistor theory of self-dual Einstein 4-manifolds was introduced by R. Pen-rose [PM73] and [Pen76] and further studied by Atiyah, Hitchin, and Singer intheir seminal paper [AHS78]. A few years later, this theory was extended both byS. Salamon [Sal82] and L. Bérard Bergery [Ber79] to quaternion-Kähler manifoldsand turned out to be closely related with the Ishihara–Konishi theory of 3-Sasakianbundles over positive quaternion-Kähler manifolds.

The aim of this section is to brie�y recall the interplay of these theories.For further details the reader is referred to the surveys [BG99] and [Sal99].

10.1.1 Quaternion-Kähler manifolds

Quaternion-Kähler Manifolds were introduced in Chapter 7. For the convenienceof the reader we brie�y review the basic de�nitions and notations.

Let GLn.H/ be the algebra of n�nmatrices with quaternionic entries, acting onthe left on H

n. Using the canonical identi�cation Hn Š R

4n, the algebra GLn.H/

266 10. Special spinors on Kähler manifolds

becomes a sub-algebra of GL4n.R/. Let Spn be the subgroup of SO4n de�ned by

Spn D SO4n \GLn.H/

and consider the embedding of the group Sp1 of unit quaternions into SO4n givenby

q 2 Sp1 �! A.q/ 2 SO4n; A.q/v ´ vq�1; v 2 R4n:

We de�ne Sp1 � Spn to be the group generated by these two subgroups of SO4n. It iseasy to check that Sp1 � Spn Š Sp1 �Z=2Z Spn. Note that Sp1 � Sp1 Š SO4. Fromnow on we assume n � 2.

Definition 10.1. A quaternion-Kähler manifold is a 4n-dimensional RiemannianmanifoldM .n � 2/ satisfying one of the following equivalent conditions:

• The holonomy ofM is contained in Sp1 � Spn.• There exists a parallel 3-dimensional sub-bundle QM of End.M/, called thequaternionic bundle, locally spanned by three endomorphisms I , J , and Ksatisfying the quaternion relations.

The equivalence of the two conditions follows from the fact that the vector sub-space of End.Hn/ given by right multiplication with imaginary quaternions is leftinvariant by the adjoint action of Sp1 � Spn.

Note that the local endomorphisms I , J , andK do not extend, in general, to thewhole manifold, unless the holonomy ofM is contained in Spn.

Theorem 10.2 ([Ber66]). Every quaternion-Kähler manifold is Einstein.

Proof. The argument goes roughly as follows. If G � SOn is the holonomy groupof a Riemannian manifold, then its curvature tensor at every point belongs to thespace R.g/ of algebraic curvature tensors, de�ned as the kernel of the Bianchi map

bW Sym2.g/ �! ƒ4.Rn/; b.! ˇ !/ D ! ^ !:The theorem then follows from the fact that

R.sp1spn/ D R.spn/˚ R;

where the one-dimensional summand R is generated by the curvature tensor of thequaternionic projective space HPn. �

Depending on the value of the Einstein constant, one can characterize more pre-cisely quaternion-Kähler manifolds in terms of holonomy: if the Einstein constantvanishes, then the manifold is hyper-Kähler and its restricted holonomy is a sub-group of Spn. If the Einstein constant is non-zero, then the manifold is irreducible,and its restricted holonomy group is a subgroup of Sp1 � Spn not contained in Spn.

A quaternion-Kähler manifold is called positive if it has positive scalar curva-ture.

10.1. An introduction to the twistor correspondence 267

10.1.2 3-Sasakian structures

We now focus our attention on 3-Sasakian structures (which were introduced in Sec-tion 8.2.2). If .S; g; �1; �2; �3/ is a 3-Sasakian manifold, the distribution V spannedby �1; �2, and �3 is integrable by (8.11), and totally geodesic by (8.13). This lastequation shows, moreover, that the integral leaves of V have constant sectional cur-vature 1. If S is compact, every maximal integral leaf is therefore isometric to somelens space S

3=� . This induces, in particular, an isometric S3-action on every com-

pact 3-Sasakian manifold.Recall that the action of a Lie group G on a smooth manifold is called regular

if the quotientM=G is smooth. A compact 3-Sasakian manifold is called regular ifthe S

3-action de�ned above is regular.There is a close relation between regular 3-Sasakian and positive quaternion-

Kähler manifolds, provided by the following theorem.

Theorem 10.3 ([Ish73], [Ish74], [IK72], and [Kon75]). The quotient of a regu-lar 3-Sasakian manifold S by the above-de�ned isometric S

3 action is a positivequaternion-Kähler manifold. Conversely, if M is a positive quaternion-Kählermanifold, the SO3-principal bundle S associated with the quaternionic bundleQM(cf. De�nition 10.1) carries a 3-Sasakian metric which turns the bundle projectionS ! M into a Riemannian submersion with totally geodesic �bers.

Proof. If S is a 3-Sasakian manifold, the quotientM ´ S=S3 is exactly the spaceof leaves of the distribution V . In the regular case, the pull-back of the tangentbundle ofM by the submersion � WS ! M can be identi�ed with V?. Every tensoron V? “constant along V” (i.e., whose Lie derivative with respect to �i vanishes)induces by projection a tensor onM .

In particular, the restriction of the metric of S to V? induces a metric on Mmaking � a Riemannian submersion. The restrictions to V? of the endomorphisms'i introduced in De�nition 8.14 satisfy the quaternion relations, but do not projecttoM . Nevertheless, every local section � of the �bration S ! M induces a localquaternionic structure fI � ; J � ; K�g onM by

I �x ´ ��..'1/�.x//; J �x ´ ��..'2/�.x//; K�x ´ ��..'3/�.x//:

It remains to check (using the properties of 3-Sasakian structures and the O’Neillformulas for Riemannian submersions) thatM is Einstein with positive scalar cur-vature and that the sub-bundle of End.TM/ spanned by I � , J � and K� does notdepend on � and is parallel with respect to the Levi-Civita connection ofM . Theconverse is similar. �

The above result shows that the study of (regular) 3-Sasakian manifolds is equiv-alent to that of positive quaternion-Kähler manifolds. For later use, let us mentionthe following lemma.


Lemma 10.4. A compact 3-Sasakian manifold is regular if and only if at least oneof the Sasakian structures is regular.

Proof. Assume that .S; g; �1; �2; �3/ is regular and let � WS ! M denote the projec-tion of S onto its quotient by the S

3 action. The manifold M is quaternion-Kählerby Theorem 10.3. Let Z denote the sphere bundle of the quaternionic bundle QMofM . We claim that the map f1WS ! Z given by x 7! ��..'1/x/ is a submersionwhose �bers are exactly the integral curves of �1. This follows easily from the factthat �1.f1/ D ��.L�1

'1/ D 0 (showing that f1 is constant along the �ow of �1)and from the fact that

.a�2 C b�3/.f1/ D ��.La�2Cb�3'1/ D ��.�2a'3 C 2b'2/;

is non-zero, whenever .a; b/ ¤ .0; 0/. For the converse statement, see [Tan71]. �

10.1.3 The twistor space of a quaternion-Kähler manifold

The manifold Z introduced in the proof of Lemma 10.4 turns out to play a centralrole in the theory of quaternion-Kähler manifolds.

Definition 10.5 ([Sal82] and [Ber79]). The unit sphere bundleZ of the quaternionicbundleQM over a quaternion-Kähler manifoldM is called the twistor space ofM .The corresponding �bration � WZ ! M is called the twistor �bration.

Theorem 10.6 ([Sal82] and [Ber79]). LetM be a quaternion-Kähler manifold andZ its twistor space.

(i) Z admits a natural complex structure such that the �bers of the twistor �bra-tion are compact complex curves of genus 0.

(ii) If M has non-zero scalar curvature, Z carries a natural complex contactstructure (cf. De�nition 10.9 below).

(iii) IfM has positive scalar curvature, Z carries a natural Kähler–Einstein met-ric such that the twistor �bration becomes a Riemannian submersion withtotally geodesic �bers.

We will not give the proof of these results here (the interested reader is referredto the original articles or to [Sal99] for details).

As a consequence of the classi�cation, in the next section, of manifolds withKählerian Killing spinors, we will obtain the converse of Theorem 10.6, stating thatevery compact Kähler–Einstein manifold with positive scalar curvature admitting acomplex contact structure is the twistor space of some positive quaternion-Kählermanifold (cf. Theorem 10.11 below and [Mor98b]).

10.2. Kählerian Killing spinors 269

10.2 Kählerian Killing spinors

In this section we study Kählerian Killing spinors on spin and Spinc Kähler mani-folds. Unless otherwise stated, we use the notations of Chapter 6.

De�nition 6.13 has the following Spinc counterpart.

Definition 10.7. Let .M 4kC2; g; J / be a Kähler manifold of odd complex dimen-sionm D 2kC1 endowedwith a Spinc structure with Spinc connectionrA. A Spinc

Kählerian Killing spinor is a section ‰ of the spinor bundle ofM satisfying

rAX‰ D 1

2X �‰ C i�

2JX � x‰; X 2 TM; (10.1)

where � ´ .�1/k .

In order to state the main result, we recall that a Hodge manifold is a compactKähler manifold with the property that the cohomology class of the Kähler form isa real multiple of an integral class.

Theorem 10.8 ([Mor98b]). Let .M 4kC2; g; J / be a simply connected Hodge man-ifold carrying a non-zero Spinc Kählerian Killing spinor‰ 2 �.†kM ˚†kC1M/.Then

(i) if k is even,M is isometric to CP2kC1;

(ii) if k is odd, either M is isometric to CP2kC1, or the Spinc structure on Mis actually a spin structure and M is isometric to the twistor space over apositive quaternion-Kähler manifold.

Proof. The proof goes roughly as follows. Using the theory of projectable spinorsfor Riemannian submersions from Section 2.4.3, one gets a real Killing spinor on anS1-bundle S overM endowedwith some natural metric and Spinc structure induced

from M . This spinor induces a parallel spinor on the Riemannian cone xS over S ,lying in the kernel of the Clifford multiplication with the Kähler form of xS . Usingthe classi�cation of Spinc manifolds carrying parallel spinors (Theorem 8.12) andthe irreducibility of xS , we must have on the one hand that the Spinc structure isa spin structure, and then that xS is hyper-Kähler, which easily translates into thegeometrical characterization forM stated in the theorem.

Let� be Kähler form ofM , de�ned as usual by �.X; Y / ´ g.JX; Y /. SinceM is Hodge, there exists some s 2 R

� such that sŒ�� 2 H 2.M;Z/. We �xs > 0 of least absolute value with this property. The isomorphism H 2.M;Z/ 'H 1.M;S1/ (cf. Proposition 3.2) guarantees the existence of some S

1-principalbundle � WS ! M whose �rst Chern class satis�es c1.S/ D s�Œ��. We claim that


S is simply connected. To see this, consider like in [BFGK91], Example 1, p. 84,the Thom–Gysin exact sequence for the S

1-�bration � :

� � � �! H 2m�2.M I Z/[c1.S/��!H 2m.M I Z/

��

�! H 2m.S I Z/ �! H 2m�1.M I Z/ �! � � � :

The last arrow is just the cap product with the homology class of a �bre. By as-sumption M is simply connected, so H 2m�1.M I Z/ D H1.M I Z/ D 0: More-over, the strong Poincaré duality implies that the �rst arrow in the above exact se-quence is onto (since c1.S/ has no integral quotient). This shows thatH 2m.S I Z/ DH1.S I Z/ D 0. In other words, the center of �1.S/ is trivial. On the other hand,the exact homotopy sequence of the S

1-�bration S ! M shows that �1.S/ is aquotient of Z, so in particular it is commutative. This proves our claim.

By construction, the S1-bundle S carries a connection whose curvature form

is 2�is��. With the notations from Section 2.4.3 we then have F D s�� andT D s�J . Theorem 2.45 says that for every positive number `, there exists aRiemannian metric on S such that � WS ! M becomes a Riemannian submer-sion, with totally geodesic �bres of length `, and such that the pull-back to S of theSpinc structure on M de�nes a Spinc structure with connection xrA satisfying thefollowing relations:

X� � .��‰/ D ��.X � ‰/; X 2 TM; (10.2)

V � .��‰/ D i��.x‰/; (10.3)

xrAX��

�‰ D ��rAX‰ � i`

4TX � x‰

�; (10.4)

xrAV �

�‰ D `

4��.F � ‰/: (10.5)

If we choose ` D 2s, then (10.1), (10.2), and (10.4) imply that

xrAX��

�‰ D 1

2X� � ��‰:

On the other hand, since ‰ is a section of †kM ˚†kC1M , we have

� � ‰ D i� x‰; (10.6)

so from (6.3), (10.3), and (10.5) we get

xrAV �

�‰ D `

4��.F �‰/ D 1

2s��.s�� ‰/ i

2��.x‰/ D 1

2V � ��‰;

10.2. Kählerian Killing spinors 271

hence ��‰ is a Spinc Killing spinor on S . Moreover, if � denotes the 1-form dualto the vertical Killing vector �eld V , then Proposition 2.43 shows that

d� D `F D 2��:

From (10.6) and (10.3) we get

�� ‰ D �V � ��‰: (10.7)

Now consider the Riemannian cone xS ´ R�C � S with the metric dt2 C t2gS

and with the pull-back Spinc structure from S . Theorem 2.41 shows that the Spinc

Killing spinor ��‰ on S induces a parallel Spinc spinor ˆ on †�� xS . The Kählerform x� of xS is given by the formula x� D t2d��2tdt^�. We claim that x� �ˆ D 0.It is enough to show this relation along the submanifold S D f1g � S of xS , sinceboth x� and ˆ are parallel on xS . By restricting to S and using (2.30), we get

x� �ˆjS D d� � ��‰ C 2� ^ dt �ˆjS D 2�� ‰ � 2�V � ��‰.10.7/D 0:

This shows that xS is a Kählermanifold which carries a parallel spinor whose Cliffordproduct with the Kähler form vanishes.

Assume from now on thatM is not isometric toCP2kC1. ThenS is not isometricto the round sphere, and xS is not �at. Lemma 8.34 thus implies that xS is irreducible.

From Theorem 8.12 we then deduce that either the Spinc structure of xS is actu-ally a spin structure, or there exists a Kähler structure I on xS such that

X �ˆ D iI.X/ �ˆ; X 2 T xS; (10.8)

and the Spinc structure of xS is the canonical Spinc structure induced by I (these twocases do not exclude each other).

Case I. If the Spinc structure of xS is a spin structure, Theorem 8.1 shows that theholonomy of xS is either SU2kC2 or SpkC1. Recall that by (6.7) the spinor bundleof the Ricci-�at Kähler manifold xS can be written as

† xS D ƒ0;0 ˚ƒ0;1 ˚ � � � ˚ƒ0;2kC2;

and the Clifford action of the Kähler form x� on each factor ƒ0;p is the scalari.2k C 2 � p/. Since x� � ˆ D 0, ˆ is a section of ƒ0;kC1. Now, it is easy to seethat SU2kC2 has no �xed point inƒ0;kC1, and SpkC1 has a �xed point inƒ

0;kC1 ifand only if k is odd.


We thus obtain that k is odd and xS is hyper-Kähler. In order to complete theclassi�cation, it remains to interpret this latter condition in terms of the originalmanifold M . From Theorem 8.20 we obtain directly that S has to be 3-Sasakian.Moreover, if S is not isometric to the round sphere, the vector �eld V de�nes oneof the Sasakian structures. To see this, we consider the Kähler structure inducedby V on xS . If this structure does not belong to the 2-sphere of Kähler structuresde�ned by the hyper-Kähler structure, then this would provide a further reductionof the holonomy of xS , a contradiction. Thus M is the quotient of the 3-Sasakianmanifold S by the �ow of one of the Sasakian vector �elds. By Lemma 10.4 andTheorem 10.3,M is the twistor space of a positive quaternion-Kähler manifold.

Case II. If xS carries a Kähler structure I satisfying (10.8) (in addition to the orig-inal Kähler structure, say xJ , de�ned by x�), we will show that xS has to be reducible,and thus obtain a contradiction.

Taking the Clifford product with x� in (10.8) and using (6.3) yields

xJX �ˆ D i xJ I.X/ �ˆ; X 2 T xS; (10.9)

so replacing X by xJX in (10.8) and using (10.9) shows that I xJ D xJI . Now it isclear that I ¤ ˙ xJ since x� � ˆ D 0 and, by (10.8), �I � ˆ D 2i.k C 1/ˆ, where�I denotes the Kähler form of I .

On the other hand, I xJ is a symmetric parallel involution of T xS , so the decom-position T xS D TCS ˚ T�S , where T˙S ´ fX I I xJX D ˙Xg gives a holonomyreduction of xS . This �nishes the proof of the theorem. �

10.3 Complex contact structures on positive Kähler–Einsteinmanifolds

As an application of Theorem 10.8, we will describe in this section the classi�cationof positive Kähler–Einstein contact manifolds obtained by C. LeBrun [LeB95] andby A. Moroianu and U. Semmelmann [KS95], [MS96], and [Mor98b].

Definition 10.9 ([Kob59]). LetM 2m be a complex manifold of complex dimensionm D 2kC 1. A complex contact structure is a family C D f.Ui ; !i/g satisfying thefollowing conditions:

(i) fUig is an open covering ofM .

(ii) !i is a holomorphic 1-form on Ui .

(iii) !i ^ .@!i/k 2 �.ƒm;0M/ is different from zero at every point of Ui .

(iv) !i D fij!j on Ui \ Uj , where fij is a holomorphic function on Ui \ Uj :

10.3. Complex contact structures on positive Kähler–Einstein manifolds 273

Let C D f.Ui ; !i/g be a complex contact structure. Then there exists an asso-ciated holomorphic line sub-bundle LC � ƒ1;0M , with transition functions ff �1

ij gand local sections !i . It is easy to see that

fZ 2 T1;0M W!.Z/ D 0; for all ! 2 LCg

is a codimension 1 maximally non-integrable holomorphic sub-bundle of T1;0M ,and conversely, every such bundle de�nes a complex contact structure. From con-dition (iii) immediately follows the isomorphism LkC1

C Š K; where K D ƒm;0M

denotes the canonical bundle ofM .Let M be a compact Kähler–Einstein manifold of odd complex dimension

m D 2k C 1 with positive scalar curvature, admitting a complex contact struc-ture C. By rescaling the metric on M if necessary, we can suppose that the scalarcurvature ofM is equal to 2m.2mC 2/, and thus the Ricci form � and the Kählerform � are related by � D .2mC 2/�. We will denote by l the integer part of k

2,

and let k D 2l C ı, where ı is 0 or 1. Our main goal is to construct, starting fromthe complex contact structure C, a Kählerian Killing spinor onM . This can only bedone for k odd. When k is even we will actually construct a Spinc Kählerian Killingspinor for a certain Spinc structure onM , determined by C.

Theorem 10.10. Let C be a complex contact structure on a Kähler–Einstein mani-fold .M; g; J / of odd complex dimension m D 2k C 1.

(i) If k is odd, then M is spin and carries a Kählerian Killing spinor ‰ 2�.†kM ˚†kC1M/.

(ii) If k is even, then the Spinc structure onM with auxiliary bundle LC carriesa Kählerian Killing spinor ‰ 2 �.†kC1M ˚†kC2M/.

Proof. The collection .Ui ; !i ^ .@!i/l / de�nes a holomorphic line bundle

Ll � ƒ2lC1;0M , and from the de�nition of C we easily obtain

Ll Š LlC1C : (10.10)

We now �x some .U; !/ 2 C and de�ne a local section ‰C of ƒ0;2lC1M ˝ LlC1C

by‰CjU ´ j�� j�2 N� ˝ �� ;

where � ´ !^ .@!/l and �� is the element corresponding to � through the isomor-phism (10.10). The fact that ‰C does not depend on the element .U; !/ 2 C showsthat it actually de�nes a global section ‰C of ƒ0;2lC1M ˝ LlC1C .

We now recall that ƒ0;�M is just the spinor bundle associated to the canonicalSpinc structure onM , whose auxiliary line bundle is K�1, so that ƒ0;�M ˝ LlC1C

is actually the spinor bundle associated to the Spinc structure onM with auxiliarybundle

L D K�1 ˝ L2lC2C Š L�k�1C ˝ L2lC2C Š L1�ı

C :


The section ‰C is thus a usual spinor for k odd, and a Spinc spinor for k even, andis always a section of †2lC1M . In particular,

� �‰C D i.m � 4l � 2/‰C D i.2ı � 1/‰C : (10.11)

We claim that the spinor �eld ‰C satis�es rZ‰C D 0; for all Z 2 �.T1;0M/

(this shows, in particular, that D�‰C D 0), and

D2‰C D D�DC‰C D l C 1

k C 1

�12Scal‰C � i� � ‰C

�; (10.12)

where as before Scal denotes the scalar curvature ofM .This claim actually follows directly from [KS95], Proposition 5, keeping inmind

that we treat simultaneously the cases k even and odd, so the coef�cients 1=2 informulas (8) and (9) of [KS95] have to be replaced by lC1

kC1 .Using (10.11), (10.12), and the fact that

� D 1

2mScal� D .2mC 2/�;

we obtain that the spinor �eld‰C is an eigenspinor ofD2 with respect to the eigen-value 16.l C 1/.l C ı/.

Note thatmC 1

4mScal D .mC 1/2:

Thus, for ı D 1 (i.e., for k odd) we getmC1 D 4lC4, so we are in the limiting caseof Kirchberg’s inequality on compact Kähler manifolds of odd complex dimension.By Theorem 6.14, the spinor �eld .4l C 4/‰C CD‰C is a Kählerian Killing spinoronM .

The case where k is even is more delicate, as there are no Spinc analogues ofKirchberg’s inequalities. One thus needs an ad hoc argument to prove that theSpinc spinor .4l C 4/‰C C D‰C is a Kählerian Killing spinor. We refer the readerto [Mor98b] for the proof of this fact. �

The main application of Theorem 10.10 is the converse to Theorem 10.6.

Theorem 10.11. Every compact Kähler–Einstein manifold of positive scalar cur-vature and odd complex dimension 2k C 1 carrying a complex contact structure isthe twistor space over a positive quaternion-Kähler manifold.

Proof. For k odd, the result follows from Theorem 10.10 (i) together with Theo-rem 10.8.

Assume now that k D 2l is even. Theorem 10.10 (ii) provides a Spinc structurewith auxiliary line bundle LC and a Kählerian Killing spinor

‰ 2 �.†2lC1M ˚†2lC2M/:

10.4. The limiting case of Kirchberg’s inequalities 275

The rest of the proof parallels that of Theorem 10.8, with a notable exception. Oneuses again projectable spinors for the submersion S ! M (where S denotes theprincipal S

1-bundle associated toLC), except that in the present situation we endowS with the spin structure inherited from the Spinc structure ofM , rather than withthe pull-back Spinc structure. A correspondingly modi�ed version of Theorem 2.45shows that ‰ induces a Killing spinor on S , and then a parallel spinor on its Rie-mannian cone xS , eventually showing that xS is hyper-Kähler. Details can be foundin [Mor98b]. �

10.4 The limiting case of Kirchberg’s inequalities

Recall Kirchberg’s inequalities (Theorem 6.1): on a compact Kähler spin mani-fold .M 2m; g; J / with positive scalar curvature Scal, any eigenvalue � of the Diracoperator satis�es

�2 � mC 1

4mScal0; if m is odd,

and�2 � m

4.m� 1/Scal0; if m is even;

where Scal0 denotes the in�mum of the scalar curvature onM .Compact Kähler spin manifolds satisfying the limiting case in one of the above

inequalities are called limiting manifolds. The aim of this section is to give theircomplete description.

If the complex dimension m is odd, Theorem 6.14 and Theorem 10.8 yield atonce

Theorem 10.12 ([Mor95]). Limiting manifolds of complex dimension 4l � 1 aretwistor spaces over positive quaternion-Kähler manifolds. The only limiting mani-fold of complex dimension 4l C 1 is CP4lC1.

Suppose now that m is even. The case m D 2 was completely solved byTh. Friedrich [Fri93], who proved that M is isometric to S

2 � T2. We assume

from now on that m > 2.

Theorem 10.13. Let M 2m be a limiting manifold of even complex dimensionm D 2l; l � 2, and let zM be its universal cover. Then:

• if l is odd, zM is isometric to the Riemannian product CP2l�1 � R2;

• if l is even, zM is isometric to the Riemannian product N � R2, where N

is the twistor space of some quaternion-Kähler manifold of positive scalarcurvature.


Proof. We give here a sketch of the proof. See [Mor97a] and [Mor99] for the details.The proof is in several steps.

Step 1. Like in the odd-dimensional case, limiting manifolds can be characterizedby the existence of special spinor �elds (see Theorem 6.15).

Applying the Dirac operator to (6.35) and using Corollary 2.8 yields

rXD‰ D �14.Ric.X/C iJRic.X// �‰: (10.13)

If the manifoldM were Einstein, (6.35) and (10.13) would imply that �‰CD‰

is a Kählerian Killing spinor. The point is that M cannot be Einstein (see Step 2below).

Step 2. It is thus necessary to study the Ricci tensor of M . A tricky argument(cf. [Mor97a], Theorem 3.1) yields that the Ricci tensor of a limiting manifold ofeven complex dimension has only two eigenvalues: the eigenvalue 0 with multiplic-ity 2, and the eigenvalue Scal=.2m� 2/ with multiplicity 2m � 2.

Step 3. The tangent bundle ofM thus splits into a J -invariant orthogonal directsum TM D E ˚ F .where E and F are the eigenbundles of TM correspondingto the eigenvalues 0 and � ´ Scal=.2m � 2/ of Ric, respectively/. Moreover, thedistributions E and F are integrable (this follows from the fact that the Ricci formis harmonic).

Let N denote a maximal leaf of the distribution F . The main point here is thatthe Ricci tensor ofN is greater than the restriction toN of the Ricci tensor ofM , andis thus positive. Since N is complete, Myers’ theorem implies that N is compact.Moreover, N being Kähler with positive de�ned Ricci tensor, a classical theoremof Kobayashi (see [Kob61], Theorem A) shows that N is simply connected.

Step 4. We now consider the restrictionˆN ofˆ ´ ‰C 2p2m�

D‰ toN . In order

to understand the nature of ˆN , we recall Hitchin’s representation of spinors (6.7):

†M ' .KM /12 ˝ƒ0;�M:

The isomorphisms of complex vector bundles

KM ' ƒm.T0;1M/ ' ƒm.E0;1 ˚ F0;1/ ' E0;1 ˝ƒm�1.F0;1/

yieldsKM jN ' E0;1jN ˝ KN :

10.4. The limiting case of Kirchberg’s inequalities 277

Similarly,ƒ0;�M jN ' ƒ0;�N ˚ .E1;0jN ˝ƒ0;�N/;

and thus

†M jN ' ..KM /12 jN ˝ƒ0;�N/˚ ..KM /

12 jN ˝ .E1;0jN ˝ƒ0;�N//:

One can check that ˆN is a section of

.KM /12 jN ˝ .E1;0jN ˝ƒ0;�N/;

and by the above, this is just the spinor bundle of some Spinc structure on N withassociated line bundle E1;0jN . In fact, we can write locally

.KM /12 jN ˝ .E1;0jN ˝ƒ0;�N/ ' ..E1;0/jN /

12 ˝ ..KN /

12 ˝ƒ0;�N/

' ..E1;0/jN /12 ˝†N;

but, of course, neither ..E1;0/jN /12 , nor †N need to exist globally on N .

Step 5. We compute the covariant derivative of ˆN as Spinc spinor on N andshow that it coincides with the restriction to N of its covariant derivative on M .Thus ˆN is a Kählerian Killing Spinc spinor on N . Moreover, N is a Hodge man-ifold: if we denote by i the inclusion N ! M and by � the Ricci form of M ,then ��N D i��, which implies �Œ�N � D i�.2�c1.M//; and thus Œ�N � is a realmultiple of i�.c1.M// 2 H 2.N I Z/.

Step 6. We then apply Theorem 10.8 and deduce that the Spinc structure on Nhas actually to be a spin structure (i.e., E1;0jN is a �at bundle on N ). This impliesthat E and F are parallel distributions at each point ofN , so they are parallel onMbecause the N ’s foliateM .

This immediately shows that the universal cover zM of a limiting manifold Mof even complex dimension is isometric to the Riemannian product N � R

2, whereN is a limiting manifold of odd complex dimension.

By taking into account the classi�cation of limiting manifolds of odd complexdimension (Theorem 10.12), we obtain the claimed result. �

Remark 10.14. Note that once Step 2 is achieved, there is an alternative way toprove that if the Ricci tensor of a compact Kähler manifold has only two eigenval-ues, both constant, then the two eigendistributions of the Ricci tensor are parallel.This was done in [ADM11] using the Weitzenböck formula applied to the curvatureoperator.

Chapter 11

Special spinors on quaternion-Kähler manifolds

In this chapter we examine the problem of the classi�cation of spin compact quater-nion-Kähler manifolds with positive scalar curvature admitting an eigenspinor forthe minimal eigenvalue

� D ˙rmC 3

mC 2

Scal

4

of the Dirac operator (cf. Chapter 7, recall that such manifolds have constant scalarcurvature). As in the Riemannian and Kählerian settings, such eigenspinors will becalled “quaternion-Kähler Killing spinors.”

However, the only positive spin quaternion-Kähler manifolds with quaternion-Kähler Killing spinors are the quaternionic projective spaces. This was provedby W. Kramer, U. Semmelmann, and G. Weingart in [KSW98a], following theapproach used in [Mor95] for the Kählerian case: any quaternion-Kähler Killingspinor on a positive spin quaternion-Kähler manifoldM 4m can be lifted to a Killingspinor on the canonical 3-Sasakian SO3-principal bundle S associated with thequaternion-Kähler structure, cf. Theorem 10.3. Now by the general cone construc-tion (cf. 2.4.2), this induces a parallel spinor on the Riemannian cone over S .Moreover, this cone is a hyper-Kähler manifold (cf. 8.2.2). Hence, using resultsof M. Wang (cf. [Wan89]), one concludes thatM 4m is isometric to the quaternionicprojective space HPm.

This method of pull-back to the 3-Sasakian SO3-principal bundle S ! M wasalready used by D. V. Alekseevsky and S. Marchiafava in [AM95] to prove the limitcase of the following result: on any positive compact quaternion-Kähler manifold,the �rst non-zero eigenvalue of the Laplace operator � is greater or equal to

mC 1

mC 2

Scal

2m;

and equality is attained if and only if the manifold is isometric to the quaternionicprojective space.

In the following, we give the proof of [KSW98a], but with a simpler formal-ism, using the results of the �rst proof of the eigenvalue estimate given in Chap-ter 7 together with the formulas for the Dirac operator on the canonical 3-SasakianSO3-principal bundle S given in [Mor96].

280 11. Special spinors on quaternion-Kähler manifolds

We will use notations and results of Chapter 7.

11.1 The canonical 3-Sasakian SO3-principal bundle

According to the survey [BG99], the existence of a 3-Sasakian structure on thecanonical SO3-principal bundle S over a quaternion-Kähler manifold was �rst stud-ied by Konishi [Kon75].

The canonical SO3-principal bundle � WS ! M is the principal frame bundleassociated with the vector bundle QM . Alternatively, considering the Sp1 : SpmreductionP of the principal bundlePSO4m

M , it can be de�ned as the SO3-associatedbundle

S D P �Sp1 :SpmSO3;

where Sp1 : Spm acts on SO3 via the covering Sp1 ! SO3 induced by the adjointrepresentation on the space of imaginary quaternions Im.H/; cf. (7.4).

The Levi-Civita principal connection on P de�nes canonically a horizontaldistribution THS , which complements the vertical distribution TVS :

TS D TVS ˚ THS:

Note that THS indeed gives rise to a principal connection � on S , and that thecovariant derivative on the associated vector bundleQM , induced by this principalconnection, is just the extension of r toQM .

According to a general result of Vilms [Vil70], given any Sp1 : Spm-invariant(indeed a SO3-invariant) metric gSO3

on SO3, there exists a unique Riemannianmetric on S such that � WS ! M is a Riemannian submersion with totally geodesic�bers isometric to .SO3; gSO3

/, and horizontal distribution THS .To de�ne such a metric on S , we renormalize the metric on M in such a way

thatScalM D 16m.mC 2/;

(as for the quaternionic projective space with its canonical metric; the reason for thisparticular choice will be explained by the result below) and consider the bi-invariantmetric on SO3 induced by .�1=8/ times the Killing form of sp1, in such a way that.i ; j ;k/ becomes an orthonormal basis of so3 ' Im.H/.

Let V1; V2; V3 be the fundamental vector �elds on S induced by the vectors

i1 ´ i ; i2 ´ j ; i3 ´ k;

of the standard basis of sp1. Those three vector �elds de�ne a 3-Sasakian structureon S (cf. Section 8.2.2 or the survey [BG99]). Let .J S1 ; J

S2 ; J

S3 / be the tensor �elds

on S de�ned byJ S˛ D �rV˛; ˛ D 1; 2; 3;

(the same symbol is used to denote the covariant derivatives onM and S ).

11.1. The canonical 3-Sasakian SO3-principal bundle 281

Then the quaternion-Kähler structure onM is related to the 3-Sasakian structureon S by the following relations.

Lemma 11.1. Let sW x 7! s.x/ D .J x˛ / be a local section of � WS ! M . Let X bea vector �eld onM and denote by X� its horizontal lift. Then

��.JS˛ .X

�s.x/// D J x˛ .Xx/;

andJ S˛ .X

�s.x// 2 TH

s.x/S:

In other words, J S˛ .X�s.x/

/ is the horizontal lift of J x˛ .Xx/.

Proof. Let x 2 M . Let fe1; : : : ; e4mg be a local orthonormal frame of TM , and forany i D 1; : : : ; 4m, let e�

i be the horizontal lift of ei . First, one gets at the point s.x/,

hJ S˛ .e�i /; e

�j i D 1

2hJ S˛ .e�

i /; e�j i � 1

2he�i ; J

S˛ .e

�j /i

D �12

hre�iV˛; e

�j i C 1

2he�i ;re�

jV˛i

D 1

2hV˛;re�

ie�j i � 1

2hre�

je�i ; V˛i

D 1

2hŒe�i ; e

�j �; V˛i

D 1

2hŒe�i ; e

�j �� Œei ; ej �

�; V˛i;

where Œei ; ej �� is the horizontal lift of Œei ; ej �.Now recall that any section J of QM , de�ned on a neighborhood of x, may

be identi�ed with the Sp1 : Spm-equivariant function fJ WS ! Im.H/ that mapsany s 2 Sx into s�1�Jx

�, viewing s as an isometry Im.H/ ! QxM . Under

this identi�cation, it is well known that reiJ corresponds to e�

i �fJ . It then followseasily that the same identi�cation mapsRQM .ei ; ej /J into

�Œe�i ; e

�j ��Œei ; ej ��

��fJ .

We apply this to one of the three sections given by the choice of the section

sW x 7�! s.x/ D .J x˛ /;

where each s.x/ may also be viewed as the isometry Im.H/ ! QxM which mapsthe basis .i˛/˛D1;2;3 into the basis .J x˛ /˛D1;2;3. On the one hand, since QM is asubbundle of the bundle End.TM/, its curvature RQM is de�ned by

RQM .X; Y / � J D ŒR.X; Y /; J �; X; Y 2 �.TM/; J 2 �.QM/:


Hence, using the basic equality (7.23), one has that

RQM .X; Y / � J x˛ D 4X

ˇ;

"123˛ˇ �xˇ .X; Y /J

x : (11.1)

On the other hand

.Œe�i ; e

�j �� Œei ; ej ��/ � fJ˛ D

X

ˇ

hŒe�i ; e

�j � � Œei ; ej ��; Vˇ iVˇ � fJ˛ :

But at the point s.x/

Vˇ � fJ˛ D d

dtfJ˛ .s.x/ � exp.tiˇ //jtD0

D d

dtexp.�tiˇ /fJ˛ .s.x// exp.tiˇ /jtD0

D �iˇ i˛ C i˛iˇ

(since fJ˛ .s.x// D i˛/

D 2"123˛ˇ i ;

whence

.Œe�i ; e

�j �� Œei ; ej ��/ � fJ˛ D 2

X

ˇ;

"123˛ˇ hŒe�i ; e

�j �� Œei ; ej �

�; Vˇ ii : (11.2)

Using the above identi�cation, one deduces from equations (11.1) and (11.2) that

hŒe�i ; e

�j � � Œei ; ej ��; V˛i D 2�x˛.ei ; ej /;

hence

hJ S˛ .e�i /; e

�j i D 1

2hŒe�i ; e

�j � � Œei ; ej ��; V˛i D �x˛.ei ; ej / D hJ x˛ .ei/; ej i:

Therefore,

��.JS˛ .e

�i // D

X

j

hJ S˛ .e�i /; e

�j i��.e

�j / D

X

j

hJ x˛ .ei/; ej iej D J x˛ .ei/:

For the second result, let V be a vertical vector �eld. rV˛V is vertical becausethe �bers of � WS ! M are totally geodesic, so

0 D hrV˛V; e�i i D �hV;rV˛e

�i i D hV; J S˛ .e�

i /i;i.e., J S˛ .e

�i / is horizontal. �

Finally, we mention that S with its 3-Sasakian structure is an Einstein space(cf. [BG99]), hence its scalar curvature ScalS is constant. This constant can be easilycomputed using O’Neill’s formulas (cf. for instance 9.37 and 9.63 in [Bes87]). One�nds

ScalS D .4mC 2/.4mC 3/:

11.2. The Dirac operator acting on projectable spinors 283

11.2 The Dirac operator acting on projectable spinors

Here we review results of the Chapter “Spineurs G-invariants sur une G-�brationprincipale” in [Mor96].

First S admits a spin structure. The decomposition TS D TVS˚THS allows toreduce the bundle PSO4mC3

S of orthonormal frames over S by considering local or-thonormal frames of TS of the form fe�

1 ; : : : ; e�4m; V1; V2; V3g, where fe�

1 ; : : : ; e�4mg

is the horizontal lift of an orthonormal local frame fe1; : : : ; e4mg of TM .

Proposition 11.2. Let PSO4mC3S be the bundle of orthonormal frames over S .

There is a bundle isomorphism

PSO4mC3S ' ��.PSO4m

M/ �SO4mSO4mC3;

considering the group SO4m as a subgroup of SO4mC3.

Let �.x/ D fe1.x/; : : : ; e4m.x/g be an orthonormal frame at x D �.s/. Then,denoting by ��.s/ the orthonormal frame

fe�1 .s/; : : : ; e

�4m.s/; V1.s/; V2.s/; V3.s/g; (11.3)

any other orthonormal frame at s has the form ��.s/ ı g, where g 2 SO4mC3, theframe ��.s/ being viewed here as the isomorphism that maps the standard basis ofR4mC3 into the orthonormal frame (11.3). Clearly, the map that sends ��.s/ ı g

into the equivalence class Œ.s; �.x//; g� does not depend on the particular choiceof �.x/, and gives the above isomorphism.

One then obtains a spin structure on S by considering the bundle

PSpin4mC3S ´ ��.PSpin4m

M/ �Spin4mSpin4mC3;

identifying the group Spin4m with a subgroup of Spin4mC3. On the other hand,the complex Clifford algebra Cl4mC3 being canonically isomorphic to the gradedtensor product Cl4m y Cl3, (cf. Proposition 1.12), the spinor space †4mC3 maybe identi�ed with †4m ˝ †3, endowed with its structure of Cl4m y Cl3-module(cf. [ABS64]): any a˝ b 2 Cl4m y Cl3 acts on a decomposable tensor of the form ˝ ' by

.a ˝ b/ � . ˝ '/ ´ .�1/deg.b/ deg. /.a � ˝ b � '/;where the structure of graded module of †4m is given by the decomposition†4m D †C

4m ˚†�4m into even and odd spinors (cf. Proposition 1.32).

This allows the following identi�cation.

Proposition 11.3. Let †4mC3S be the spinor bundle of S . Then

†4mC3S ' ��.†M/˝†3S;

where †3S denotes the trivial bundle on S with �bre †3.


Proof. Let s 2 S . Any element ‰s of the �ber over s of†4mC3S is represented bya quadruple �

s; Q��.s/; u;X

i

i ˝ 'i

�;

where Q��.s/ is an element of the �ber of PSpin4mM over �.s/, u 2 Spin4mC3, and

X

i

i ˝ 'i 2 †4mC3 ' †4m ˝†3:

Any other quadruple .s; Q� 0�.s/

; u0;Pj

0j ˝ '0

j /, represents ‰s if and only if there

exist u0 2 Spin4m and u1 2 Spin4mC3 such that Q� 0�.s/

D Q��.s/u0, u0 D u�10 uu1,

andu1�Pj

0j˝'0

j DPi i˝'i , whereu1 is viewed as an element ofCl4m y Cl3.

Viewing u and u0 also as elements of Cl4m y Cl3, setX

a

�a ˝ !a D u �X

i

i ˝ 'i andX

b

�0b ˝ !0

b D u0 �X

j

0j ˝ '0

j :

Since u0 2 Spin4m, one has

X

b

.u0 � �0b/˝ !0

b D u0 �X

b

�0b ˝ !0

b DX

a

�a ˝ !a;

hence we can de�ne

F.‰s/ ´X

a

.s; Œ Q��.s/; �a�/˝ .s; !a/;

since the right-hand side in the above equation does not depend on the quadruplerepresenting ‰s .

The map‰s 7�! F.‰s/

is then a bundle morphism

F W†4mC3S �! ��.†M/˝†3S:

This is actually a bundle isomorphism: it is easy to verify that the inverse mor-phism maps a decomposable element of the form

.s; Œ Q��.s/; �/˝ .s; '/

into the equivalence class of the quadruple

.s; Q��.s/; 1; ˝ '/: �


This identi�cation allows us to introduce the following particular spinor �eldson S .

Definition 11.4. Any couple .‰; '/with‰ 2 �.†M/ and ' 2 †3 de�nes a spinor�eld on S called projectable and denoted by ‰ ˝ '.

Proposition 11.5. Let X� be the horizontal lift of a vector �eld X on M , and Vbe the fundamental vertical �eld on S induced by a vector v 2 so3 ' R

3. TheirClifford multiplication on a projectable spinor ‰ ˝ ' is given by

X� � .‰ ˝ '/ D .X � ‰/˝ '

and

V � .‰ ˝ '/ D x‰ ˝ .v � '/;

where x‰ is the image of‰ by the conjugation map (cf. De�nition 1.34), x‰ D !C4m�‰.

Proof. Let s 2 S and x D �.s/. Let � D fe1; : : : ; e4mg be a local section of thebundle PSO4m

S de�ned in a neighborhood of x. Let Q� be a section of PSpin4mM

which projects onto � . Denote also by Q� the section of PSpin4mC3S given by

s 7�! Œ.s; Q�.�.s//; 1/�:

This section projects onto the local section of PSO4mC3S , s 7! ��.s/.

Finally, let 2 †4m be de�ned by ‰.x/ D Œ Q�.x/; �. Viewing �.x/ (resp.��.s/) as the isometry that sends the standard basis of R4m (resp. R4mC3) intofe1.x/; : : : ; e4m.x/g (resp. fe�

1 .s/; : : : ; e�4m.s/; V1.s/; V2.s/; V3.s/g), one has at the

point s

e�i � .‰ ˝ '/ ' Œ Q�.s/; ��.s/�1.e�

i .s// � ˝ '�

D Œ Q�.s/; �.x/�1.ei.x// � ˝ '�

' Œ Q�.x/; �.x/�1.ei.x// � �˝ '

D .ei �‰/˝ ':

On the other hand, considering the decomposition D C C � induced by thesplitting †4m D †C

4m ˚ †�4m into even and odd spinors, for any ˛ D 1; 2; 3, one

gets

V˛ � .‰ ˝ '/ ' .�1/deg Œ Q�.s/; ˝ ��.s/�1.V˛.s// � '�D Œ Q�.s/; . C � �/˝ i˛ � '�' Œ Q�.x/; C � ��˝ .i˛ � '/

D x‰ ˝ .i˛ � '/: �


Proposition 11.6 ([Mor96]). Let s be a local section of � WS ! M and x 2 M .Denote by fe1; : : : ; e4mg a local orthonormal frame of TM such that .rei /x D 0,and for any i D 1; : : : ; 4m, let e�

i be the horizontal lift of ei . Introduce the 2-forms�S˛ on S , locally de�ned by

�S˛ D 1

2

X

i

e�i ^ J S˛ .e�

i /:

Then for any projectable spinor ‰ ˝ ' on S one has

re�i.‰ ˝ '/ D .rei

‰/˝ ' C 1

2

X

˛

J S˛ .e�i / � V˛ � .‰ ˝ '/; (11.4)

rV˛.‰ ˝ '/ D �12�S˛ � .‰ ˝ '/ � 1

2.�1/mV˛ � .x‰ ˝ '/; (11.5)

Hence,

DS .‰ ˝ '/ D .D‰/˝ ' C 1

2

X

˛

�S˛ � V˛ � .‰ ˝ '/C 3

2.�1/m.x‰ ˝ '/:

Proof. Consider the local section � D fe�1 ; : : : ; e

�4m; V1; V2; V3g of PSO4mC3

S . LetQ� be a section of PSpin4m

M which projects onto fe1; : : : ; e4mg. It induces a sectionof PSpin4mC3

S , also denoted by Q� , which projects onto � . Now, the spinor �eld‰ islocally de�ned by ‰ D Œ Q�; �, where WU � M ! †4m, hence ‰ ˝ ' is locallyde�ned by Œ Q�; . ı �/˝ '�. Therefore,

re�i.‰ ˝ '/ D 1

2

X

j<k

hre�ie�j ; e

�kie�

j � e�k � .‰ ˝ '/C Œ Q�; e�

i . ı �/˝ '�

C 1

2

X

j;˛

hre�ie�j ; V˛ie�

j � V˛ � .‰ ˝ '/:

But since � WS ! M is a Riemannian submersion, and since Œe�i ; e

�j � � Œei ; ej �

� isa vertical vector, it follows that

hre�ie�j ; e

�ki D 1

2.�he�

i ; Œe�j ; e

�k �i C he�

j ; Œe�k ; e

�i �i C he�

k ; Œe�i ; e

�j �i/

D 1

2.�he�

i ; Œej ; ek��i C he�

j ; Œek; ei ��i C he�

k ; Œei ; ej ��i/

D 1

2.�hei ; Œej ; ek�i C hej ; Œek; ei �i C hek ; Œei ; ej �i/

D hreiej ; eki:


One the other hand,

hre�ie�j ; V˛i D �he�

j ;re�iV˛i D he�

j ; JS˛ .e

�i /i;

hence,

re�i.‰ ˝ '/ D

�12

X

j<k

hreiej ; ekiej � ek �‰

�˝ ' C Œ Q�; ei. /�˝ '

C 1

2

X

j

hJ S˛ .e�i /; e

�j ie�

j � V˛ � .‰ ˝ '/

D .rei‰/˝ ' C 1

2

X

˛

J S˛ .e�i / � V˛ � .‰ ˝ '/;

i.e., (11.4). In the same way,

rV˛.‰ ˝ '/ D 1

2

X

j<k

hrV˛e�j ; e

�kie�

j � e�k � .‰ ˝ '/C Œ Q�; V˛. ı �/˝ '�

C 1

2

X

ˇ<

hrV˛Vˇ ; V iVˇ � V � .‰ ˝ '/:

But since the �bres of the Riemannian submersion � WS ! M are totally geodesic,rV˛Vˇ is a vertical vector and

hrV˛Vˇ ; V i D 1

2.�hV˛; ŒVˇ ; V �i C hVˇ ; ŒV ; V˛�i C hV ; ŒV˛; Vˇ �i/ D "123˛ˇ ;

hence

rV˛Vˇ DX

"123˛ˇ V : (11.6)

Consequently,

X

ˇ<

hrV˛Vˇ ; V iVˇ � V � .‰ ˝ '/ D 1

2

X

ˇ;

"123˛ˇ Vˇ � V � .‰ ˝ '/:

Now, by considering an equivalent representation if necessary, we may always as-sume that the volume element in Cl3 acts on †3 in such a way that

V1 � V2 � V3 � .‰ ˝ '/ D .�1/m x‰ ˝ '


(cf. Proposition 1.32), therefore

.�1/mV˛ � .x‰ ˝ '/ D V˛ � .V1 � V2 � V3/ � .‰ ˝ '/

D 1

2V˛ �

X

ˇ;

"123˛ˇ V˛ � Vˇ � V � .‰ ˝ '/

D �12

X

ˇ;

"123˛ˇ Vˇ � V � .‰ ˝ '/

(11.7)

and �nallyX

ˇ<

hrV˛Vˇ ; V iVˇ � V � .‰ ˝ '/ D .�1/mC1V˛ � x‰ ˝ ':

On the other hand, the right invariance of horizontal vectors implies ŒV˛; e�i � D 0,

hence, since the connection is torsion free,

rV˛e�i D re�

iV˛ D �J S˛ .e�

i /; (11.8)

and

rV˛.‰ ˝ '/ D �14

X

j

e�j � J S˛ .e�

j / � .‰ ˝ '/

� 1

2.�1/mV˛ � x‰ ˝ ';

which yields (11.5). The third result is easily obtained from (11.4) and (11.5). �

11.3 Characterization of the limiting case

Observe �rst that the volume element !C4m commutes with the fundamental 4-form

�, hence, by the Schur lemma, the restriction of !C4m to †rM is either Id or �Id,

since any �ber †rM is an irreducible Sp1 � Spm-space equivalent to †4m;r . Con-sidering for instance a maximal vector in†4m;r , it is easy to verify that, form even,!C4mj†4m;r

D Id if r is even, and !C4mj†4m;r

D �Id if r is odd, whereas the contraryholds whenm is odd.

The limiting case in the main estimate provides two spinor �elds

‰0 2 �.†0M/ and ‰1 D D‰0 2 �.†1M/;

(cf. Section 7.4). Proceeding heuristically, we begin by considering the action ofthe Dirac operator DS on the two projectable spinors ‰0 ˝ ' and ‰1 ˝ ', ' beinga �xed non-zero vector in †3. First, since by Lemma 7.5,

�S˛ � .‰0 ˝ '/ D .�˛ �‰0/˝ ' D 0;

and by the above remark ‰0 D .�1/m‰0, one gets the following lemma.

11.3. Characterization of the limiting case 289

Lemma 11.7.

DS.‰0 ˝ '/ D ‰1 ˝ ' C 3

2.‰0 ˝ '/:

Knowing that ‰1 D .�1/mC1‰1, one also gets the following lemma.

Lemma 11.8.

DS .‰1 ˝ '/ D 4m.mC 3/‰0 ˝ ' C 1

2

X

˛

�S˛ � V˛ � .‰1 ˝ '/ � 3

2.‰1 ˝ '/:

We now computeDS

�X

˛

�S˛ � V˛ � .‰1 ˝ '/�:

For that, we need the following formulas.

Lemma 11.9. Assume that the local frame feig is such that reiej .x/ D 0. Then at

the point s.x/,

re�jJ S˛ .e

�i / D ıijV˛ �

X

ˇ;

"123˛ˇ hJ Sˇ .e�i /; e

�j iV ; (11.9)

and

rVˇJ S˛ .e

�i / D ı˛ˇe

�i �

X

"123˛ˇ JS .e

�i /: (11.10)

Proof. Since each V˛ is a Sasakian structure (cf. De�nition 8.14), using the argu-ments of the above proof, one has

re�jJ S˛ .e

�i / D �re�

j.re�

iV˛/

D �r2e�j;e�

iV˛ � rr

e�je�

iV˛

D ıijV˛ �X

k

hre�je�i ; e

�kire�

kV˛ �

X

ˇ

hre�je�i ; Vˇ irVˇ

V˛

D ıijV˛ CX

ˇ;

"123˛ˇ he�i ; J

Sˇ .e

�j /iV :

On the other hand,

rVˇJ S˛ .e

�i / D �rVˇ

.re�iV˛/ D �r2

Vˇ;e�iV˛ � rrVˇ

e�iV˛:

But r2Vˇ;e

�i

V˛ D 0, by (8.9), and

rrVˇe�

iV˛ D rr

e�iVˇV˛ D J S˛ .J

Sˇ .e

�i /;

by (11.8), hence the second result. �


From the relations obtained above one deduces

Lemma 11.10. Assume that the local frame feig is such that reiej .x/ D 0. Then

at the point s.x/

re�j�S˛ D e�

j � V˛ CX

ˇ;

"123˛ˇ JSˇ .e

�j / � V ; (11.11)

and

rVˇ�S˛ D 0: (11.12)

Proof. As already noted,

re�je�i D

X

k

hre�je�i ; e

�kie�

k CX

ˇ

hre�je�i ; Vˇ iVˇ

DX

ˇ

he�i ; J

Sˇ .e

�j /iVˇ ;

whence

re�j�S˛ D 1

2

X

i

re�je�i � J S˛ .e�

i /C 1

2

X

i

e�i � re�

jJ S˛ .e

�i /

D 1

2

�X

ˇ;i

he�i ; J

Sˇ .e

�j /iVˇ � J S˛ .e�

i /C e�j � V˛

CX

i;ˇ;

"123˛ˇ he�i ; J

Sˇ .e

�j /ie�

i � V �

D 1

2

�X

ˇ

Vˇ � J S˛ .J Sˇ .e�j //C e�

j � V˛ CX

ˇ;

"123˛ˇ JSˇ .e

�j / � V

�

D e�j � V˛ C

X

ˇ;

"123˛ˇ JSˇ .e

�j / � V :

On the other hand, using (11.8), one gets

rVˇ�S˛ D 1

2

X

i

rVˇe�i � J S˛ .e�

i /C 1

2

X

i

e�i � rVˇ

J S˛ .e�i /

D �12

X

i

J Sˇ .e�i / � J S˛ .e�

i / � 2mı˛ˇ �X

"123˛ˇ �S :


The tensorial form of the �rst term in the right-hand side of the above equationshows that we can assume that feig is an adapted local orthonormal frame (i.e.,such that any ei has the form ˙Jˇej ). Then, by Lemma 11.1, any e�

i has then theform ˙Jˇ .e�

j /, therefore

rVˇ�S˛ D 1

2

X

i

e�i � J S˛ .J Sˇ .e�

i // � 2mı˛ˇ �X

"123˛ˇ �S D 0: �

Lemma 11.11. We have

DS�X

˛

�S˛ � V˛ � .‰1 ˝ '/�

D 6‰1 ˝ ' C 9

2

X

˛

�S˛ � V˛ � .‰1 ˝ '/C 2X

˛

V˛ � DS˛ .‰1 ˝ '/;

where DS˛ is the operator locally de�ned by

DS˛ .‰ ˝ '/ D

X

i

J S˛ .e�i / � .rei

‰ ˝ '/:

Proof. With the notations of the proof of Proposition 11.6, using (11.4), (11.9),and (11.11) one gets

re�i

�X

˛

�S˛ � V˛ � .‰1 ˝ '/�

D �3e�i � .‰1 ˝ '/

CX

˛;ˇ;

"123˛ˇ JSˇ .e

�i / � V � V˛ � .‰1 ˝ '/

�X

˛

�S˛ � J S˛ .e�i / � .‰1 ˝ '/

CX

˛

�S˛ � V˛ � .rei‰1 ˝ '/

� 1

2

X

˛;ˇ

�S˛ � J Sˇ .e�i / � V˛ � Vˇ � .‰1 ˝ '/:

Now, by (11.7), for any ˛, ˇ 2 f1; 2; 3g and any projectable spinor ‰ ˝ ',X

"123˛ˇ V � .‰ ˝ '/ D �12.�1/m

X

;�;�

"123˛ˇ "123 ��V� � V� � .x‰ ˝ '/

D �12.�1/m

X

."123˛ˇ /2.V˛ � Vˇ � Vˇ � V˛/ � .x‰ ˝ '/

D .�1/m.ı˛ˇ � 1/V˛ � Vˇ � .x‰ ˝ '/:

(11.13)


Hence since ‰1 D .�1/mC1‰1, one gets, using again (11.7),

re�i

�X

˛

�S˛ � V˛ � .‰1 ˝ '/�

D �3e�i � .‰1 ˝ '/

C 2X

ˇ

J Sˇ .e�i / � Vˇ � .‰1 ˝ '/

� 1

2

X

˛

�S˛ � J S˛ .e�i / � .‰1 ˝ '/

CX

˛

�S˛ � V˛ � .rei‰1 ˝ '/

� 1

2

X

˛;ˇ;

"123˛ˇ �S˛ � J Sˇ .e�

i / � V � .‰1 ˝ '/:

Using the straightforward formula e�i ��S˛ D �S˛ � e�

i � 2J S˛ .e�i /, and noting that,

by Lemma 11.1

X

˛

.�S˛ ��S˛ / � .‰1 ˝ '/ DX

˛

.�˛ ��˛ � ‰1/˝ '

D �12‰1 ˝ ';

(11.14)

�S˛ � .‰0 ˝ / D 0

(cf. the proof of Lemma 11.7), and, by (7.10),

X

˛;ˇ;

"123˛ˇ �S˛ ��Sˇ D 4�S ; (11.15)

it followsX

i

e�i � re�

i

�X

˛

�S˛ � V˛ � .‰1 ˝ '/�

D 12.‰1 ˝ '/C 2X

˛

V˛ � DS˛ .‰1 ˝ '/

CX

˛;ˇ;

"123˛ˇ JS˛ .e

�i / � J Sˇ .e�

i / � V � .‰1 ˝ '/:

In view of the tensorial form of the last term in the right-hand side of the aboveequation, we may suppose that feig is an adapted local orthonormal frame (i.e., suchthat any ei has the form ˙J˛ej ). Using once again Lemma 11.1, any e�

i has then


the form ˙J˛.e�j /, and we getX

˛;ˇ;

"123˛ˇ JS˛ .e

�i / � J Sˇ .e�

i / � V � .‰1 ˝ '/

D �X

˛;ˇ;

"123˛ˇ e�i � Jˇ .J S˛ .e�

i // � V � .‰1 ˝ '/

D 2X

˛;ˇ; ;�

"123˛ˇ "123˛ˇ��

S� � V � .‰1 ˝ '/

D 4X

˛

�S˛ � V˛ � .‰1 ˝ '/:

So �nallyX

i

e�i � re�

i

�X

˛

�S˛ � V˛ � .‰1 ˝ '/�

D 12.‰1 ˝ '/C 2X

˛

V˛ � DS˛ .‰1 ˝ '/

C 4X

˛

�S˛ � V˛ � .‰1 ˝ '/:

On the other hand, using (11.5), (11.6), (11.7), and (11.12), the expression

E ´X

ˇ

Vˇ � rVˇ

�X

˛

�S˛ � V˛ � .‰1 ˝ '/�

reads

E D �X

˛;ˇ;

"123˛ˇ �S˛ � Vˇ � V � .‰1 ˝ '/

� 1

2

X

˛;ˇ

�S˛ ��Sˇ � Vˇ � V˛ � .‰1 ˝ '/

C 1

2

X

˛

�S˛ � V˛ � .‰1 ˝ '/

D �2X

˛

�S˛ � V˛ � .‰1 ˝ '/C 1

2

X

˛

�S˛ ��S˛ � .‰1 ˝ '/

C 1

2

X

˛;ˇ;

"123˛ˇ �S˛ ��Sˇ � V � .‰1 ˝ '/C 1

2

X

˛

�S˛ � V˛ � .‰1 ˝ '/

D �6.‰1 ˝ '/C 1

2

X

˛

�S˛ � V˛ � .‰1 ˝ '/;

again by making use of (11.14) and (11.15). �


We now have

Lemma 11.12.

DS�X

˛

V˛ � DS˛ .‰1 ˝ '/

�D 2.m � 1/.mC 4/

X

˛

�S˛ � V˛ � .‰1 ˝ '/

� 13

2

X

˛

V˛ � DS˛ .‰1 ˝ '/:

Proof. Using (11.9) and (11.13), one can write the term

F ´X

i

e�i � re�

i

�X

˛

V˛ � DS˛ .‰1 ˝ '/

�

as

F D �X

˛

�S˛ � DS˛ .‰1 ˝ '/C

X

˛;ˇ;

"123˛ˇ V˛ � Vˇ � DS .‰1 ˝ '/

�X

˛;i;j

V˛ � e�i � J S˛ .e�

j / � .reirej‰1 ˝ '/

C 1

2

X

˛;ˇ; ;i;j

"123˛ˇ e�i � J S˛ .e�

j / � J Sˇ .e�i / � V � .rej‰1 ˝ '/:

Now, from Lemma 11.1 and Lemma 7.10,X

˛

�S˛ � DS˛ .‰1 ˝ '/ D 0:

From (11.7), using once again Lemma 11.1 and the fact thatD˛‰1 is a local sectionof †0M ˚†2M , one gets

X

˛;ˇ;

"123˛ˇ V˛ � Vˇ � DS .‰1 ˝ '/ D �2

X

˛

V˛ � DS˛ .‰1 ˝ '/: (11.16)

By Lemma (11.1),

X

˛;i;j

V˛ � e�i � J S˛ .e�

j / � .reirej‰1 ˝ '/ D

X

˛

V˛ ��X

i;j

ei � J˛ej � reirej‰1

�˝ ';

but we saw in (7.33) that

D2.{‰1/ D 4.mC 4/.m � 1/{‰1; (11.17)

where {‰1 is locally de�ned by {‰1 DP˛ J˛ ˝�˛ �‰1.


But since D.{‰1/ D �2P˛ J˛ ˝ D˛‰1 (cf. (7.31)), equality (11.17) gives

(at the point x)X

i;j

ei � J˛ej � reirej‰1 D �2.mC 4/.m� 1/�˛ �‰1;

whenceX

˛;i;j

V˛ �e�i �J S˛ .e�

j / � .reirej‰1˝'/ D �2.mC4/.m�1/

X

˛

V˛ ��S˛ � .‰1˝'/:

Finally, using that for ˛ ¤ ˇ

X

i

e�i � J S˛ .e�

j / � J Sˇ .e�i / D �2

X

i

hJ S˛ .e�j /; J

Sˇ .e

�i /ie�

i � 2�Sˇ � J S˛ .e�j /

D �2X

"123˛ˇ JS .e

�j / � 2�Sˇ � J S˛ .e�

j /;

one gets

1

2

X

˛;ˇ; ;i;j

"123˛ˇ e�i � J S˛ .e�

j / � J Sˇ .e�i / � V � .rej‰1 ˝ '/

D 2X

˛

V˛ � DS˛ � .‰1 ˝ '/ �

X

˛;ˇ;

"123˛ˇ V˛ ��Sˇ � DS .‰1 ˝ '/:

Now using Lemma 7.10 and Lemma 11.1, the last term in the right-hand side of theabove equation is �8

P˛ V˛ � DS

˛ � .‰1 ˝ '/, and so

X

i

e�i � re�

i

�X

˛

V˛ � DS˛ .‰1 ˝ '/

�

D 2.m� 1/.mC 4/X

˛

V˛ ��S˛ � .‰1 ˝ '/ � 8X

˛

V˛ � DS˛ .‰1 ˝ '/:

On the other hand, using (11.5), (11.6), (11.10), and the de�nition of DS˛ , one gets

rVˇ

�X

˛

V˛ � DS˛ .‰1 ˝ '/

�D �

X

˛;

"123˛ˇ V˛ � DS .‰1 ˝ '/

� 1

2

X

˛

V˛ ��Sˇ � DS˛ .‰1 ˝ '/

� 1

2

X

˛

V˛ � Vˇ � DS˛ .‰1 ˝ '/:


Therefore,X

ˇ

Vˇ � rVˇ

�X

˛

V˛ � DS˛ .‰1 ˝ '/

�DX

˛;ˇ;

"123˛ˇ V˛ � Vˇ � DS .‰1 ˝ '/

� 1

2

X

˛;ˇ

Vˇ � V˛ ��Sˇ � DS˛ .‰1 ˝ '/

� 1

2

X

˛;ˇ

Vˇ � V˛ � Vˇ � DS˛ .‰1 ˝ '/:

Now, we already noted in (11.16) thatX

˛;ˇ;

"123˛ˇ V˛ � Vˇ � DS .‰1 ˝ '/ D �2

X

˛

V˛ � DS˛ .‰1 ˝ '/:

On the other hand, by (11.13), since D˛‰1 is a local section of †0M ˚†2M onehas

�12

X

˛;ˇ

Vˇ � V˛ ��Sˇ � DS˛ .‰1 ˝ '/ D 1

2

X

˛

�S˛ � DS˛ .‰1 ˝ '/

C 1

2

X

˛

V˛ ��X

ˇ;

"123˛ˇ �Sˇ � DS

.‰1 ˝ '/�

D 4X

˛

V˛ � DS˛ .‰1 ˝ '/;

since, as we already noted,X

˛

�S˛ � DS˛ .‰1 ˝ '/ D 0

and X

ˇ;

"123˛ˇ �Sˇ � DS

.‰1 ˝ '/ D 8X

˛

DS˛ .‰1 ˝ '/:

Finally, since

�12

X

˛;ˇ

Vˇ � V˛ � Vˇ � DS˛ .‰1 ˝ '/ D �1

2

X

˛

V˛ � DS˛ .‰1 ˝ '/;

one obtains

X

ˇ

Vˇ � rVˇ

�X

˛

V˛ � DS˛ .‰1 ˝ '/

�D 3

2

X

˛

V˛ � DS˛ .‰1 ˝ '/: �


We may now state the main result. It is a straightforward computation usingLemmas 11.7, 11.8, 11.11, and 11.12.

Proposition 11.13. Let ˆ be the spinor �eld on S de�ned by

ˆ D 2.mC 3/.‰0 ˝ '/C‰1 ˝ ' � 1

2

X

˛

V˛ ��S˛ � .‰1 ˝ '/

� 1

2.mC 4/

X

˛

V˛ � DS˛ .‰1 ˝ '/:

Then

DSˆ D 4mC 3

2ˆ D

r4mC 3

4mC 2

ScalS4

ˆ;

i.e., ˆ is a Killing spinor on S .

Remark 11.14. The heuristic construction of the above Killing spinor on S seemsrather “miraculous”. By reviewing the basic argument in [KSW98a], we will ex-plain why it is not. We will see that the above spinor �elds,

X

˛

V˛ ��S˛ � .‰1 ˝ '/ andX

˛

V˛ � DS˛ .‰1 ˝ '/;

are indeed related to the two sections of the twisted bundle QM ˝ †M , {‰1 andD.{‰1/, which appear in the study of the limiting case of the estimate in Section 7.4.

The approach developed in [KSW98a] is based on the following remark. Onthe one hand, the limiting case of the estimate brings into play two spinor �elds be-longing, respectively, to �.†0M/ and �.†1M/, together with a section of a vectorbundle Pƒm�2

ı E , with �ber isomorphic toƒm�2ı E. On the other hand, the determi-

nation of the �rst eigenvalue of the Dirac operator on .HPm; can/ (cf. Section 15.4),brings into play the irreducible fundamental representation of the group SpmC1 inthe space ƒmı EmC1, where

EmC1 ´ .C C j C/mC1 ' HmC1:

But the decomposition of this representation under the action of Sp1 � Spm (viewedas a subgroup of SpmC1), is given by

ƒmı EmC1 D ƒmı E ˚ .H ˝ƒm�1ı E/˚ ƒm�2

ı E

'†4m;0 ˚ †4m;1 ˚ƒm�2ı E;

(11.18)

(cf. for instance the branching rules given in [Tsu81]).


In [KSW98a]), the authors give an explicit description of the inclusion

ƒmı E ˚ .H ˝ƒm�1ı E/˚ƒm�2

ı E ,�! ƒmı EmC1;

and then use it to construct a section of a bundle with �bre isomorphic toƒmı EmC1,which appears to be a spinor �eld on S verifying the Killing equation.

In order to understand the relation between ƒmı EmC1 and the spinor represen-tation †4.mC1/, recall that the restriction of the homomorphism (cf. (7.2))

Sp1 � SpmC1 �! SO4.mC1/; .q; A/ 7�! .x 2 HmC1 7�! Ax Nq/

to the subgroup SpmC1, gives the standard inclusion SpmC1 � SO4.mC1/.Now, the spinor space †4.mC1/ decomposes under the action of Sp1 � SpmC1

as (cf. (7.6))

†4.mC1/ DmC1M

rD0†4.mC1/;r ;

where†4.mC1/;r ' SrH ˝ƒmC1�r

ı EmC1:

Hence, since SpmC1 acts trivially on SrH , each †4.mC1/;r decomposes under theaction of SpmC1 into rC1 copies of the representationƒmC1�r

ı EmC1. In particular,

(a) †4.mC1/;mC1 decomposes into .mC 2/ trivial representations;

(b) †4.mC1/;1 decomposes into two copies of ƒmı EmC1.

Hence, the fundamental representation of SpmC1 in the space ƒmı EmC1 is con-

tained (with multiplicity 2) in†4.mC1/;1. Notemoreover that, as it was noted before,†4mC4;1 is contained in either †C

4.mC1/ or †�4.mC1/, hence the fundamental repre-

sentation of SpmC1 in the space ƒmı EmC1 can be seen as contained in the spinorrepresentation †4mC3 ' †C

4.mC1/ ' †�4.mC1/.

Now from the de�nition of the spin structure of S given above, together withthe description of the spin structure of M given in Section 7.2, it follows that thespinor bundle †4mC3S is indeed an Sp1 � Spm-bundle. Hence, by (11.18), oneconcludes that the bundles ��.†0M/, ��.†1M/, ��.V1;1M/, and ��.V0;2M/,where V1;1M and V0;2M are the subbundles ofQM˝†2M , with �ber respectivelyisomorphic to H ˝ƒm�1

ı E and ƒm�2ı E, can be embedded into the spinor bundle

†4mC3S ' ��.†M/˝†3S .By considering explicit embeddings, it can be proved that, up to a scalar mul-

tiple, the spinor �elds ‰0, ‰1 correspond respectively to the projectable spinors‰0 ˝ ' and ‰1 ˝ ', whereas the two sections {‰1 and D.{‰1/ of the twistedbundle QM ˝†M correspond respectively to

X

˛

V˛ ��S˛ � .‰1 ˝ '/ andX

˛

V˛ � DS˛ .‰1 ˝ '/:

11.4. Conclusion 299

11.4 Conclusion

We may now conclude as in [KSW98a]. The Killing spinor ˆ induces a parallelspinor on the Riemannian cone xS over S . Moreover, xS is a hyper-Kähler manifold(cf. Theorem 8.20). Now, due to a result ofWang [Wan89], on an hyper-Kähler man-ifold of dimension 4.mC 1/ there are exactly mC 2 linearly independent parallelspinors, corresponding to the mC 2 trivial representations induced by the decom-position of †4.mC1/;mC1 under the group SpmC1 (cf. Remark 11.14).

By the general description of spinors on Riemannian cones, cf. Section 2.4.2,together with Remark 11.14 above, the parallel spinor induced by ˆ is indeed asection of †4.mC1/;1 xS . This forces the holonomy to be further reduced. But fromthe Berger–Simons theorem (cf. for instance Theorem 10.90 in [Bes87]), this is onlypossible if xS is reducible or locally symmetric.

In the �rst case, xS has to be �at, by a result of Gallot [Gal79]. This is also the caseif xS is locally symmetric, because any hyper-Kähler manifold is Ricci-�at and anyhomogeneous Ricci-�at manifold is �at (cf. for instance Theorem 7.61 in [Bes87]).Therefore xS is �at. Since S is simply connected (cf. [BG99], Proposition 4.4.1),this implies that S is isometric to the standard sphere S

4mC3. Hence, by a result ofIshihara [Ish73],M is isometric to the quaternionic projective space.

Part IV

Dirac spectra of model spaces

Dirac spectra of model spaces 303

In the study of the spectrum of the Dirac operator, it is natural and useful to considerexamples, and the archetypal examples in Riemannian Geometry are the symmetricspaces. For a symmetric space, the spectrum of the Dirac operator can be (theoret-ically) computed by classical harmonic analysis methods.

The aim of this part is to present a method for the determination of the spectrumof the Dirac operator on a spin, compact, simply connected and irreducible sym-metric space, and to give the explicit computation of the spectrum in those threecases:

(1) spheres .Sn; can/, n � 2;

(2) complex projective spaces .CPm; can/,m D 2q C 1, q � 0;

(3) quaternionic projective spaces .HPm; can/, m � 1.

Note that the holonomy of those manifolds is, respectively, SOn, U2qC1 andSp1 : Spm, so they are standard examples of, respectively, Riemannian, Kähler andquaternion-Kähler manifolds.

Chapter 12

A brief survey on representation theory of compact

groups

In this chapter we sketch the theory of �nite-dimensional complex representationsof a compact and connected Lie group G, that is Lie groups homomorphisms

�WG �! GL.V /;

where V is a �nite-dimensional C-vector space.We assume known the basic terminology on representation theory. Our aim is

to give a brief survey on the construction of the theory in order to consider thefollowing problems:

• Describe all irreducible representations (up to equivalence) of G.

• Give the decomposition of an arbitrary representation of G into irreduciblerepresentations.

Results and detailed proofs can be found, for example, in [Ada69], [BtD85],[Die75], [FH91], [GW98], [Kna01], [Hum72], and [Žel73].

12.1 Reduction of the problem to the study of irreduciblerepresentations

Because of the compactness of the group, to describe all �nite-dimensional repre-sentations of G it is enough to know its irreducible representations (up to equiva-lence). This is due to the following fundamental result.

Theorem 12.1. Every �nite-dimensional representation of G is unitary, and hencecompletely reducible.

Proof. Choose a Hermitian product on the Lie algebra g D TeG and construct aright-invariant Riemannian metric on G by right translations:

Rg WG �! G; Rgx ´ xg:

306 12. A brief survey on representation theory of compact groups

This provides us with a right-invariant volume form dg. We normalize the metricso that vol.G/ D

RGdg D 1. Let .�; V / be a �nite-dimensional representation. Let

h�; �i be any Hermitian product on V and let h�; �iG be the Hermitian product on Vde�ned by

hv1; v2iG DZ

G

h�.g/v1; �.g/v2idg; .v1; v2/ 2 V � V:

By the right-invariance of dg, the representation � is unitary for this Hermitianproduct.

Suppose that V is not irreducible. Then it admits a non-trivial G-invariant sub-space U . But the orthogonal complement U? for the Hermitian product h�; �iG ,is also a G-invariant subspace of V , so V decomposes into the sum of G-spacesV D U ˚ U?. If either U or U? is not irreducible, V can be further decomposed.The process must eventually stop because V is �nite dimensional. �

This theorem applies, in particular, to the adjoint representation ofG, yielding anAdG-invariant Hermitian product on g. The right-invariant metric on G de�ned bythis product is then bi-invariant, and its volume form dg is thus the Haar measureon G. Note also that the averaging construction continues to hold when V is anin�nite-dimensional Hilbert space.

A simple but crucial fact in representation theory is the Schur lemma.

Lemma 12.2. Let �; �0 be irreducible unitary representations ofG andAWV ! V 0

a linear map commuting with the G-actions. If V and V 0 are not isomorphic asrepresentation spaces, then A must be 0. If � D �0 and dim.V / < 1, then A mustbe a constant multiple of the identity.

Proof. The kernel of A and the orthogonal complement of its image are both rep-resentation spaces of G. Since the two representations are assumed irreducible, itfollows that A is either 0 or an isomorphism of G-spaces, proving the �rst claim. If� D �0; V D V 0, the endomorphismAmust have an eigenspace for some eigenvalue�. Since A commutes with the G action, this eigenspace is a sub-representation, soit must be the whole of V . �

For any unitary representation .�; V / (where V is a complex Hilbert space ofpossibly in�nite dimension) and any orthonormal basis feig of V , the matrix of�.g/ is unitary for any g 2 G. Let

m�ij WG �! C; m

�ij .g/ ´ h�.g/ej ; eii;

be the (smooth) functions de�ned by considering the coef�cients of this matrix.These functions are called matrix coef�cients of the representation �. With respectto the usual Hermitian scalar product .�; �/ on L2.G/, the matrix coef�cients verifythe following orthogonality properties.

12.1. Reduction of the problem to the study of irreducible representations 307

Theorem 12.3. Let � and �0 be two irreducible representations of G .possibly ofin�nite dimension/. Their matrix coef�cients satisfy the following properties:

(i) .m�ij ; m�0

k0;l 0/ D 0, if � and �0 are not equivalent.

(i) If � D �0 and dim.V / D n� < 1, then .m�ij ; m�

kl/ D 1

n�ıikıjl .

Proof. Let .�; V / and .�0; V 0/ be two inequivalent irreducible representations ofG.The Schur lemma implies that the space of G-invariant homomorphisms V ! V 0,

HomG.V; V0/ ´ fA 2 Hom.V; V 0/I �0.g/ ıA D A ı �.g/; for all g 2 Gg;

is trivial because any G-invariant homomorphism V ! V 0 has to be an isomor-phism.

Now to eachA 2 Hom.V; V 0/ ' V �˝V 0 can be associated, using the averagingtechnique of the above proof, the G-invariant endomorphism

yA DZ

G

�0.g�1/ ıA ı �.g/dg:

But since HomG.V; V 0/ D f0g, any such yA has to be zero. Applying this to thebasis fe�

i ˝ e0j g of V � ˝ V 0, where feig (resp. fe0

j g) is a basis of V (resp. V 0), inwhich the matrix of �.g/ (resp. �0.g/) is unitary for any g 2 G, one gets the �rstset of orthogonality relations.

On the other hand, by applying the averaging technique to A 2 Hom.V; V /,since by the Schur lemma any vector in HomG.V; V / is a scalar multiple of identity,one gets

yA DZ

G

�.g�1/ ı A ı �.g/dg D �Id;

where � 2 C. Note thattr yA D trA D � dimV:

Applying this to the basis fe�i ˝ ej g of V � ˝ V , one gets the second set of orthog-

onality relations. �

Instead ofmatrix coef�cients which involve the choice of a basis, representationscan be uniquely characterized, up to equivalence, by a single function on G.

Definition 12.4. Let � be a representation of G. The (smooth) function

��WG �! C; g 7�! ��.g/ ´ tr.�.g//

is called the character of the representation �. Characters of irreducible represen-tations are called irreducible characters.


The following results are easily deduced from the de�nition.

Proposition 12.5. Let � and �0 be two representations of G.

(i) If � and �0 are equivalent, then �� D ��0 , so �� depends only on the equiva-lence class of �.

(ii) Characters are class functions, i.e., they verify

��.hgh�1/ D ��.g/; g; h 2 G;

(iii) ��˚�0 D �� C ��0 .

(iv) ��˝�0 D ��0 .

(v) If � is unitary, �� D ��:

By Theorem 12.3 we get the following proposition.

Proposition 12.6. Let �� and ��0 be two irreducible characters. Then

.��; ��0/ D

8<:0 if � and �0 are not equivalent;

1 if � D �0:

This implies that irreducible characters are linearly independent and

Corollary 12.7. If two irreducible characters satisfy �� D ��0 , then � and �0 areequivalent.

Corollary 12.8. The character �� of every representation � satis�es .��; ��/ 2 Z,.��; ��/ � 1, with equality if and only if � is irreducible.

Proof. Let � DLi ni�i be the decomposition of � into irreducible components,

where ni 2 N denotes the multiplicity of �i in �. Then .��; ��/ DPi n2i , hence

the result. �

Proposition 12.9. Let G and H be compact Lie groups. Let �G ; �0G and �H ; �0

H

be irreducible representations of G, respectively of H . Then the tensor products�G ˝ �H and �0

G ˝ �0H are irreducible representations of G � H . Moreover we

have �G ˝ �H � �0G ˝ �0

H if and only if �G � �0G and �H � �0

H .Conversely, every irreducible representation ofG �H is equivalent to a unique

tensor product of this form.


Proof. From Proposition 12.5 (iv) and Proposition 12.6, one has

.��G˝�H; ��0

G˝�0H/ D

Z

G�H��0

G.g/��0

H.h/��G

.g/��H.h/dgdh

DZ

G

��0G.g/��G

.g/dg �Z

H

��0H.h/��G

.h/dh

D .��G; ��0

G/.��H

; ��0H/:

Hence, by Corollary 12.8, �G ˝ �H is an irreducible .G �H/-representation andthe �rst part of the proposition follows.

Let .�; V / be an irreducible .G � H/-representation. Identifying H with theclosed subgroup feg �H ofG �H , the space V can be decomposed into the directsum V D

Li Vi , where the Vi are the irreducible H -subspaces of V .

Now the space HomH .Vi ; V / of H -invariant homomorphisms Vi ! V is aG-space. Indeed, for any g 2 G, for any A 2 HomH .Vi ; V /, set

g � A ´ �.g; e/ ıA:

It is easy to see that .g � A/ 2 HomH .Vi ; V /. Thus the space HomH .Vi ; V / can bedecomposed into the direct sum HomH .Vi ; V / D

Lj Wij , where the Wij are the

irreducible G-subspaces of HomH .Vi ; V /. Consider then the endomorphism

Fij WWij ˝ Vi �! V

.A˝ v/ 7�! A.v/:

It is easy to check that Fij 2 HomG�H .Wij ˝ Vi ; V /. Hence, since Wij ˝ Vi andV are irreducible .G �H/-spaces, the Schur lemma implies that Fij is either zeroor an isomorphism.

Now HomH .Vi ; V / contains the canonical injection AWVi ,! V . According tothe decomposition HomH .Vi ; V / D

Lj Wij , A splits as A D

Pj Aj . Let vi be a

non-zero vector in Vi . Necessarily one of the vectors Aj .vi / D Fij .Aj ; vi/ has tobe non-zero, thus the corresponding Fij is an isomorphism, and V is equivalent toWij ˝ Vi . �

Let RG be the set of equivalence classes of irreducible �nite-dimensional com-plex representations of G, and let ˆG be the vector space generated by the matrixcoef�cientsm�ij of all the � 2 RG . By choosing different bases in V , the matrix co-ef�cients change by linear combinations, so ˆG does not depend on these choices.

By Theorem 12.1,ˆG is a subalgebra of the algebra C1.G/ of smooth functionson G (indeed, the product of two matrix coef�cients m�ij , m

�0

k0l 0is a matrix coef�-

cient of the tensor product � ˝ �0, which factors as the direct sum of irreduciblerepresentations).


Theorem 12.10 (Peter–Weyl). Let G be a compact Lie group.

(1) The functions . 1n�m�ij /, with � 2 RG , form aHilbert basis of the Hilbert space

L2.G/.

(2) Every continuous function onG is uniformly approximable by functions fromˆG .

Proof. We �rst give the proof of the second statement of the theorem under theassumption that G has a faithful representation (such as classical matrix groups orspin groups), because in this case the result can be simply deduced from the Stone–Weierstraß theorem.

First, the constant function f0.g/ � 1 is the matrix coef�cient of the trivialrepresentation on the space C, hence constant functions belong to ˆG . For any

m�ij 2 ˆG , .�; V / 2 RG , the function m�ij is also in ˆG , because it is a matrix

coef�cient of the dual representation .�; V �/, hence ˆG is closed under complexconjugation. Finally, the existence of a faithful representation ensures thatˆG sepa-rates points ofG. Thus, from the Stone–Weierstraß theorem, ˆG is dense in C0.G/for the supremum norm topology.

Let us now turn to the general case. There exists another algebra structure onC1.G/, with respect to the convolution product

.f � f 0/.x/ ´Z

G

f .xy/f 0.y�1/dy DZ

G

f .y/f 0.y�1x/dy:

The algebra C1.G/ is endowed with the usual L2 inner product with respect to thebi-invariant metric on G of volume 1,

.f; f 0/ DZ

G

f .x/f 0.x/dx:

Finally, there is aG�G action on C1.G/, called the regular representation, derivedfrom the (mutually commuting) left and right actions of G on itself:

.g; g0/f ´ LgRg0f; .Lgf /.x/ D f .g�1x/; .Rgf /.x/ D f .xg/:

This action preserves the L2 inner product, and is compatible with the convolution

LgRg0.f � f 0/ D .Lgf / � .Rg0f 0/:

Thus there is an induced Hermitian action ofG�G on the norm-completion L2.G/of C1.G/. Note also that the Laplacian � on G commutes with the G �G action.Since the eigenspaces of the Laplacian are �nite-dimensional and span L2.G/ byCorollary 4.37,G�G preserves these �nite-dimensional eigenspaces and hence itsaction on L2.G/ splits into �nite-dimensional representations.


Let �WG ! Aut.V / be an irreducible unitary representation. Then the algebraEnd.V / is also a Hermitian representation of G � G:

.g; g0/A D �.g/A�.g0�1/; hA;Bi D dim.V / tr.AB�/:

By Proposition 12.9, this representation is irreducible (since End.V / D V ˝ V �).We de�ne a map from End.V / to C1.G/ by

�V WEnd.V / �! C1.G/; A 7�! FA; FA.x/ D nV tr.�.x�1/A/:

where nV D dim.V /. We also de�ne

pV W C1.G/ �! End.V /; f 7�! Af DZ

G

f .x/�.x/dx:

Directly from the de�nitions, these two maps commute with the G �G actions.

Lemma 12.11. Let �V ; �V0be irreducible representations ofG. For allA 2 End.V /

one has pV 0 ı �VA D A if �V D �V0, and zero if �V and �V

0are inequivalent.

Proof. We write

pV 0 ı �V WEnd.V / �! End.V 0/;

pV 0.�V .A// D nV

Z

G

tr.�.x�1/A/�V0.x/dx 2 End.V 0/:

By Schur lemma and Proposition 12.9, this .G � G/-equivariant map vanishes ifV is not equivalent to V 0, while if V D V 0 it is a constant multiple of the identitytransformation of End.V /, i.e., there exists �V 2 C such that pV .�V .A// D �VA

for all A. To compute the constant �V we apply pV ı �V to the identity IV andcompute its trace:

tr.pV .�V .IV /// D nV

Z

G

tr.�.x�1// tr.�.x//dx D nV k��k2L2 D nV ;

where in the last equality we have used Corollary 12.8. Therefore, �V D 1. �

It is easy to see that pV and �V are compatible with the algebra structures (thecomposition of linear maps, respectively the convolution product). Moreover, weclaim that �V preserves the inner products, more precisely, for two representations�V ; �V

0we have

.F VA ; FV 0

B / DnV nV 0

Z

G

tr.�V .x�1/A/tr.�V 0.x�1/B/dx

D nV nV 0 tr

�B�

Z

G

tr.�V .x�1/A/�V0.x/dx

�

D nV 0 tr.B�pV 0 �V .A//:


By Lemma 12.11, this is 0 if V is not equivalent to V 0, respectively nV tr.B�A/ ifV D V 0, thus proving the claim.

It follows that �V embeds End.V / into C1.G/ as an irreducible representa-tion AV of G � G. Since the L2 orthogonal projection on every eigenspace of� commutes with the G � G action, it follows that �V .End.V // lives inside someeigenspace of the Laplacian.

The complex conjugates of the matrix coef�cients of the representation V arein fact, by de�nition, a linear basis of AV . Thus for inequivalent representationsV; V 0, the spaces AV ;AV 0 are orthogonal in L2.G/. Clearly,

LV 2R.G/AV is just

the space of matrix coef�cients ˆG .We will show that

LV 2R.G/AV is dense in L2.G/. Otherwise, there would

exist some orthogonal complement of it inside some eigenspace of �. Call B this�nite-dimensional Hilbert space. It consists of smooth functions, L2-orthogonal tothe matrix coef�cients of every �nite-dimensional representation of G. It is more-over a G � G representation space, hence also closed under the left regular actionofG, i.e., if f 2 B, then for every x 2 G, the mapLxf sending g to f .x�1g/ alsobelongs to B. Let f; f 0 2 B be of norm 1, living in some irreducible component B0of B with respect to the right action of G on C1.G/. Then

.Rgf; f0/ D

Z

G

f .xg/f 0.x/dx D�Z

G

f 0.x/Lx�1.f /dx

�.g/:

As a function of g, this integral belongs to the complete linear space B. We reachnow a contradiction: on the one hand this function is a matrix coef�cient for theirreducible component B0 of the right regular representation, on the other hand,since it belongs to B, it is orthogonal to every irreducible matrix coef�cient. Thusall the matrix coef�cients of the representation B0 vanish, which contradicts thecharacter identity of Proposition 12.6.

For the second claim of the theorem, by Stone–Weierstraß, every continuousfunction on G can be uniformly approximated by smooth functions. The decom-position of a smooth function into eigenmodes (eigenfunctions of the Laplacian)converges in every Sobolev norm. By the Sobolev embedding theorem

Hn2 C�.G/ ,�! C0.G/I

such a convergence holds in C0 (i.e., uniform) norm. �

For a more “classical” proof in the general case, see [BtD85], Theorem (3.1),p. 134.

Corollary 12.12. RG is a countable set.

Note that for a general compact Lie group, the existence of a faithful represen-tation is deduced from the Peter–Weyl theorem.


Theorem 12.13. Every compact Lie group G has a �nite-dimensional faithful rep-resentation.

Proof. Consider the eigenspacesE� of the Laplacian insideL2.G/, which are �nite-dimensional representation spaces for the left regular action of G, and at the sametime the spaces of matrix coef�cients of these representations:

E� D End.V 1� /˚ � � � ˚ End.V k.�/�

/:

Let

L2N ´M

��NE�;

and set

VN ´M

��N;j�k.�/

Vj

�

the corresponding representation. The kernel of the representation VN is the set ofg 2 G such that all the matrix coef�cients of VN take the same values at g andat 1 2 G, or equivalently the algebra of functions VN does not separate 1 and g.The set of such g is clearly a closed subgroup, and becomes smaller asN increases.If the intersection of these nested Lie groups contains some point g other than 1,it would mean that L2N does not separate 1 and g for any N , hence by the Peter–Weyl theorem C0.G/ would not separate these two points, which is absurd. Hencethe intersection is f1g. Now note that a decreasing sequence of compact Lie groupsis stationary after a �nite number of steps. Thus there exists N such that VN isfaithful. �

For another proof see, for instance, [BtD85], Theorem (4.1), p. 136.

Remark 12.14. Using the Peter–Weyl theorem, we can also prove that any contin-uous irreducible representation of a compact Lie group on a Hilbert space is �nitedimensional. Indeed, by Proposition 12.3, the matrix coef�cients of an in�nite-dimensional representation must be L2-orthogonal to every matrix coef�cient ofany �nite-dimensional representation, thus by Peter–Weyl they must vanish, whichimplies the absurd conclusion that the representation itself is 0. See also [BtD85],Corollary (5.8), p. 141, or [Žel73], Lemma 7, p. 77.

The existence of a faithful representation allows us to describe any irreduciblerepresentation as acting on a space of tensors.


Theorem 12.15. Let �1 be a �nite-dimensional faithful representation of G. Everyirreducible representation of G is contained in a tensor product of the form

�m;n ´ �1 ˝ � � � ˝ �1„ ƒ‚ …m

˝ �1 ˝ � � � ˝ �1„ ƒ‚ …n

; m; n 2 N;

(�0;0 being the trivial representation on the space C).

Proof. Suppose there exists an irreducible representation � which does not satisfythe above condition. Let m�ij be the matrix coef�cients of �. From Theorem 12.1

and Theorem 12.3, any m�ij should be orthogonal to any matrix coef�cient of therepresentations �m;n,m; n 2 N. But it is not hard to see that the matrix coef�cientsof the representations �m;n,m; n 2 N, de�ne a subalgebra of C0.G/ which satis�esall the hypothesis of the Stone–Weierstraß theorem. Thus, all the m�ij should beequal to 0, a contradiction. �

Corollary 12.16. LetH be a closed subgroup ofG (hence a compact Lie subgroupof G). Every irreducible representation ofH is contained in the restriction toH ofa representation of G.

Proof. Let �1 be a faithful representation of G. Then �1jH is a faithful represen-tation of H . By Theorem 12.15, any irreducible representation of H is containedin a tensor product of �1jH and its conjugate, which is clearly the restriction of arepresentation of G. �

Theorem 12.10 admits the

Corollary 12.17. (i) Every continuous class function1 on G is uniformly approx-imable by a linear combination of irreducible characters.

(ii) The irreducible characters ��, � 2 RG , form a Hilbert basis of the centerof L2.G/ with respect to the convolution product.

The previous results allow one to endow the Z-module ZRG generated by RGwith a commutative ring structure given by identifying an equivalence class with itscharacter.

Classically, ZRG is called ring of equivalence classes of representations of G.

Note that a linear combination of representations in RG with integer coef�cients isthe class of a representation only if all the integers are non-negative. Accordingly,the elements of Z

RG are also called virtual representations.

The following result is straightforward.

1The de�nition of “class function” was given in Proposition 12.5 (ii).

Irreducible representations of a maximal torus 315

Proposition 12.18. Let G be a compact and connected Lie group and let

� WG �! H

be an onto homomorphism of Lie groups (thus H is compact and connected too).Any irreducible representation of H is obtained as a factor through the quotient ofan irreducible representation � of G with the property that for all g 2 Ker� , wehave �.g/ D Id.

12.2 Reduction of the problem to the study of irreduciblerepresentations of a maximal torus

Definition 12.19. A torus of G is a closed, connected, and Abelian subgroup ofG. A torus of G is said to be maximal if there is no other torus of G containing itstrictly.

Note that every closed, connected, and Abelian subgroup ofG is a compact con-nected Abelian Lie group, hence is isomorphic to a torus T n, hence the terminology.

Theorem 12.20. The Lie group G is the union of its maximal tori.

Proof. Let g be the Lie algebra of G. The proof uses the fact that the exponen-tial map expW g ! G is onto. Indeed, the compact group G posses a bi-invariantmetric (see the remark after Theorem 12.1) for which the (Riemannian) exponentialmap Expe at the point e coincides with the group exponential exp (cf. for instanceExample 2.90 in [GHL87]). But since G is compact and connected, it results fromthe Hopf–Rinow theorem that Expe is onto, hence exp is onto too.

Let g 2 G. Since exp is onto, there exists X 2 g such that exp.X/ D g. Thusg belongs to the one-parameter subgroup fexp.tX/; t 2 Rg. The Lie algebra of thissubgroup being Abelian, is contained in a maximal Abelian subalgebra t of g.

Now let T be the connected Abelian Lie subgroup of G whose Lie algebra is t.This group is necessarily closed: otherwise, its closure xT would be a connectedcompact and Abelian Lie subgroup of G whose Lie algebra t0 satis�es t t0, con-tradicting the maximality of t. Hence, T is a maximal torus such that g 2 T . �

Theorem 12.21. Let T be a maximal torus ofG. For any g 2 G, there exists s 2 Gsuch that sgs�1 2 T .

Proof. First recall that since T is isomorphic to a torus T n, it has a generator, that isan element u such that the group fuk; k 2 Zg is dense in T (cf. for instance [BtD85],Theorem (4.13), p. 38). Take g0 2 G. Since the exponential map of G is onto(cf. the proof of Theorem 12.20), there exist a X0 2 g and a Z in the Lie algebrat of T such that exp.X0/ D g0 and exp.Z/ D u. Note that the one-parametersubgroup fexp.tZ/I t 2 Rg is dense in T .


Let h�; �i be an Ad.G/-invariant scalar product on g (such a scalar product on the�nite-dimensional representation space g exists as a consequence of Theorem 12.1since G is compact), and let k � k be the norm associated with it. Now G beingcompact, there exists an s 2 G such that the function on G de�ned by

g 7�!k Ad.g/ �X0 �Z k2

admits a minimum at the point s. Therefore, the tangent map at s of this function iszero, whence

d

dtfk Ad.exp.tX/s/ �X0 �Z k2gtD0 D 0; X 2 g:

Setting Y0 ´ Ad.s/ �X0, one gets

0 D 2hŒX; Y0�; Y0 � Zi:

But by the Ad.G/-invariance of the scalar product, this implies that

hX; ŒY0; Z�i D 0; X 2 g;

hencead.Z/ � Y0 D 0:

This in turn yields

Adexp.tZ/ Y0 D et ad.Z/ � Y0 D Y0; t 2 R:

Hence, since the one-parameter subgroup fexp.tZ/; t 2 Rg is dense in T ,

Ad.t / � Y0 D Y0; t 2 T

and soŒX; Y0� D 0; X 2 t:

But since t is a maximal Abelian subalgebra of g, this forces Y0 to be in t (otherwiset ˚ hY0i would be an Abelian subalgebra of g containing t strictly). Thus

sg0s�1 D exp.Ad.s/ � X0/ D exp.Y0/ 2 T: �

Corollary 12.22. Any two maximal tori of G are conjugate. Thus all maximal toriof G have same dimension, called the rank of G.

Proof. Let T1 and T2 be two maximal tori. Let t1 be a generator of T1. By The-orem 12.21, there exists s 2 G such that st1s�1 2 T2. Since t1 is a genera-tor of T1, this implies that sT1s�1 � T2. But sT1s�1 is a maximal torus, hencesT1s

�1 D T2. �


Corollary 12.23. Every irreducible representation of G is uniquely de�ned, up toequivalence, by the restriction of its character to a maximal torus T .

Proof. For � 2 RG , let �� be its character. By Theorem 12.21, for any g 2 G, thereexists s 2 G such that sgs�1 2 T , hence

��.g/ D ��.sgs�1/ D ��jT .sgs�1/: �

This last result is interesting since irreducible representations of tori are explic-itly known. Let T be a torus ofG, and let t be its Lie algebra. The exponential mapexpW t ! T is an onto homomorphism of Lie groups. Furthermore, its kernel

�T ´ exp�1.e/;

is a lattice of t.

Theorem 12.24. Let T be a torus of G and t its Lie algebra.

(i) The representation space of every irreducible representation of T is one-dimensional.

(ii) The irreducible characters of T are the Lie groups homomorphisms

�WT �! U1

such that�.expX/ D e2i�'.X/; X 2 t;

where ' 2 t� has the property that for all X 2 �T , we have '.X/ 2 Z.

Proof. Let �WT ! GL.V / be an irreducible representation of T . Since T isAbelian,

�.t/ D �.s/ ı �.t/ ı �.s�1/; t; s 2 T:Hence, for any t 2 T , �.t/ is a T -invariant C-automorphism of V . The Schurlemma then implies that

�.t/ D �.t/IdV

with �.t/ 2 C, t 2 T . So every one-dimensional subspace of V is T -invariant, andsince V is irreducible, V has to be one-dimensional.

Furthermore, since � is unitary, one has for any t 2 T , �.t/�.t/ D 1, hence�.t/ 2 U1. Now the character � of � is the Lie group homomorphism

T �! U1; t 7�! �.t/:

Considering the tangent map of � at the point e, ��W t ! u1 D iR, one gets

�.expX/ D e��.X/; X 2 t:

Let ' D 12i��. Note that ' 2 t� and satis�es '.X/ 2 Z, for any X 2 �T .

Conversely, if ' 2 t� satis�es this condition, then the map 2i�' factors into anirreducible character of T . �


The linear forms ' 2 t� that verify the condition of the theorem form a lattice��T of t�. This lattice is called dual of the lattice �T .

Definition 12.25. The set PT .G/ D 2i��T is a lattice of the complexi�ed Lie

algebra t�C

of t�, called the weight lattice of G relative to the maximal torus T .The elements of PT .G/ are called weights.

Remark 12.26. Note that elements of i t� are often also called weights. In this case,if there is some ambiguity, weights of PT .G/ will be said to be integral weights.

From now on, we �x a maximal torus T of G and denote the correspondingweight lattice by P.G/. Let � be the irreducible character of an element � 2 RG .From the results above it follows that

�jT DX

ˇ2P.G/n.ˇ/eˇ ;

where n.ˇ/ 2 N, which is zero except for a �nite number of values of ˇ, is themultiplicity in �jT of the representation

eˇ WT �! U1; t 7�! eˇ.X/;

X being any element in exp�1.t /. If n.ˇ/ is a positive integer, ˇ is said to becontained in � with multiplicity n.ˇ/.

Remark 12.27. Note that, conversely, Corollary 12.16 implies that any weightˇ 2 P.G/ is contained in the character of some irreducible representation of G.

Now as �jT is the restriction to T of a class function, one has

�jT .gtg�1/ D �jT .t /; g 2 G; gTg�1 � T:

(Note that the condition gTg�1 � T indeed implies gTg�1 D T , since gTg�1 isa maximal torus of G). Hence �jT is invariant under the action of the normalizerN.T / of T ,

N.T / ´ fg 2 GI gTg�1 D T g;which acts on T by inner automorphisms

Ig WT �! T; t 7�! gtg�1:

Since Ig D Id if g 2 T , this action factors into a smooth action of the groupN.T /=T on T , given by

w � t D Ig.t /;

where g 2 N.T / is a representative of w 2 N.T /=T .


Theorem 12.28. The group

WG ´ N.T /=T

is a �nite group, called theWeyl group of G.

Indeed it would be more correct to call it the Weyl group of G correspondingto T , but this does not matter since from Corollary 12.22, the Weyl groups corre-sponding to two maximal tori are isomorphic.

Proof. The group N.T / is a closed group of G, hence a Lie subgroup of G.For any g 2 N.T /, Ad.g/ is an automorphism of the Lie algebra t of T . But Tbeing isomorphic to a torus T n ' R

n=Zn, any automorphism of t can be identi�edwith an isomorphism of Rn which factors through the quotient, hence an elementof GL.n;Z/.

Thus, the set Aut.t/ ' GL.n;Z/ being discrete, the image of the component ofidentity N.T /e of N.T / by the homomorphism

Ad jN.T /WN.T / �! Aut.t/;

reduces to the identity of Aut.t/. Hence for any X in the Lie algebra of N.T /,ad.X/ � t D f0g, so X 2 t (otherwise t ˚ hXi would be an Abelian subalgebra of gcontaining t stictly, contradicting the maximality).

From this, one deduces that the Lie algebra of N.T / is equal to t, henceN.T /e D T , since those two connected subgroups ofG have the same Lie algebra.Now N.T /, being closed in G, is compact, so WG D N.T /=N.T /e is compactand discrete, hence �nite. �

Now considering the differential at e of the action of N.T / on T , we get anaction of WG on t given by

w �X ´ Ad.g/.X/; w 2 WG ;

where g 2 N.T / is a representative of w 2 N.T /=T . We consider the naturalextension of this action to the complexi�ed Lie algebra tC of t, and de�ne a leftaction of WG on t�

Cby

.w � ˇ/.X/ ´ ˇ.w�1 �X/; w 2 WG ; ˇ 2 t�C; X 2 tC:


Lemma 12.29. The weight lattice P.G/ is invariant under the action of the WeylgroupWG .

Proof. Let w 2 WG , ˇ 2 P.G/, and X 2 �T . One has

.w � ˇ/.X/ D ˇ.w�1 �X/ D ˇ.Ad.g�1/ �X/;

where g 2 N.T / is some representative of w. Now, since

exp.Ad.g�1/ �X/ D g�1.expX/g D e;

we have Ad.g�1/.X/ 2 �T , hence ˇ.Ad.g�1/ �X/ 2 2i�Z. Thus w � ˇ 2 P.G/.�

Consider now the decomposition of the restriction to T of an irreducible char-acter � of G,

�jT DX

ˇ2P.G/n.ˇ/eˇ :

Since � is invariant under the action of the Weyl group, for any w 2 WG and anyt 2 T , considering some element X in exp�1.t /, one deduces from the precedinglemma that

�jT .t / D �jT .w � t /

DX

ˇ2P.G/n.ˇ/eˇ .w � t /

DX

ˇ2P.G/n.ˇ/eˇ.w�X/

DX

ˇ2P.G/n.ˇ/e.w

�1�ˇ/.X/

DX

ˇ2P.G/n.w � ˇ/eˇ .t /:

Irreducible characters being linearly independent, we get n.w � ˇ/ D n.ˇ/, for anyw 2 WG and any ˇ 2 P.G/. Hence considering the set P.G/=WG of orbits of thegroupWG in P.G/, one can write

�jT DX

…2P.G/=WG

n.…/S.…/;

where S.…/ DPˇ2… eˇ and n.…/ D n.ˇ/, ˇ being some element in….

Description in terms of dominant weights 321

We may thus conclude the following consequence of the above considerations:

Proposition 12.30. Let ZR.T /WG

be the subring of ZR.T / formed by the sums

X

ˇ2P.G/n.ˇ/eˇ ;

where n.ˇ/ 2 Z is such that n.ˇ/ D 0, except for a �nite number of values of ˇ,and n.w � ˇ/ D n.ˇ/, for any w 2 WG .

Restriction to T induces an injective homomorphism from the representationring Z

R.G/ of G to ZR.T /WG

.

The next step in the theory is then to characterize which elements in ZR.T /WG

arerestrictions to T of irreducible characters ofG. Here come into play two fundamen-tal objects in the theory: the adjoint representation ofG, and the representation ringof SU2.

12.3 Characterization of irreducible representations by meansof dominant weights

12.3.1 Restriction to semi-simple simply connected groups

From now on, the Lie group G will be assumed to be semi-simple and simply con-nected. This assumption can be made thanks to the following result on the structureof compact and connected groups.

Definition 12.31. A Lie group is said to be semi-simple if the Killing form of itsLie algebra is non-degenerate.

Theorem 12.32. Any compact connected Lie groupG has a �nite covering isomor-phic to a product T n �K, where T n is a torus and K a compact semi-simple andsimply connected Lie group.

For a proof see for instance [Die75], (21.6.9), p. 40. It is proved that the universalcovering zG of G is isomorphic to a product of the form R

n � K, where K is acompact semi-simple and simply connected Lie group. Now, coverings of G are ofthe form zG=D, whereD is a discrete subgroup of the center R

n �Z of zG, Z beingthe (�nite) center ofK. Thus consideringD D Z

n � feg, one gets a �nite coveringof G isomorphic to T

n �K.

Remark 12.33. So we only consider compact semi-simple simply connected Liegroups. Indeed, with the notations of the above theorem, if the irreducible repre-sentations of K are known, then irreducible representations of the product T

n �Kare known by Theorems 12.9 and 12.24. Then, the irreducible representations ofGcan be deduced from Proposition 12.18.


12.3.2 Roots and their properties

Let AdWG ! GL.g/ be the adjoint representation of G, and AdCWG ! GLC.gC/

its extension to the complexi�ed Lie algebra gC of g, gC D g ˚ ig. Denote byT a maximal torus of G. According to the results of the previous section, fromthe decomposition into irreducible parts of the representation AdC jT it follows thatthere exists a family of distinct weights �1; : : : ; �p in P.G/, such that

gC D g0 ˚� pM

iD1g�i

�; (12.1)

where

g0 D fY 2 gCI for all X 2 t; ŒX; Y � D 0g;and

g�iD fY 2 gCI for all X 2 t; ŒX; Y � D �i .X/Y g:

Of course, tC � g0, but in fact one has

Lemma 12.34. g0 D tC and this property characterizes the maximal tori of G.

Proof. Let Z D X C iY , X; Y 2 g, be an element of g0. Since ŒU; Z� D 0 for anyU 2 t, one deduces that, for any U 2 t, ŒU; X� D 0 and ŒU; Y � D 0, hence X andY belong to the centralizer of t, which is equal to t, since T is a maximal torus. �

Definition 12.35. The weights �i , 1 � i � p, are called roots of G.

Lemma 12.36. The Weyl group of G permutes the roots.

Proof. Let w 2 WG and let g 2 N.T / be a representative of w. For any t 2 T ,one has

AdC.w � t / D AdC.gtg�1/ D AdC.g/AdC.t /AdC.g

�1/:

Therefore, the two representations t ! AdC.t / and t ! AdC.w � t / are equivalent,hence have the same weights. Furthermore, considering the extension of the actionofWG to gC, for any X 2 t and Xi 2 g�i

, one has

ŒX;w �Xi � D ŒX;AdC.g/ �Xi �

D AdC.g/.ŒAdC.g�1/ �X;Xi �/

D AdC.g/.�i .w�1 �X/Xi /

D ..w � �i /.X//w �Xi ;

and sow � g�i

D gw��i: �


Lemma 12.37. If �i is a root of G, so is .��i /, and one has g�iD g��i

.

Proof. For any root �i , any Xi 2 g�i, and any X 2 t, one has

ŒX; Xi � D �i .X/Xi ;

henceŒX; Xi � D ŒX; Xi � D ��i .X/Xi ;

since �i 2 2i�t�. So for any root �i , .��i / is a root and g�i� g��i

, as claimed. �

Now let �j be a root ofG and let Xj be a non-trivial vector of g�j . Consider thetwo vectors

Yj D Xj CXj ; Zj D i.Xj � Xj /;

of g. As Xj ranges over a basis (over C) of g�j , they form a basis (over R) ofg \

�g�j ˚ g��j

�.

Let 'j 2 t�, de�ned by �j D 2i�'j . For any X 2 t,

ŒX; Yj � D 2�'j .X/Zj ; ŒX; Zj � D �2�'j .X/Yj : (12.2)

Lemma 12.38. For any root �j , letHj ´ ŒYj ; Zj �. ThenHj 2 t and 'j .Hj / > 0.Furthermore, fHj ; Yj ; Zj g is a basis of a subalgebra of g, isomorphic to the Lie

algebra su2.

Proof. From relations (12.2), one has for any X 2 t that

ŒX;Hj � D ŒX; ŒYj ; Zj �� D ŒYj ; ŒX; Zj �� ŒZj ; ŒX; Yj �� D 0;

henceHj belongs to the centralizer of t, which is equal to t. Let h�; �i be anAd.G/-in-variant scalar product on g (for instance, the Killing form of g sign-changed); thenfor any X 2 t,

hX;Hj i D hX; ŒYj ; Zj �i D hŒX; Yj �; Zj i D 2�'j .X/k Zj k2:

First note that this equality implies, by choosing X 2 t such that 'j .X/ 6D 0, thatHj is non-zero. Now applying this equality to X D Hj gives 'j .Hj / > 0. Finally,from relations (12.2) it follows

ŒHj ; Yj � D 2�'j .Hj /Zj and ŒHj ; Zj � D �2�'j .Hj /Yj ;

so letting

Hj D 2

ajHj ; Yj D 2

pajYj ; Zj D 2

pajZj ;

withaj D 2�'j .Hj / > 0;


one gets

ŒHj ;Yj � D 2Zj ; ŒHj ;Zj � D �2Yj ; ŒYj ;Zj � D 2Hj :

These are the same multiplication rules as for the Lie algebra su2. �

We now consider the scalar product on g given by the Killing form of g sign-changed. (In fact, one could use any Ad.G/-invariant scalar product on g). Thisscalar product being Ad.G/-invariant, its restriction h�; �i to t is invariant under theaction of the Weyl groupWG . We consider the canonical isomorphism ] between tand its dual t� given by

]WX 2 t �! X] 2 t�; X]WY 7�! X].Y / D hX; Y i:

We endow t� with the scalar product h; i which makes ] an isometry:

hX]; Y ]i ´ hX; Y i; X; Y 2 t:

This scalar product is invariant under the action of WG . In fact, for any w 2 WG

and X; Y 2 t, one has w � X] D .w �X/], so

hw �X];w � Y ]i D h.w �X/]; .w � Y /]i D hw �X;w � Y i D hX; Y i D hX]; Y ]i:

Finally, we consider the natural extension of this scalar product on i t�, given by

hiˇ; iˇ0i ´ hˇ; ˇ0i; ˇ; ˇ0 2 t�:

Using Lemma 12.38 and the explicit description of irreducible representations ofSU2, one gets the following “fundamental” properties of roots.

Theorem 12.39. (i) Roots span i t�.

(ii) For any pair of roots .�i ; �j /, �j � 2h�i ;�j ih�i ;�i i �i is a root. .In other words,

for any root �i , the set ˆ of roots is invariant under re�ections wrt the hyperplaneperpendicular to �i /.

(iii) For any pair of roots .�i ; �j /, one has 2h�i ;�j ih�i ;�i i 2 Z.

(iv) If �i is a root, then ˙�i are the only scalar multiples of �i which are alsoroots.

As it was mentioned before, the proof of this result is based on the descriptionof irreducible representations of SU2, so we �rst consider this group.


12.3.3 Irreducible representations of the group SU2

Since SU2 is simply connected, the map � 7! �� is a one-to-one correspondencebetween �nite-dimensional complex representations of SU2 and �nite-dimensionalcomplex representations of its Lie algebra su2.

The map �� 7! �C� (extension to the complexi�ed Lie algebra of su2) is a one-

to-one correspondence between �nite-dimensional complex representations of su2and �nite-dimensional complex representations of sl2;C, which is the complexi�edLie algebra of su2. (The inverse map is simply �C

� 7! �C� jsu2

.)Hence the map � 7! �C

� gives a one-to-one correspondence between irreduciblerepresentations of SU2 and irreducible �nite-dimensional complex representationsof sl2;C.

Notations 12.40. For any representation .�; V / of SU2, and for any X 2 sl2;C andv 2 V , we will simply write X � v instead of �C

� .X/ � v.

Let

H D�1 0

0 �1

�; XC D

�0 1

0 0

�; X� D

�0 0

1 0

�

be the standard basis of sl2;C. One has

ŒH;XC� D 2XC; ŒH;X�� D �2X�; ŒXC; X�� D H: (12.3)

Theorem 12.41. For any integer m � 0, there exists an irreducible representationofSU2 in a complex vector space of dimensionmC1, having a basis fe0; e1; : : : ; emgsuch that

H �ek D .m�2/ek ; XC�ek D .m�kC1/ek�1; X��ek D .kC1/ekC1; (12.4)

setting e�1 D emC1 D 0.Every (complex) irreducible representation of SU2 is equivalent to one of the

indicated representations.

Proof. First if .�; V / and .�0; V 0/ are two representations of SU2 such that bothV and V 0 have a basis fekg, resp. fe0

kg, k D 0; : : : ; m, verifying (12.4), then they

are equivalent via the C-isomorphism V ! V 0 that sends the basis fekg into thebasis fe0

kg.

Now let �W SU2 ! GL.V / be a (�nite-dimensional complex) irreducible repre-sentation. Consider the Lie subgroup of SU2 given by

T D²�

ei� 0

0 e�i�

�I � 2 R

³I

its Lie algebra is R.iH/.


By (12.3), T is a maximal torus of SU2 (see (12.1) and Lemma 12.34).The decomposition of �jT into irreducible parts induces the decomposition of Vinto weight spaces:

V DM

�2ƒV�;

where ƒ is a �nite subset of C and

V� D fv 2 V IH � v D �vg:

Take v� 2 V�. Then by (12.3), one has

XC � v� 2 V�C2 and X� � v� 2 V��2:

Since V is �nite dimensional and since the sum V DL�2ƒ V� is direct, there

exists a �0 2 ƒ such that V�0¤ f0g and V�0C2 D f0g.

Consider a non-zero vector v0 2 V�0. It satis�es XC � v0 D 0. Such a weight

vector is said to be a “maximal vector” of the representation. Set

e�1 ´ 0; ej ´� 1j Š

�.X�/j � v0; j 2 N:

It is easy to check that

(a) H � ej D .�0 � 2j /ei ;

(b) X� � ej D .j C 1/ejC1;

(c) XC � ej D .�0 � j C 1/ej�1.

From (a), one deduces that the non-zero ej are linearly independent. But V is�nite dimensional. Hence, there exists a smallest integer m for which em 6D 0, andemCj D 0 for all j > 0. It is easy to see from relations (a)–(c) that the subspaceof V with basis fe0; e1; : : : ; emg is a sl2;C-submodule of V . But V is an irreduciblesl2;C-module, hence this subspace has to be V . Finally, taking j D m C 1 in (c)gives �0 D m. Note that weights are integers, that m is the “highest” one and that,up to a scalar, V admits a unique maximal vector, namely e0.

Conversely, for any m 2 N, relations (12.4), as it was noted before, de�ne arepresentation of sl2;C on a complex space Lm of dimension m C 1, with basisfe0; : : : ; emg. Let U be a non-trivial invariant sl2;C-submodule of Lm and denoteby U D

L�2ƒ U� its decomposition into weight spaces under the action ofH . By

the �rst formula in (12.4), one has necessarily ƒ � f�m;�mC 2; : : : ; m� 2;mg,hence one of the ej belongs to U . But the two last formulas in (12.4) imply that allthe vectors of the basis fej g belong toU . HenceU D Lm, soLm is irreducible. �


Remarks 12.42. (i) Let .V; �/ be a representation of SU2. Decomposing � intoirreducible components, one deduces from (12.4) that the weights of � are integers(and an integer occurs as a weight as many times as its negative).

Furthermore, since weights of an irreducible representation are all simultane-ously even integers or odd integers, the number of summands in the decompositionof � is dimV0 C dimV1, where V0 (resp. V1) is the weight space corresponding tothe weight 0 (resp. 1).

(ii) Let �W SU2 ! GLC.V /, V D C2, be the usual (faithful) representation of

SU2. Recall Theorem 12.15: irreducible representations of SU2 are contained intensor products of � and its conjugate.

For any m 2 N, let Sm.V / be the space of symmetric tensors of order m (notethat it has dimension .m C 1/). This is (up to equivalence) the irreducible repre-sentation with highest weight m since the basis fe0; e1; : : : ; emg of Sm.V / givenby

ek D 1

kŠ.m � k/Š�1m�k _ �2k;

where f�1; �2g is the standard basis of C2, satis�es (12.4).

12.3.4 Proof of the fundamental properties of roots

Proof of Theorem 12.39.

Step 1. First, we show that roots (extended to t�C) span t�

C. Assuming this is not

the case, there existsX 2 tC such that �j .X/ D 0 for any root �j . Then ŒX; Y � D 0

for any Y 2 gC, hence X belongs to the center of gC. But G is supposed to besemi-simple, hence g and then gC are semi-simple, so the center of gC is trivial:contradiction.

Step 2. Now recall that for any root �j , the vectors Hj , Yj , Zj introduced inLemma 12.38 de�ne a subalgebra of g isomorphic to su2. Let hj , x

Cj , x�

j be thefollowing vectors of gC:

hj D �iHj ; xCj D �1

2.Zj C iYj /; x�

j D 1

2.Zj � iYj /:

It is easy to verify that

hj 2 i t; xCj 2 g�j ; x�

j 2 g��j ;

�j .hj / D 2;

Œhj ; xCj � D 2xC

j ; Œhj ; x�j � D �2x�

j ; ŒxCj ; x

�j � D hj ;


sosj ´ hhj ; xC

j ; x�j i;

is a Lie subalgebra of gC isomorphic to sl2;C. Now, for any pair of roots .�j ; �k/,one has

Œhj ; Y � D �k.hj /Y; Y 2 g�k;

so �k.hj / is a weight of the representation adC restricted to sj ' sl2;C. Hence, byRemark 12.42 i), �k.hj / 2 Z.

Step 3. Using the Jacobi identity it is easy to verify that

Œg�j ; g�k� � g�j C�k

if �j C �k is a root, (12.5)

and Œg�j ; g�k� D f0g otherwise. For any root �j , let

Mj ´ tC ˚�M

c2Cgc�j

�;

whereC is the set of scalars c such that c�j is a root. By (12.5),Mj is a sj -invariantsubspace of gC (for the representation adC jsj

). By Step 2, its weights under theaction of hj are 0 and c�j .hj / D 2c. So by Remark 12.42 i), c has to be an integralmultiple of 1

2.

Now note that the hyperplane Ker �j (which complements hhj i in tC) and thesubspace sj D hhj ; xC

j ; x�j i are sj -invariant subspaces ofMj , and they exhaust the

occurrences of the weight 0 for the action of hj onMj . Observe that sj acts triviallyon Ker �j and that sj is an irreducible sj -module (by formulas (12.4)).

By Remark 12.42, one sees that the only irreducible sj -modules with even high-est weight contained in Mj are (with the notations of the proof of Theorem 12.41)L0 (with multiplicity dim tC � 1) and L2 (with multiplicity 1), so the only evenweights of hj onMj are 0 and ˙2. Hence one deduces that

for any root �j , 2�j is not a root.

Consequently,

for any root �j ,1

2�j cannot be a root.

From this last result, one sees that 1 cannot occur as a weight of hj on Mj . FromRemark 12.42, one then concludes that

Mj D Ker �j ˚ sj ;

which establishes the property (iv) of roots. Moreover, this implies:

for any root �j , dim g�j D 1:


Step 4. For any root �j , let �j be the map

�j W t�C �! t�C; � 7�! � � �.hj /�j :

Since �j .hj / D 2, �j is an involution, distinct from the identity (since �j .�j / D��j ). We claim that it leaves invariant the set of roots ˆ. To see this, consider, forany i ¤ j , the space

Nj i ´M

k2Kg�j Ck�i

;

where K is the set of integers k such that �j C k�i is a root. By (12.5), Nj i is ansi -invariant subspace of gC (for the representation adC jsi

). Its weights under theaction of hi are �j .hi /C2k. Moreover, by Step 3, the corresponding weight spacesg�j Ck�i

are one-dimensional, and by Step 1, �j .hi / 2 Z.Now since the weights �j .hi /C 2k are simultaneously all even or odd integers

and since 0 or 1 (but not both) can occur only once, as weights of this form, onededuces from Remark 12.42 thatNj i is irreducible. Its highest (resp. lowest) weightmust be �j .hi/ C 2p , (resp. �j .hi / � 2q), where p (resp. q) is the largest integerk 2 K for which �j C k�i (resp �j � k�i ) is a root.

Moreover, since the other weights have the form �j .hi / C 2k, �q � k � p

(cf. (12.4)), �j Ck�i is a root for any k such that�q � k � p. Now, relations (12.4)imply

�j .hi / � 2q D �.�j .hi /C 2p/;

hence, �j .hi/ D q�p. So, �i .�j / D �jC.p�q/�i is a root, since�q � p�q � p.Let F be the R-vector subspace of i t� spanned by the roots �i , endowed with

the induced scalar product h�; �i. Since �j .hi / 2 Z, any �i leaves F invariant. Nowconsider ˇ 2 F such that hˇ; �i i D 0. By the de�nition of the scalar product,

hˇ; �i i D �i�i .Xˇ /;

where Xˇ is the vector in t corresponding to ˇ via the canonical isomorphism

t �! i t�; X 7�! iX]:

So Xˇ 2 Ker �i . Using the Ad.G/-invariance of the scalar product, one gets

hXˇ ;Hii D 1

2hXˇ ; ŒYi ;Zi �i

D 1

2hŒXˇ ;Yi �;Zi i

D � i2�i .Xˇ /kZik2

D 0:


Consequently,

ˇ.hi/ D �iˇ.Hi / D hXˇ ;Hii D 0:

Hence, �i .ˇ/ D ˇ, for any ˇ 2 �?i . Thus �i jF is the re�ection across the hyper-

plane �?i . This implies that

�j .hi / D 2h�i ; �j ih�i ; �ii

;

which establishes properties (ii) and (iii).

Step 5. Since the �j span t�C(cf. Step 1), there exists a basis of t�

Cof the form

f�1; �2; : : : ; �lg, so any ˇ 2 i t� can be written uniquely as ˇ DPljD1 cj �j , where

cj 2 C. Besides, for any k D 1; : : : ; l , one has ˇ.hk/ DPljD1 cj �j .hk/.

This is a system of l equations with l unknowns cj and with real coef�cients(ˇ.hk/ 2 R, �j .hk/ 2 Z). The matrix corresponding to this system is non-degener-ate (from the result before, it is, up to a scalar, the matrix of the C-linear extensionof the scalar product h�; �i of i t� to t�

C, in the basis .�j /), so by Cramer’s formulas

cj 2 R. Thus F D i t�, hence (i). �

Definition 12.43. Let .E; h�; �i/ be an Euclidean space. Any �nite subset ˆ of Ewhose elements �i verify properties (i)–(iv) of Theorem 12.39 is called a root systemof E.

Definition 12.44. Letˆ be a root system of an Euclidean space .E; h�; �i/. A subset� of ˆ is called a basis of ˆ if it satis�es the two following properties:

(i) � is a basis of E;

(ii) every � 2 ˆ can be written as � DPni�i , where �i 2 � and ni are integers,

all of the same sign.

Roots belonging to a basis are called simple roots.

Theorem 12.45. Every root system has a basis.

Definitions 12.46. A vector in E is said to be regular if 2 E n�S

�2ˆ�?�,

singular otherwise. Any connected component of E n�S

�2ˆ�?� is called a Weyl

chamber.


Idea of the proof of Theorem 12.45. Let 2 E be regular and let

ˆC D f� 2 Î h�; i > 0g and ˆ�

D f� 2 Î h�; i < 0g:

One hasˆ D ˆC [ˆ�

. An element � inˆC is said to be decomposable if it can be

written as � D �1C�2with �1; �2 2 ˆC . Otherwise, � is said to be indecomposable.

It is then shown (cf. for instance [Hum72], p. 48), that the set�. / of indecom-posable elements in ˆC

is a basis and that any basis of ˆ has this form. For thisreason, roots belonging to a basis are called simple roots. �

Definition 12.47. Let � D f�1; �2; : : : ; �lg be a basis of a root system ˆ in anEuclidean space .E; h�; �i/. Any root � 2 ˆ such that � D

PliD1 ni�i , where

ni 2 N, is called a positive root. The set of positive roots is denoted by ˆC.

Nowwe get back to “our” root systemˆ de�ned by the choice of amaximal torusT in G. By the previous results, it has a basis � D f�1; �2; : : : ; �lg, l D dim.i t�/.

Remark 12.48. From the proof of Theorem 12.39 (Step 2), for any simple root�i there exists hi 2 i t, xC

i 2 g�iand x�

i 2 g��isuch that spanfhi ; xC

i ; x�i g is a

subalgebra of gC isomorphic to sl2;C. In fact, those vectors h1; : : : ; hl , form a basisof tC.

Indeed, let c1; : : : ; cl 2 C such thatPliD1 cihi D 0. Applying each simple root

�i to this equation, one obtains a system of l equations in l unknowns ci whose ma-

trix��i .hj / D 2

h�i ;�j ih�j ;�j i

��is non-singular, cf. the proof of Theorem 12.39 (Step 5).

Hence all the ci are 0. Thus, the hi , 1 � i � l , are linearly independent andthey form a basis, since l D Card� D dim.i t�/ D dimC.tC/.

The existence of simple roots allows to give a simple characterization of theWeyl group.

Theorem 12.49. The Weyl group WG is spanned by re�ections �i across thehyperplanes �?

i , where �i 2 �.

Proof. We �rst prove that any such re�ection is an element of WG . Let �i be asimple root and consider spanfHi ;Yi ;Zig, the corresponding subalgebra of g iso-morphic to su2, introduced in Lemma 12.38. Let gi D exp.�

2Yi /. We claim that gi

de�nes an element wi 2 WG whose action on roots is the re�ection �i across thehyperplane �?

i . One has Ad.gi / D exp.�2ad.Yi //. Now

ad.Yi / �X D �ŒX;Yi � D �i�i .X/Zi D 0; X 2 Ker �i ;

soAd.gi / �X D X; X 2 Ker �i :


On the other hand, it is easy to verify that, for all k 2 N,

.adYi /2k � Hi D .�1/k22kHi ;

and

.adYi /2kC1 � Hi D .�1/kC122kC1Hi :

Therefore,

Ad.gi / � Hi D exp��2ad.Yi/

�� Hi

DX

k�0

.�=2/k

kŠ.adYi /

k � Hi

D .cos�/Hi � .sin�/Zi

D �Hi :

Thus, since Ad.gi / leaves invariant the space t D Ker �i ˚ hHi i, one concludesthat gi 2 N.T /.

Furthermore, since Hi is orthogonal to Ker �i (cf. Step 4 in the proof of Theo-rem 12.39), the action on t of the element wi 2 WG de�ned by gi is the re�ectionacross H ?

i . Now, by the results of Step 4 in the proof of Theorem 12.39, Ker �icorresponds to �?

i under the canonical isomorphism t ! i t�, X 7! iX]. Hencethe action of wi on i t� is the re�ection across �?

i . So the group W spanned by the�i corresponding to simple roots satis�es W � WG .

Conversely, letw 2 WG . Let be a regular element in i t� such that� is the set�. / of indecomposable elements in ˆC

, cf. the proof of Theorem 12.45. Since wpermutes the roots, cf. Lemma 12.36, it is easy to see that �0 ´ w �� is the basisof ˆ de�ned by the set �.w � / of indecomposable elements in ˆC

w� .We �rst prove that there exists � 2 W such that � ��0 D �. To see this, observe

that for any �i corresponding to a simple root, one has

�i .�/ 2 ˆC; � 2 ˆC; � ¤ �i :

Indeed, by property (ii) of Theorem 12.39, �i .�/ D � � 2h�;�i ih�i ;�i i�i is a root, and

furthermore a positive root, since it has to belong to either ˆC or ˆ�.From this last result, one deduces easily that the half-sum of the positive roots,

ı ´ 1

2

X

�2ˆC

�;

is such that for any simple root �i , �i .ı/ D ı � �i . Now, since the groupW � WG

is �nite, there exists � 2 W such that h�.w � /; ıi is as big as possible.


By the de�nition of � , one gets for any simple root �i ,

h�.w � /; ıi � h�i�.w � /; ıiD h�.w � /; �i.ı/iD h�.w � /; ıi � h�.w � /; �i i:

Henceh�.w � /; �ii � 0; �i 2 �:

But h�.w � /; �i i D hw � ; ��1.�i /i ¤ 0, since w � is regular. Therefore,

h�.w � /; �ii > 0; �i 2 �:

This implies ˆC�.w� / D ˆC

.µ ˆC/, hence �.�.w � / D �. /.µ �/. So� ��0 D � as required.

We now prove that �w is the neutral element e ofW. Letw0 ´ �w, and denoteby g0 a representative of w0 and by X the vector in t corresponding to under thecanonical isomorphism t ! i t�, X 7! iX]. For any simple root �j , one has

�i�j .X / D h�j ; i > 0;

and since w0 �� D �, this implies

�i�j .w0 �X / D hw0�1 � �j ; i > 0:

Consider X DPm�1kD0 w

0k � X , with m the order of w0 in WG . By de�nition,w0 �X D X . This implies that g0 belongs to the group

ZG.X/ ´ fg 2 GIAd.g/ �X D Xg:

It can be shown that this compact subgroup of G, which contains T , is actuallyconnected, hence its Lie algebra is

B.X/ ´ fY 2 gI ŒY; X� D 0g:

Now let Y 2 B.X/. Thanks to decomposition (12.1) and Lemma 12.34, Y canbe uniquely written as

Y D Y 0 CX

�2ˆY� ; Y 0 2 tC; Y� 2 g� :

ThenŒX; Y � D 0 D

X

�2ˆ�.X/Y� :


But we noted before that �i�j .X/ > 0, for any simple root �j . Now, since any rootbelongs either to ˆC or ˆ�, this implies that �.X/ ¤ 0, for any root � . Hence allthe Y� are 0, and then Y D Y 0 2 g \ tC D t.

This proves the inclusion B.X/ � t, which implies B.X/ D t. We then getZG.X/ D T , since these two connected subgroups of G have same Lie algebras.

Finally g0 2 T , so w0 D �w D e and w D ��1 2 W . �

The following result is a �rst step in the characterization of weights of G bymeans of simple roots.

Theorem 12.50. Letˇ be a weight .in the “integral” sense/ ofG, that isˇ 2 P.G/.Then

2hˇ; �i ih�i ; �ii

2 Z; �i 2 �:

Proof. Any ˇ 2 P.G/ can be written as ˇ DP�i 2� ci�i . Therefore,

2hˇ; �j ih�j ; �j i D

X

i

ci2h�i ; �j ih�j ; �j i :

Let �j be a simple root, and sj D spanfhj ; xCj ; x

�j g the Lie subalgebra of gC is

isomorphic to sl2;C introduced in the proof of Theorem 12.39. From the proof ofthis same theorem, we have

ˇ.hj / DX

i

ci�i .hj / DX

i

ci2h�i ; �j ih�j ; �j i :

Hence

2hˇ; �j ih�j ; �j i D ˇ.hj /:

But ˇ is contained in the character of an irreducible representation � of G(cf. Remark 12.27), hence ˇ.hj / is a weight of the representation �C

� restrictedto sj . By Remark 12.42 i), one has ˇ.hj / 2 Z. �

12.3.5 Dominant weights

Proceeding further in this direction, we are going to characterize irreducible rep-resentations of G in terms of certain weights, called dominant weights. Since Gis supposed to be simply connected, irreducible representations of G are in one-to-one correspondence with complex �nite-dimensional representations of its Liealgebra g.


This allows to derive the study of irreducible representations of G from thatof irreducible complex �nite-dimensional representations of its Lie algebra g. Weadopt this “algebraic” point of view in the following, using a method which may beseen as a generalization of the method we used in the study of the representationsof SU2.

We begin by introducing a (partial) order � in i t�.

Definition 12.51. For ˇ; ˇ0 2 t�, we will say that

ˇ0 � ˇ if ˇ � ˇ0 DX

�i 2�ci�i ; where the ci ’s belong to R

C.

Since G is simply connected, the map � 7! �C� is a one-to-one correspondence

(as it was remarked in the case G D SU2), between complex �nite-dimensionalrepresentations of G and complex �nite-dimensional representations of the com-plexi�ed Lie algebra gC. Indeed, considering the enveloping algebra U.gC/ ofgC, the map � 7! U

��C

��is a one-to-one correspondence between complex �nite-

dimensional representations of G and complex �nite-dimensional representationsof U.gC/.

Moreover, this one-to-one correspondence maps irreducible representations ofG into irreducible �nite-dimensional complex representation of U.gC/. Recall thatif fXi ; i D 1; : : : ; N g, is a basis of gC, then U.gC/ can be seen as the algebraspanned by (formal) monomials of the form

Xk1

i1Xk2

i2� � �Xkp

ip; 1 � i1 < i2 < � � � < ip � N; ki 2 N;

with the (associative) multiplication induced by the relations

XiXj �XjXi D ŒXi ; Xj �:

If �WG ! GLC.V / is a representation, a monomial of the form Xk1

i1Xk2

i2� � �Xkp

ipacts on V by the endomorphism

.�C� .Xi1//

k1 ı � � � ı .�C� .Xip//

kp :

Indeed, it seems “natural” to consider such an endomorphism in the study of therepresentation, however it is not in general an element of �C

� .gC/, a reason to intro-duce this trick of enveloping algebra. Moreover, note that there is a �ne descriptionof irreducible G-representation spaces in terms of cyclic modules on U.gC/, seeTheorem 12.58 below.

Notations 12.52. For any representation .�; V / ofG, X 2 gC and v 2 V , we writeX � v instead of �C

� .X/ � v.


Lemma 12.53. If .�; V / is an irreducible representation of G and ˇ a weight of �,we denote by

Vˇ D fv 2 V I for all X 2 t; ��.X/ � v D ˇ.X/vg;

the corresponding weight space. For any root �i , one has

1. either ˇ C �i is not a weight and then g�i� Vˇ D f0g or

2. ˇ C �i is a weight and then g�i� Vˇ � VˇC�i

.

Proof. Let X 2 t, Xi 2 g�i. For any v 2 V ,

X � .Xi � v/ D Xi � .X � v/C ŒX; Xi � � v D .ˇ.X/C �i .X//.Xi � v/:

Hence the result. �

Let � be an irreducible representation of G. Since it has only a �nite number ofweights, there exists a weight � such that � C �i is not a weight for any positiveroot �i . From the above lemma it follows that

Xi � v D 0; �i 2 ˆC; Xi 2 g�i; v 2 V�:

Definition 12.54. A weight vector of a representation �WG ! GLC.V / is said tobe a maximal vector, if it is annihilated by all the g�i

, where �i 2 ˆC.

Remark 12.55. The above argument shows that every representation (irreducibleor not) has a maximal vector. Thus if a representation has, up to a scalar multiple,only one maximal vector, it is irreducible.

Remark 12.56. By (12.5) and the fact that any positive root can be written as a sumof simple roots �1 C � � � C �k in such a way that each partial sum �1 C � � � C �iis a root (cf. for instance Lemma 10.2A in [Hum72]), one concludes that for anypositive root � , any element in g� belongs to the Lie subalgebra of gC generated bythe g�i

, i D 1; : : : ; l , corresponding to simple roots.Hence, in order for a vector of a G-space to be maximal it suf�ces that it is

annihilated by all the g�icorresponding to simple roots.

Theorem12.57. Denote by �WG ! GLC.V / a representation ofG .not necessarilyirreducible/, and by vC a maximal vector of � with weight �. Let V C be the cyclicmodule U.gC/ � vC. Then

(i) every weight � of V C has the form � D � �P�i 2� ni�i , where ni 2 N,

hence � � �;

(ii) dimV� D 1, that is, the weight � has multiplicity 1 in V C;

(iii) V C is an irreducible G-space.


Proof. Let �1; : : : ; �n be the positive roots of G. From the proof of Theorem 12.39(Step 2) it follows that for any positive root �i there exist hi 2 i t, x�i

2 g�iand

x��i2 g��i

such that spanfhi ; x�i; x��i

g is a subalgebra of gC isomorphic to sl2;C.From decomposition (12.1), Lemma 12.34, and Remark 12.48, using moreover

the fact that dim g� D 1, for any root � , (cf. the proof of Theorem 12.39, Step 3),one deduces that the hi , i D 1; : : : ; l (corresponding to simple roots) and the x�i

,x��i

, i D 1; : : : ; n, form a basis of gC. Hence the enveloping algebra U.gC/ has abasis of the form

xa1

��1� � �xan

��nhc1

1 � � �hcl

lxb1

�1� � �xbn

�n; (12.6)

where the ai , bi , and ci belong to N.Now, since vC is a maximal vector, the action of a basis vector of U.gC/ of the

form (12.6) on vC gives 0, if one of the integer bi is nonzero. Furthermore, sincevC is a weight vector, the action of a basis vector of U.gC/ of the form h

c1

1 � � �hcl

l

on vC gives a scalar multiple of vC. Hence V C is spanned by vectors of the form

xa1

��1� � �xan

��n� vC:

Now by the proof of Lemma 12.53, if such a vector is nonzero, then it is a weightvector for the weight � D ��

PniD1 ai�i . Expressing each positive root �i in terms

of simple roots, one gets (i).Moreover, note that up to a scalar, the only vector of the above form of weight �

is vC, hence (ii). Now suppose V C is not irreducible; then it decomposes into thesum of two G-invariant subspaces: V C D V C

1 ˚ V C2 . According to this decompo-

sition, one gets vC D vC1 C vC

2 . If both vC1 and vC

2 are nonzero, then they are twolinearly independent (maximal) vectors for the weight �, contradicting (ii). Henceone of vC

1 and vC2 has to be zero. If, for instance, vC

2 D 0, then vC D vC1 , so

V C D U.gC/ � vC1 � V C

1 ;

hence V C1 D V C and V C

2 D f0g. Thus, V C is irreducible. �

Theorem 12.58. If �WG ! GLC.V / is an irreducible representation, then V has,up to a scalar multiple, only one maximal vector vC, and V D V C ´ U.gC/ � vC.

Proof. Let vC be a maximal weight vector of V . Since V is irreducible, V C D V .Uniqueness of vC results from Theorem 12.57 (ii). �

Definition 12.59. Let �WG ! GLC.V / be an irreducible representation of G. LetvC be the maximal vector of � and � the corresponding weight. This weight is saidto be the dominant weight of the representation �.

This dominant weight � has the following property.


Theorem 12.60. Let � be the dominant weight of an irreducible representationof G. For any simple root �i ,

2h�; �i ih�i ; �ii

2 N:

Proof. Let �i be a simple root and spanfhi ; x�i; x��i

g the corresponding Lie subal-gebra of gC (it was introduced in the proof of Theorem 12.57). If vC is the maximalvector of �, then by de�nition, x�i

� vC D 0, since x�i2 g�i

.

Moreover, 2 h�;�i ih�i ;�i i D �.hi/, see the proof of Theorem 12.50. Therefore,

hi � vC D �.hi /vC and x�i

� vC D 0;

so by Theorem 12.41 and its proof, �.hi/ must belong to N. �

Theorem 12.61. Let � 2 i t�. If for any simple root �j , the real number 2h�;�j ih�j ;�j i

is a non negative integer, then � is the dominant weight of an irreducible represen-tation of G. Hence the map which assigns to each equivalence class of irreduciblerepresentations of G its dominant weight is a one-to-one correspondence onto theset

ƒC D²� 2 i t�W for all �j 2 �; 2 h�; �j i

h�j ; �j i 2 N

³:

Elements of ƒC are called dominant weights.

In particular, the vectors ˇk 2 i t�, k D 1; : : : ; l , de�ned by 2 hˇk ;�j ih�j ;�j i D ıjk , for

any �j 2 �, are dominant weights of irreducible representations of G.

Definition 12.62. The irreducible representations �k , k D 1; : : : ; l , of G with cor-responding dominant weights ˇk de�ned by the condition

2hˇk ; �j ih�j ; �j i D ıjk ; �j 2 �;

are called fundamental representations ofG. Their corresponding dominant weightsj are called fundamental weights.

Note that, by de�nition, fundamental weights de�ne a basis fˇ1; : : : ; ˇlg of i t�.Thus any ˇ 2 i t� can be written as

ˇ DlX

jD1cj j ; cj 2 R:

But for each �j 2 �, we have 2 hˇ;�j ih�j ;�j i D cj , hence

ˇ DlX

jD12

hˇ; �j ih�j ; �j i j :


Applying this to a weight ˇ, we conclude from Theorem 12.50 that

Proposition 12.63. Fundamental weights de�ne a basis fˇ1; ˇ2; : : : ; ˇlg of theweight lattice P.G/. Moreover, a vector ˇ D

PnjD1mj j 2 i t� is a dominant

weight if and only if the mj ’s are non-negative integers.

For the proof of Theorem12.61 see [Hum72], Sections 20.3 and 21.2, or [Die75],pp. 116–120. Indeed, instead of this rather abstract proof, we prefer to show howto construct an irreducible representation with dominant weight ˇ D

PljD1mj j ,

mj 2 N, assuming that the fundamental representations are explicitly known.

Remark 12.64. Suppose that fundamental representations �1; �2; : : : ; �l are explic-itly known.

Let ˇ1; ˇ2; : : : ; ˇl , be the corresponding fundamental weights, V1; V2; : : : ; Vlthe corresponding representation spaces, and v1; v2; : : : ; vl the corresponding max-imal vectors. Let ˇ 2 i t� given by

ˇ DlX

jD1mj j ;

where mj 2 N. In the space

W D Sm1V1 ˝ Sm2V2 ˝ � � � ˝ SmlVl ;

where for any j , Smj Vj denotes the space of symmetric tensors of ordermj overVj ,consider the vector

v D vm1

1 ˝ vm2

2 ˝ � � � ˝ vml

l:

By construction, this is a maximal vector of W corresponding to the weight ˇ.So by Theorem 12.57, the cyclic module V D U.gC/ � v is an irreducible G-spaceofW with dominant weight ˇ.

In the following we give the explicit description of the fundamental represen-tations of the groups SUn, Spinn and Spn. Hence this remark can be seen as a“proof” of Theorem 12.61 in those cases (quotation marks because the existence offundamental representations is indeed asserted by Theorem 12.61!).

12.3.6 The Weyl formulas

By now, we are able to answer the two questions pointed out in the introductionof this chapter. However, a more precise answer can be given to the question thatfollows Proposition 12.30: which elements in Z

R.T /WG

are restrictions to T of irre-ducible characters of G?


This answer (which is a more explicit formulation of the one-to-one correspon-dence established in Theorem 12.61) is given by the Weyl character formula. Thisformula is obtained using an “analytic” approach (which is Weyl’s originalapproach, cf. [Wey46]), dealing more with the groups themselves rather than theirLie algebras. In spite of its importance (it plays for instance a central role in thereferences [Ada69] and [BtD85]), we only give this formula since only one of itscorollaries, the Weyl dimension formula, cf. Theorem 12.67 below, will be usefulfor the problems we are dealing with.

Theorem 12.65 (Weyl character formula). Let ˇ be a dominant weight and �� thecharacter of the corresponding irreducible representation ofG. Then the restrictionto T of �� satis�es

�G��jT DX

w2WG

.detw/ew�.ˇCı/; (12.7)

where ı is the half-sum of the positive roots and�G is the function on T de�ned by

�G.t / DY

�2ˆC

�e

12 �.X/ � e� 1

2�.X/�

D eı.X/Y

�2ˆC

.1� e��.X//; (12.8)

X being any element in exp�1.t /.

Remark 12.66. First, note that since

ı D ˇ1 C � � � C ˇl ;

where the ˇi are the fundamental weights ofG, ı is an integral weight ofG (cf. thecomments following Theorem 12.61), hence the function �G is well de�ned on T .Now observe that �G.t / ¤ 0 if and only if

�.X/ … 2i�Z; X 2 exp�1.t /; � 2 ˆ:

An element t 2 T satisfying this condition is said to be regular. Otherwise it is saidto be singular.

Thus the value of ��jT at regular elements can be obtained by multiplying bothsides of equation (12.7) by ��1

G . Now if t is a singular element, then �G.t / D 0,so the right-hand side of equation (12.7) has also to be zero at the point t .Let us check this. Indeed, suppose that there exists X 2 exp�1.t / such that�i .X/ 2 2i�Z for some root �i and let �i 2 WG be de�ned by the re�ectionacross �?

i , cf. Theorem 12.49.


For any integral weight ˇ, one has

X

w2WG

.detw/ew�ˇ.X/ D 1

2

X

w2WG

.detw/ew�ˇ.X/ C 1

2

X

w2WG

.det �iw/e�iw�ˇ.X/;

where X is any element in exp�1.t /. But from Theorem 12.50 and its proof, since2

hw�ˇ;�i ih�i ;�i i �i .X/ 2 2i�Z, one gets e�iw�ˇ.X/ D ew�ˇ.X/. Hence, since det �iw D

� detw, the right-hand side of equation (12.7) vanishes at t .Thus the value of ��jT at singular elements cannot be obtained directly from

equation (12.7). In fact, the set Treg of regular elements is dense in T (since anygenerator of T has to be regular), so the value of ��jT at singular elements can beobtained from the value at regular elements by a continuous extension argument.

For a proof of Theorem 12.65 following Weyl’s original approach, see [Ada69]Chapter 6, [BtD85] Chapter VI, [Die75] Chapter 15, or for classical groups [GW98](Section 7.4). Indeed there is also a (rather abstract) algebraic proof of this result,cf. for instance [Hum72] (Section 24.3), [Žel73] (Paragraph 123), or for classicalgroups [GW98] (Section 7.3).

Note that, taking ˇ D 0 (which is the dominant weight of the trivial representa-tion) in formula (12.7) yields

�G DX

w2WG

.detw/ew�ı : (12.9)

From the Weyl character formula one obtains a useful formula that gives thedimension of the irreducible representation corresponding to a dominant weight interms of the dominant weight itself.

Theorem 12.67 (Weyl dimension formula). Let ˇ be the dominant weight of anirreducible representation � of G. The dimension n� of the representation is givenby

n� D

Y

�2ˆC

hˇ C ı; �iY

�2ˆC

hı; �i;

where ı is the half-sum of the positive roots.

Proof. By de�nition, n� D ��.e/, but, as it was noted in the above remark, ��.e/cannot be computed directly from (12.7) since e is singular. For this, we �rst com-pute �� at the point exp.sXı/, where Xı is the vector in t corresponding to ı underthe canonical isomorphism

t �! i t�; X 7�! iX]; (12.10)


and then let s tend to 0. Note that, by de�nition, for any integral weight ˇ,

�iˇ.Xı/ D hˇ; ıi:

Note moreover that since ı is the sum of the fundamental weights, one has

h�i ; ıi > 0; �i 2 �;

so�.Xı/ ¤ 0; � 2 ˆ;

hence exp.sXı/ is regular for s suf�ciently small s ¤ 0. Now, for any integralweight ˇ and for any w 2 WG , one gets from the WG-invariance of the scalarproduct that

.w � ˇ/.Xı/ D ihw � ˇ; ıi D ihˇ;w�1ıi D .w�1 � ı/.Xˇ /;

where Xˇ 2 t corresponds to ˇ under the canonical isomorphism 12.10. Hence

X

w2WG

.detw/e.w�ˇ/.sXı / DX

w2WG

.detw/e.w�1 �ı/.sXˇ/

DX

w2WG

.detw/e.w�ı/.sXˇ /:

But from (12.9), this is the value of �G at the point exp.sXˇ /. Using the �rstexpression of �G in (12.8), one gets

�G.exp.sXˇ // DY

�2ˆC

.e12 ihˇ;�is � e� 1

2 ihˇ;�is/:

As s tends to 0, this expression is equivalent toY

�2ˆC

ihˇ; �is:

Thus when s tends to 0, ��.exp.sXı// is equivalent to

Y

�2ˆC

ihˇ C ı; �is

Y

�2ˆC

ihı; �is;

which completes the proof. �

Irreducible representations of the classical groups 343

12.4 Application: irreducible representations of the classicalgroups SUn, Spinn, and Spn

12.4.1 Irreducible representations of the groups SUn and Un, n � 3

Irreducible representations of the group SUn. Let T0 be the torus of Un, i.e.,

T0 D fdiag. matrix.eiˇ1; : : : ; eiˇn/I ˇ1; : : : ; ˇn 2 Rg: (12.11)

Its Lie algebra is

t0 D fdiag. matrix .iˇ1; : : : ; iˇn/I ˇ1; : : : ; ˇn 2 Rg:

We denote by fy1; : : : ; yng the basis of t0� given by

yk Œdiag. matrix.iˇ1; : : : ; iˇn/� D ˇk :

We consider the torus T of SUn de�ned by T D T0 \ SUn. Its Lie algebra is

t D°diag. matrix.iˇ1; : : : ; iˇn/I ˇ1; : : : ; ˇn 2 R;

nX

kD1ˇk D 0

±:

We denote by .x1; : : : ; xn�1/ the basis of t� given by

xk Œdiag. matrix.iˇ1; : : : ; iˇn/� D ˇk :

Let f OxkgkD1;:::;n�1 be the basis of i t� de�ned by Oxk D ixk . A vector � 2 i t� suchthat � D

Pn�1kD1 �k Oxk is denoted by

� D .�1; : : : ; �n�1/: (12.12)

Since the Lie algebra un of the group Un splits into

un D sun ˚ c;

c D iRIdn being the Lie algebra of the center of Un, each vector � 2 i t� can beidenti�ed with a vector in i t�0 which is identically zero on c. Thus, we can alsoexpress � in the basis f Oyk � iykgkD1;:::;n, of i t�0:

� DnX

kD1�k Oyk and

nX

kD1�k D 0:

In this case we denote � by

� D Œ�1; : : : ; �n�: (12.13)


The correspondence between the two expressions (12.12) and (12.13), is given by

�i D �i � �n ”

8ˆ<ˆ:

�i D �i C �0; i D 1; : : : ; n� 1;

�n D �0 ´ � 1n

n�1X

iD1�i :

(12.14)

We consider the SUn-invariant scalar product on t given by

hX; Y i D �ReŒTr.XY /� D � 1

2nB.X; Y /; X; Y 2 t;

where B is the Killing form of sun. Its extension to i t� is given by

h�;�0i DnX

iD1�i�

0i ; � D Œ�1; : : : ; �n�; �

0 D Œ�01; : : : ; �

0n� 2 i t�: (12.15)

The decomposition of sun;C ' sln;C into root spaces (see (12.1)) is easily obtained.One gets

sun;C D tC ˚� M

1�i;j�ni 6Dj

gij

�;

where gij D CEij and fEij g is the usual basis of Mn.C/. The corresponding roots�ij are given by

�ij D Oxi � Oxj ; 1 � i 6D j � n � 1;

�in D ��ni D Ox1 C � � � C 2 Oxi C � � � C Oxn�1; 1 � i � n� 1:

Using Lemma 12.34, we �rst note that T is a maximal torus of SUn. Therefore, then � 1 roots

�i D Oxi � OxiC1; 1 � i � n � 2; �n�1 D Ox1 C � � � C 2 Oxn�1;

form a basis� of the root systemˆ corresponding to this maximal torus T . Indeed,we have

�ij D ��j i D �i C � � � C �j�1; 1 � i < j � n � 1;

�in D ��ni D �i C � � � C �n�2 C �n�1; 1 � i � n � 1:Note that

�i D .0; : : : ; 0; 1„ƒ‚…i

; �1„ƒ‚…iC1

; 0; : : : ; 0/

D Œ0; : : : ; 0; 1„ƒ‚…i

; �1„ƒ‚…iC1

; 0; : : : ; 0�; 1 � i � n � 2;


and

�n�1 D .1; : : : ; 1; 2/ D Œ0; : : : ; 0; 1;�1�:

Thus from (12.15), the re�ection across �?i is given by

� D Œ�1; : : : ; �n� 7�!Œ�1; : : : ; �iC1; �i ; : : : ; �n�:

From Theorem 12.49 we conclude that the Weyl groupWSUn acts on i t� by

� D Œ�1; : : : ; �n� 7�!Œ��.1/; : : : ; ��.n/�; � 2 Sn:

Hence,WSUn ' Sn:

Furthermore, we have

h�i ; �j i D 0; 1 � i; j � n � 1; ji � j j 6D 0; 1;

h�i ; �i i D 2; 1 � i � n � 1;h�i ; �iC1i D �1; 1 � i � n � 2:

We then conclude from Theorem 12.61 the following result.

Proposition 12.68. A vector ˇ D .k1; : : : ; kn�1/ D Œp1; : : : ; pn� 2 i t� is a domi-nant weight of SUn if and only if

ki 2 N; i D 1; : : : ; n� 1; and k1 � k2 � � � � � kn�1;

which is equivalent to

pi � piC1 2 N; i D 1; : : : ; n� 1:

It is easy to see using (12.14) and (12.15) that the fundamental weights of SUnare given by

ˇi D Ox1 C Ox2 C � � � C Oxi ; i D 1; 2; : : : ; n� 1:

Let � be the standard representation of SUn on Cn. Its restriction to the maximal

torus T splits into the direct sum of n irreducible representations in the spaces Cei ,where feig is the usual basis of C

n, with corresponding weight Oxi .Since only one of those weights is dominant, namely Ox1, the representation � is

irreducible with dominant weight Ox1 D ˇ1, thus � is the fundamental representation�1 corresponding to the fundamental weight ˇ1.

Note also that there is only one maximal vector annihilated by all the matricesEi;iC1, 1 � i � n � 1, corresponding to simple roots, namely e1, so we may alsoconclude from Remark 12.55 that � is irreducible.


We now consider the fundamental representations �j , j D 2; : : : ; n � 1. FromTheorem 12.15 we know that each one of them is contained in some tensor productof the faithful representation � and its conjugate.

The simplest example of such a tensor product is the representation ˝2� whosespace T 2

Cn is the space of tensors of order 2 over C

n. This space splits intoT 2

Cn D ƒ2Cn˚S2Cn, whereƒ2Cn (resp. S2Cn) is the space of anti-symmetric

(resp. symmetric) tensors of order 2 over Cn.

Each of those spaces is SUn-invariant. Considering the one of lowest dimension,we get the representation

ƒ2�W SUn �! GLC.ƒ2.Cn//:

Its restriction to the maximal torus T splits into the direct sum of�n2

�irreducible

representations in the spaces C.ei ^ ej /, 1 � i < j � n, with correspondingweight Oxi C Oxj .

Only one of those weights is a dominant weight, namely Ox1C Ox2 D ˇ2, soƒ2� isirreducible with dominant weight ˇ2. Henceƒ2� is the fundamental representation�2 corresponding to the fundamental weight ˇ2.

Considering now each representation

ƒj �W SUn �! GLC.ƒj .Cn//; j D 3; : : : ; n � 1;

and determining its weights under the action of the maximal torus as above, onesees that only one of those weights is a dominant weight, namely

Ox1 C � � � C Oxj D j ;

hence,ƒj� is irreducible with dominant weight j . Thus for any j D 2; : : : ; n�1,ƒj � is the fundamental representation corresponding to the fundamental weight j .

Note that, with the help of Theorem 12.67, we can already know the dimensionnj D

�nj

�of �j , an indication to look at the representations ƒj �.

Irreducible representations of the group Un. Consider the homomorphism ofLie groups

� W T1 � SUn �! Un; .ei� ; A/ 7�! ei�A:

Note that � is a �nite covering with kernel isomorphic to the group of nth-rootsof unity. So we can apply Proposition 12.18: irreducible representations of Un areirreducible representations of T

1 � SUn which factor through the quotient. Thus,let us consider an irreducible representation Q� of T

1 � SUn.


By Theorem 12.9, the representation Q� is the tensor product of an irreduciblerepresentation � of T

1 and an irreducible representation � of SUn. The representa-tion � has the following form (see Theorem 12.24):

�W ei� 7�! eim� ; m 2 Z:

On the other hand, the representation � has a dominant weight of the form

.k1; : : : ; kn�1/;

where the ki are non-negative integers satisfying

k1 � k2 � � � � � kn�1

(see Proposition 12.68). By Remark 12.64 and the description of fundamental rep-resentations, the vector space corresponding to � is the cyclic module U.sun;C/ � v,where

v D ek1�k2

1 ˝ .e1 ^ e2/k2�k3 ˝ � � � ˝ .e1 ^ � � � ^ en�2/kn�2�kn�1

˝ .e1 ^ � � � ^ en�1/kn�1 :

The kernel of � is the group spanned by the elements

.e2ik�

n ; diag. matrix.e� 2ik�n ; : : : ; e� 2ik�

n //; k D 0; 1; : : : ; n� 1:

Therefore, a necessary and suf�cient condition for Q� to factor through the quotientis

.e2ik�

n /m

�Œdiag. matrix.e� 2ik�n ; : : : ; e� 2ik�

n /� � v D v; k D 0; 1; : : : ; n� 1:

This is equivalent to

e2ik�

n .m�Pn�1iD1 ki / D 1; k D 0; 1; : : : ; n� 1:

Hence, Q� factors through the quotient if and only if

m �n�1X

iD1ki is a multiple of n.

In this case, one can write

Q�.g/ D .detg/m=n�..detg/�1=ng/; g 2 Un;

since the right-hand side in this equality does not depend on the choice of thenth-root of detg.


12.4.2 Irreducible representations of the groups Spinn

and SOn, n � 3

For any n � 3, we denote by feig1�i�n the standard basis of Rn. We identify Un

with a subgroup of SO2n, also denoted by Un, using the injective homomorphismwhich maps a matrix in Mn.C/ with coef�cients zjk D xjk C iyjk into the matrixin M2n.R/ 0

BBB@

ŒZ11� : : : ŒZ1n�

:::: : :

:::

ŒZn1� : : : ŒZnn�

1CCCA

formed by the .2 � 2/-block matrices

Zjk D�xjk �yjkyjk xjk

�:

We also identify SO2n with a subgroup of SO2nC1, also denoted by SO2n, usingthe injective homomorphism

M2n.R/ 3 A 7�!�A 0

0 1

�2 M2nC1.R/:

Let T be the torus of Spinn,

T D° Œn=2�Y

kD1.cosˇk C sinˇke2k�1 � e2k/I ˇk 2 R

±:

Its image �.T / under the group homomorphism �W Spinn ! SOn is the image ofthe maximal torus of Un, see (12.11), under the above injective homomorphism.The Lie algebra of T is

t D° Œn=2�X

kD1ˇke2k�1 � e2k I ˇk 2 R

±:

We denote by .x1; : : : ; xŒn=2�/ the basis of t� given by

xk

� Œn=2�X

kD1ˇke2k�1 � e2k

�D ˇk:


We consider the Spinn-invariant scalar product on t de�ned by

hX; Y i D �.1=2/TrŒ��.X/��.Y /� D � 1

2.n� 2/B.X; Y /; X; Y 2 t;

where B is the Killing form of spinn. We have

he2k�1 � e2k ; e2l�1 � e2li D 4ıkl :

The canonical isomorphism t ! t� is given by e2k�1 � e2k 7! 4xk . So we have

hxk ; xli D .1=4/ıkl :

Thus, in order to have an orthonormal basis of .i t�; h�; �i/, we consider

Oxk ´ 2ixk ; k D 1; 2; : : : ; Œn=2�:

A vector � 2 i t� such that

� DŒn=2�X

kD1�k Oxk ;

in the basis f Oxkg, is denoted by

� D .�1; : : : ; �Œn=2�/:

So

h�;�0i DŒn=2�X

kD1�k�

0k ; � D .�1; : : : ; �Œn=2�/; �

0 D .�01; : : : ; �

0Œn=2�/ 2 i t�:

(12.16)

Irreducible representations of the group Spin2m, m � 2. Let .uk; vk/, k D 1,2; : : : ; m, be the Witt basis of C

2m de�ned by

uk D 1

2.e2k�1 � ie2k/ and vk D 1

2.e2k�1 C ie2k/:

One has

.ui ; uj / D .vi ; vj / D 0;

.ui ; vj / D 1

2ıij ; 1 � i; j � m;

where .�; �/ is the C-linear extension to C2m of the standard scalar product on R

2m.


After some computations one gets the decomposition of the complexi�ed Lie alge-bra spin2m;C into root spaces (see (12.1)):

spin2m;C D tC ˚� M

1�i<j�m.g�

Cij

˚ g��Cij

˚ g�

�ij

˚ g��ij

/�;

where

g�

Cij

D Cui � uj ; g��Cij

D Cvi � vj ; g�

�ij

D Cui � vj ; g��ij

D Cvi � uj ;

the corresponding roots being

˙�Cij D ˙. Oxi C Oxj /; ˙��

ij D ˙. Oxi � Oxj /; 1 � i < j � m:

By Lemma 12.34, T is a maximal torus of Spin2m. Therefore, we conclude that them roots

�i D Oxi � OxiC1; 1 � i � m � 1;

�m D Oxm�1 C Oxm;

form a basis� of the root system ˆ corresponding to this maximal torus T . Indeedwe have

��ij D �i C �iC1 C � � � C �j�1

and

�Cij D �i C � � � C �j�1 C 2�j C 2�jC1 C � � � C 2�m�2 C �m�1 C �m:

Note that

�i D .0; : : : ; 0; 1„ƒ‚…i

; �1„ƒ‚…iC1

; 0; : : : ; 0/; 1 � i � m � 1;

�m D .0; : : : ; 0; 1; 1/:

Thus, by (12.16), the re�ection across �?i is given by

� D .�1; : : : ; �m/ 7�! .�1; : : : ; �iC1; �i ; : : : ; �m/;

and the re�ection across �?m is given by

� D .�1; : : : ; �m/ 7�! .�1; : : : ;��m;��m�1/:


So from Theorem 12.49 we conclude that the Weyl groupWSpin2macts on i t� by

� D .�1; : : : ; �m/ 7�! .�1��.1/; : : : ; �m��.m//; � 2 Sm;

where �i D ˙1, �1�2 : : : �m D 1. Hence Card.WSpin2m/ D 2m�1.mŠ/.

Furthermore, we have

h�i ; �iC1i D �1; 1 � i � m � 2; (12.17)

h�i ; �ii D 2; 1 � i � m; (12.18)

h�i ; �j i D 0; 1 � i � m � 3; j � i C 2; (12.19)

h�m�2; �mi D �1; (12.20)

h�m�1; �mi D 0: (12.21)

We then conclude from Theorem 12.61

Proposition 12.69. A vector ˇ D .k1; k2; : : : ; km/ of i t� is a dominant weight ifand only if

k1 � k2 � � � � � km�1 � jkmj;and the ki are all simultaneously integers or half-integers.

By the above results, the (fundamental) weights of the fundamental representa-tions �i of Spin2m are given by

ˇi D Ox1 C Ox2 C � � � C Oxi ; i D 1; 2; : : : ; m� 2;

ˇm�1 D .1=2/. Ox1 C Ox2 C � � � C Oxm�1 � Oxm/;

ˇm D .1=2/. Ox1 C Ox2 C � � � C Oxm�1 C Oxm/:

We already know that the spin representation of Spin2m splits into two irreduciblerepresentations (see De�nition 1.33), so we �rst have a look at this representation.

Consider the complex spinor representation

�2mW Spin.2m/ �! GLC.†2m/: (12.22)

As it was mentioned at the end of Chapter 1, the space †2m can be de�ned in thefollowing way. Let

w ´ v1 � v2 � � �vm.2 Cl2m/ and †2m ´ Cl2m �w:

For any subset I � f1; 2; : : : ; mg, let

uI � w ´ ui1 � ui2 � � �uik � w;


where i1 < i2 < � � � < ik are the elements of I . The vectors uI � w form a basis of†2m, so dim†2m D 2m. Hence, the homomorphism �2mW Cl2m ! HomC.†2m/,de�ned for all ' 2 Cl2m by

�2m.'/W†2m �! †2m; � w 7�! ' � � w;

is an irreducible representation of Cl2m. Hence �2mjSpin2mis the complex spin

representation of Spin2m. Observe that for any j D 1; : : : ; m,

�2m.e2j�1 � e2j / � .uI � w/ D �iuI � w; if j … I ,

and

�2m.e2j�1 � e2j / � .uI � w/ D iuI � w; if j 2 I .

Thus the restriction of �2m to the maximal torus T splits into the direct sum of 2m

irreducible representations in the spaces

CuI � w;

with weights1

2.�1; : : : ; �m/; �i D ˙1:

Note that only two of those weights are dominant, namely

1

2.1; : : : ; 1;�1/ D ˇm�1

and

1

2.1; : : : ; 1; 1/ D ˇm;

and that they correspond respectively to the two maximal vectors

u1 � u2 � � �um�1 � w and u1 � u2 � � �um � w;

annihilated by the root vectors ui � viC1, 1 � i � m� 1, um�1 � um, correspondingto simple roots. So we recover the fact that the spin representation splits into twoirreducible representations. The corresponding spaces are the cyclic modules

U.spin2m;C/ � .u1 � u2 � � �um�1 � w/ and U.spin2m;C/ � .u1 � u2 � � �um � w/:

Consider the complex volume element ! ´ ime1 � � � e2m in Cl2m. Since ! com-mutes with the action of Spin2m and since

! � .u1 � u2 � � �um�1 � w/ D .�1/mC1.u1 � u2 � � �um�1 � w/


and

! � .u1 � u2 � � �um � w/ D .�1/m.u1 � u2 � � �um � w/;

we recover the decomposition of †2m into negative and positive spinor spaces: ifm is even, the fundamental representation �m�1 is ��

2m and the fundamental repre-sentation �m is �C

2m; the contrary if m is odd.On the other hand, Spin2m has a natural complex representation � on the space

C2m, obtained by considering the composition

�W Spin2m��! SO2m �! SO2m;C :

Its restriction to the maximal torus T splits into the direct sum of 2m irreduciblerepresentations in the spaces Cui , Cvi , i D 1; : : : ; m, with respective weights Oxiand � Oxi .

Since only one of these weights is dominant, namely Ox1, � is irreducible withdominant weight Ox1 D ˇ1, thus the fundamental representation �1 is �. (Note alsothat the only maximal vector is u1).

Inspired by the description of the fundamental representations of SUn, we nowconsider the representations

ƒk�W Spin2m �! GLC.ƒk.C2m//; 2 � k � m � 2:

Note that, with the help of Theorem 12.67, we already know the dimensionnk D

�2mk

�of �k , an indication to consider ƒk�.

Indeed those representations are irreducible. The “classical” proof of this resultis given in Theorem 1.26. It is easy to see that the vector u1 ^ � � �ûk is a maximalvector with weight ˇk , hence ƒk� is the fundamental representation �k .

We are going to retrieve the result by proving that the representation ƒk� hasonly one maximal vector.

We introduce the following basis of ƒk.C2m/: for any triple .A; B; C / of dis-joint subsets of f1; : : : ; mg such that

CardAC CardB C 2CardC D k;

we set

wA;B;C D ua1^ � � �ûap ^ vb1

^ � � �^ vbqûc1

^ vc1^ � � �ûcr ^ vcr ; (12.23)

where a1 < � � � < ap (resp. b1 < � � � < bq , c1 < � � � < cr ) are the elements of A(resp. B , C ).

Under the action of the maximal torus T ,ƒk� splits into the direct sum of�2mk

�

irreducible representations in the spaces CwA;B;C , with corresponding weights

Oxa1C � � � C Oxap � Oxb1

� � � � � Oxbq:


By Proposition 12.69, the dominant weights are

Ox1 C � � � C Oxk�2l ; l D 0; : : : ; Œk=2� (0 if k is even and l D k=2),

with corresponding weight vectors

wA;;;C ; A D f1; : : : ; k � 2lg (A D ; if k is even and l D k=2),

and C being a subset of fk � 2l C 1; : : : ; mg with l elements. But there is only onemaximal vector, namely

wA;;;; D u1 ^ � � � ^ uk :

Indeed, for any l D 1; : : : ; Œk=2�, any vector in the weight space corresponding toOx1 C � � � C Oxk�2l (0 if k is even and l D k=2) is of the form

v DX

C

�CwA;;;C ; �C 2 C;

where A D f1; : : : ; k � 2lg (A D ; if k is even and l D k=2), the sum being takenover the distinct subsets C of fk � 2l C 1; : : : ; mg with l elements.

Now let c; d 2 fk � 2l C 1; : : : ; mg such that c < d (note that the conditionk � m�2 implies the existence of such a couple .c; d/). Recall that uc �ud is a rootvector for the positive root Oxc C Oxd . It is easily checked that for any given subsetC of fk � 2l C 1; : : : ; mg with l elements, one has (perhaps up to a sign which hasno importance for the proof)

.ƒk�/C

� .uc � ud / �wA;;;C D 0;

if c … C and d … C , or c 2 C and d 2 C ,

.ƒk�/C

� .uc � ud / �wA;;;C D wA[fc;dg;;;Cnfcg;

if c 2 C and d … C , and

.ƒk�/C

� .uc � ud / �wA;;;C D wA[fc;dg;;;Cnfdg;

if c … C and d 2 C . Therefore,

.ƒk�/C

� .uc � ud / � v DX

C 0

.�C 0[fcg C �C 0[fdg/wA[fc;dg;;;C 0 ; (12.24)

where the sum is taken over the distinct subsets C 0 of fk � 2l C 1; : : : ; mgnfc; dgwith l � 1 elements.


Consider now the root vector uc �vd corresponding to the positive root Oxc � Oxd .Then for any given subset C of fk � 2l C 1; : : : ; mg with l elements (perhaps up toa sign which has no importance for the proof) one has

.ƒk�/C

� .uc � vd / �wA;;;C D 0;

if c … C and d … C , or c 2 C and d 2 C ,

.ƒk�/C

� .uc � vd / �wA;;;C D wA[fcg;fdg;Cnfcg;

if c 2 C and d … C ,

.ƒk�/C

� .uc � vd / �wA;;;C D �wA[fcg;fdg;Cnfdg;

if c … C and d 2 C . Consequently,

.ƒk�/C

� .uc � ud / � v DX

C 0

.�C 0[fcg � �C 0[fdg/wA[fcg;fdg;C 0 ; (12.25)

where the sum is taken over the distinct subsets C 0 of fk � 2l C 1; : : : ; mgnfc; dgwith l � 1 elements.

From (12.24) and (12.25), one sees that if v is annihilated by the root vectorscorresponding to the positive roots, then necessarily �C 0[fcg D �C 0[fdg D 0, forany subset C 0 of fk � 2l C 1; : : : ; mgnfc; dg with l � 1 elements. As c and d arearbitrary, this implies that �C D 0 for any subset C of fk � 2l C 1; : : : ; mg with lelements. Hence v D 0, so there is no maximal vector corresponding to

Ox1 C � � � C Oxk�2l ; l D 1; : : : ; Œk=2�:

Then, since ƒk� has only one maximal vector, namely

wA;;;; D u1 ^ � � � ^ uk ;

it is irreducible. This maximal vector corresponds to the weight

Ox1 C � � � C Oxk D ˇk ;

and so ƒk� is the fundamental representation �k .


Irreducible representations of the group SO2m. Considering the two-fold cov-ering �W Spin2m ! SO2m, one deduces from Proposition 12.18, that irreduciblerepresentations of SO2m correspond to irreducible representations of Spin2m whichfactor through the quotient.

Let � be an irreducible representation of Spin2m with dominant weight ˇ D.k1; k2; : : : ; km/. From Remark 12.64 and the description of fundamental repre-sentations, it follows that the vector space corresponding to � is the cyclic moduleU.spin2m;C/ � v, where

v D u1k1�k2 ˝ � � � ˝ .u1 ^ � � � ^ um�2/

km�2�km�1

˝ .u1 � u2 � � �um�1 � w/km�1�km ˝ .u1 � u2 � � �um � w/km�1Ckm :

A necessary and suf�cient condition for � to factor through the quotient is that�.�1/ � v D v. But

�.�1/ � v D .�1/2km�1v:

Hence � factors through the quotient if and only if km�1 is an integer. Therefore,the irreducible representations of SO2m are in one-to-one correspondence with thedominant weights .k1; k2; : : : ; km/ of Spin2m such that all the ki are integers.

Irreducible representations of the groups Spin2mC1 and SO2mC1, m � 1. Thedecomposition of the complexi�ed Lie algebra spin2mC1;C into root spaces, seeequation (12.1), is given by

spin2mC1;C D tC ˚� M

1�i<j�m.g�

Cij

˚ g��Cij

˚ g�

�ij

˚ g��ij

/�

˚� mM

iD1.g Oxi

˚ g� Oxi/�;

where

g�

Cij

D Cui � uj ; g��Cij

D Cvi � vj ; g�

�ij

D Cui � vj ; g��ij

D Cvi � uj ;

g OxiD Cui � e2mC1; g� Oxi

D Cvi � e2mC1;


˙�Cij D ˙. Oxi C Oxj /;˙��

ij D ˙. Oxi � Oxj /; 1 � i < j � m;˙ Oxi ; 1 � i � m:

Hence T is a maximal torus of Spin2mC1. Furthermore, the m roots

�i D Oxi � OxiC1; 1 � i � m � 1; �m D Oxm;


form a basis � of the root system ˆ corresponding to this maximal torus T , sincewe have

��ij D �i C �iC1 C � � � C �j�1;

�Cij D �i C � � � C �j�1 C 2�j C 2�jC1 C � � � C 2�m;

Oxi D �i C �iC1 C � � � C �m:

Note that

�i D .0; : : : ; 0; 1„ƒ‚…i

; �1„ƒ‚…iC1

; 0; : : : ; 0/; 1 � i � m � 1;

�m D .0; : : : ; 0; 0; 1/:

Thus by (12.16), the re�ection across �?i is given by

� D .�1; : : : ; �m/ 7�! .�1; : : : ; �iC1; �i ; : : : ; �m/;

and the re�ection across �?m is given by

� D .�1; : : : ; �m/ 7�! .�1; : : : ; �m�1;��m/:

Using Theorem 12.49, we conclude that the Weyl groupWSpin2mC1acts on i t� by

� D .�1; : : : ; �m/ 7�! .�1��.1/; : : : ; �m��.m//; � 2 Sm; �i D ˙1:

Hence, Card.WSpin2mC1/ D 2m.mŠ/. Furthermore we have

h�i ; �iC1i D �1; 1 � i � m � 1;

h�i ; �i i D 2; 1 � i � m � 1;

h�i ; �j i D 0; 1 � i � m � 2; j � i C 2;

h�m; �mi D 1:

We then conclude from Theorem 12.61

Proposition 12.70. A vector ˇ D .k1; k2; : : : ; km/ of i t� is a dominant weight ifand only if

k1 � k2 � � � � � km�1 � km � 0;

and the ki are all simultaneously integers or half-integers.


The fundamental weights are

ˇi D Ox1 C Ox2 C � � � C Oxi ; i D 1; 2; : : : ; m � 1;

ˇm D 1

2. Ox1 C Ox2 C � � � C Oxm�1 C Oxm/:

Recall that the complex spin representation �2mC1 can be de�ned as follows. Thereexists an isomorphism between Cl2mC1 and Cl02mC2, given by

ei1 � � � eik 7�! ei1 � � � eik ; if k is even,

ei1 � � � eik 7�! ei1 � � � eik � e2mC2; if k is odd.

Under this isomorphism, one obtains, using the representations �C2mC2 and �

�2mC2

of Cl2mC2, two inequivalent irreducible representations of Cl2mC1.The spin representation �2mC1 is obtained by restricting to Spin2mC1 any of

those two representations (seeDe�nition 1.33). With the notations used on page 351,one chooses for instance †2mC1 to be the space generated by the vectors

uI � wmC1 2 †2mC2; I � f1; 2; : : : ; mC 1g; cardI even;

withwmC1 ´ v1 � v2 � � �vmC1:

Since for any j D 1; : : : ; m, if j … I ,

�2mC1.e2j�1 � e2j / � .uI � wmC1/ D �2mC2.e2j�1 � e2j / � .uI � wmC1/

D �iuI � wmC1

and, if j 2 I ,

�2mC1.e2j�1 � e2j / � .uI � wmC1/ D �2mC2.e2j�1 � e2j / � .uI � wmC1/

D iuI � wmC1;

the restriction of �2mC1 to the maximal torus T splits into the direct sum of 2m

irreducible representations in the spaces

CuI � wmC1; cardI even, with weights1

2.�1; : : : ; �m/; �i D ˙1:

Since only one of these weights is dominant, namely

1

2.1; : : : ; 1/ D ˇm;

we recover that �2mC1 is irreducible. Hence, �2mC1 is the fundamental representa-tion corresponding to the weight ˇm.


With the same arguments used in the study of the fundamental representation ofSpin2m, it can be proved that the fundamental representation corresponding to ˇk ,k D 1; : : : ; m� 1, is

ƒk�W Spin2mC1 �! GLC.ƒk.C2mC1//;

where � is the natural representation of Spin2mC1:

Spin2mC1 �! SO2mC1 �! SO2mC1;C ,�! GLC.C2mC1/:

(With the same notations as on page 353, consider the basis of ƒkC2mC1 obtained

by taking the vectors wA;B;C of the basis of ƒk.C2m/, together with the vectorswA;B;C ^ e2mC1, such that cardAC cardB C 2 cardC D k � 1).

Finally, one can show that the irreducible representations of SO2mC1 arein one–to–one correspondence with the dominant weights .k1; k2; : : : ; km/ ofSpin2mC1 such that all the ki are integers, using the same arguments as in the de-scription of the irreducible representations of the group SO2m.

An application to the decomposition ofCn˝†n into irreducible modules. This

decomposition (which may be obtained with a more elementary method) leads tothe notion of the twistor operator (see Section 2.3.3). We consider the case n even,n D 2m (the case n odd can be studied using analogous arguments), and use nota-tions of page 351.

Under the action of the maximal torus T , the vector space C2m ˝ †2m splits

into the direct sum of m2mC1 irreducible representations

Cui ˝ uI � w and Cvi ˝ uI � w; i D 1; : : : ; m;

with corresponding weights1

2.�1; : : : ; �m/;

where �i D ˙1, except one of them which can be equal to ˙3. Thus, the dominantweights occurring in this decomposition are

• 12.3; 1; : : : ; 1/, corresponding to the weight-vector

u1 ˝ u1 � u2 � � �um � w;

• 12.3; 1; : : : ; 1;�1/, corresponding to the weight-vector

u1 ˝ u1 � u2 � � �um�1 � w;

• 12.1; : : : ; 1/, corresponding to the weight vectors

ui ˝ u1 � u2 � � � bui � � �um � w; i D 1; : : : ; m;

the notation bui meaning that the term ui is omitted, and


• 12.1; : : : ; 1;�1/, corresponding to the weight vectors

ui ˝ u1 � u2 � � � bui � � �um�1 � w; i D 1; : : : ; m� 1;

andvm ˝ u1 � u2 � � �um � w:

It is easy to check that the two weight vectors

u1 ˝ u1 � u2 � � �um � w and u1 ˝ u1 � u2 � � �um�1 � w;

are maximal vectors. Hence the two modules

V. 32 ;

12 ;:::;

12 /

´ U.spin2m;C/ � .u1 ˝ u1 � u2 � � �um � w/

and

V. 32 ;

12 ;:::;

12 ;� 1

2 /´ U.spin2m;C/ � .u1 ˝ u1 � u2 � � �um�1 � w/;

appear with multiplicity one in the decomposition.Let v be a weight vector corresponding to 1

2.1; : : : ; 1/:

v DmX

kD1�kuk ˝ u1 � u2 � � �cuk � � �um � w; �k 2 C:

For any j D 2; : : : ; m, one has

�C� .u1 � vj / � v D ..�1/j�1�1 � �j /u1 ˝ u1 � u2 � � � buj � � �um � w:

Thus, in order for v to be a maximal vector, one must have, for any j D 2; : : : ; m,�j D .�1/j�1�1: So, in order for v to be a maximal vector, it has to be a scalarmultiple of the vector

mX

kD1.�1/kC1uk ˝ u1 � u2 � � �cuk � � �um � w:

Now, it is easy to verify that the above vector is a maximal vector. So, up to a scalar,there exists one and only one maximal vector corresponding to 1

2.1; : : : ; 1/. Hence

there is a module V. 12 ;

12 ;:::;

12 /, isomorphic to †C

2m, in the decomposition and it hasmultiplicity 1.

Similarly, it can be proved that, up to a scalar, there exists one and only onemaximal vector corresponding to 1

2.1; : : : ; 1;�1/, namely

.�1/mvm ˝ u1 � u2 � � �um � w Cm�1X

kD1.�1/kC1uk ˝ u1 � u2 � � �cuk � � �um�1 � w:


Hence, there is a module V. 12 ;

12 ;:::;

12 ;� 1

2 /, isomorphic to †�

2m, in the decompositionand it has multiplicity 1. Thus, since there are four and only four linearly indepen-dent maximal vectors, corresponding to the four weights above, we get the decom-position of C

2m ˝†2m into irreducible modules:

C2m ˝†2m D V. 1

2 ;12 ;:::;

12 /

˚ V. 12 ;

12 ;:::;

12 ;� 1

2 /˚ V. 3

2 ;12 ;:::;

12 /

˚ V. 32 ;

12 ;:::;

12 ;� 1

2 /:

Let be the Clifford multiplication. Since is a Spin2m-equivariant operator,it follows from the Schur lemma that restricted to any irreducible component ofC2m ˝†2m is either 0 or an isomorphism. Now, it is easy to verify that

.u1 ˝ u1 � u2 � � �um � w/ D .u1 ˝ u1 � u2 � � �um�1 � w/ D 0;

� mX

kD1.�1/kC1uk ˝ u1 � u2 � � �cuk � � �um � w

�D mu1 � u2 � � �um � w;

and

�.�1/mvm ˝ u1 � u2 � � �um � w C

m�1X

kD1.�1/kC1uk ˝ u1 � u2 � � �cuk � � �um�1 � w

�

D mu1 � u2 � � �um�1 � w:

From this, one deduces that restricted to V. 12 ;

12 ;:::;

12 /

is an isomorphism onto†C2m,

restricted to V. 12 ;

12 ;:::;

12 ;� 1

2 /is an isomorphism onto †�

2m, and

Ker D V. 32 ;

12 ;:::;

12 /

˚ V. 32 ;

12 ;:::;

12 ;� 1

2 /:

12.4.3 Irreducible representations of the group Spn

We consider Hn as a right vector space over H. The canonical R-basis of H is

denoted by .1; i ; j ;k/ and H is identi�ed with C ˚ j C via the map

a1 C bi C cj C dk 7�! .aC ib/˚ j .c � id/:We consider the matrix representation of the group

Spn D fA 2 Mn.H/I t xAA D I g;and introduce the following basis of the Lie algebra spn:

Mrs D Ers �Esr ; 1 � r < s � n;

Nrs D .Ers CEsr /i ; 1 � r � s � n;

Prs D .Ers CEsr /j ; 1 � r � s � n;

Qrs D .Ers CEsr /k; 1 � r � s � n;


where fErsgr;sD1;:::;n is the canonical basis of Mn.H/. Let T be the torus of Spn:

T D fdiag. matrix .eiˇ1; : : : ; eiˇn/Iˇ1; ˇ2; : : : ; ˇn 2 Rg:

Its Lie algebra is

t D fdiag. matrix.iˇ1; : : : ; iˇn/Iˇ1; ˇ2; : : : ; ˇn 2 Rg :

We denote by fx1; : : : ; xng the basis of t� given by

xk Œdiag. matrix.iˇ1; : : : ; iˇn/� D ˇk :

A vector � 2 i t� such that � DPnkD1 �k Oxk , in the basis f Oxk � ixkgkD1;:::;n, is

denoted by� D .�1; : : : ; �n/: (12.26)

We consider the Spn-invariant scalar product on t given by

hX; Y i D �ReŒTr.XY /� D � 1

4.nC 1/B.X; Y /; X; Y 2 t;

where B is the Killing form of spn. Its extension to i t� is given by

h�;�0i DnX

kD1�k�

0k; � D .�1; : : : ; �n/; �

0 D .�01; : : : ; �

0n/ 2 i t�: (12.27)

Some computations lead to the decomposition of the complexi�ed Lie algebra spn;Cinto root spaces; see (12.1):

spn;C D tC ˚� M

1�r�s�m.g�

Crs

˚ g��Crs/�

˚� M

1�r<s�m.g�

�rs

˚ g��rs/�;

where

g�

Crs

D C.Prs � iQrs/; g��Crs

D C.Prs C iQrs/; 1 � r � s � n;

g�

�rs

D C.Mrs � iNrs/; g��rs

D C.Mrs C iNrs/; 1 � r < s � n;


˙�Crs D ˙. Oxr C Oxs/; 1 � r � s � n;

˙��rs D ˙. Oxr � Oxs/; 1 � r < s � n:


By Lemma 12.34, T is a maximal torus of Spn. Therefore, we get that the n roots

�i D Oxi � OxiC1; 1 � i � n� 1;

�n D 2 Oxn;

form a basis� of the root systemˆ corresponding to this maximal torus T . Indeed,we have

��ij D �i C �iC1 C � � � C �j�1;

�Cij D �i C � � � C �j�1 C 2�j C 2�jC1 C � � � C 2�n�1 C �n; 1 � i < j � n;

�Ci i D 2�i C � � � C 2�n�1 C �n; 1 � i � n� 1:

Note that

�i D .0; : : : ; 0; 1„ƒ‚…i

; �1„ƒ‚…iC1

; 0; : : : ; 0/; 1 � i � n � 1;

and

�n D .0; : : : ; 0; 0; 2/:

Thus, by (12.27), the re�ection across �?i is given by

� D .�1; : : : ; �n/ 7�! .�1; : : : ; �iC1; �i ; : : : ; �n/;

and the re�ection across �?n is given by

� D .�1; : : : ; �n/ 7�! .�1; : : : ; �n�1;��n/:

So from Theorem 12.49, we conclude that the Weyl groupWSpnacts on i t� by

� D .�1; : : : ; �n/ 7�! .�1��.1/; : : : ; �n��.n//; � 2 Sn; �i D ˙1:

Hence Card.WSpn/ D 2n.nŠ/. Furthermore we have

h�i ; �iC1i D �1; 1 � i � n � 2;

h�n�1; �ni D �2;

h�i ; �i i D 2; 1 � i � n � 1;

h�n; �ni D 4;

h�i ; �j i D 0; 1 � i � n � 1; ji � j j 6D 0; 1:

We then conclude by Theorem 12.61.


Proposition 12.71. A vector ˇ D .k1; : : : ; kn/ of i t� is a dominant weight if andonly if

k1 � � � � � kn � 0;

and all the ki are integers.

From the above results, the (fundamental) weights of the fundamental represen-tations �i of Spn are given by

ˇi D Ox1 C � � � C Oxi ; i D 1; 2; : : : ; n:

In order to describe the fundamental representations, we identify Hn with C

2n,by the isomorphism .C ˚ j C/n ! C2n, given by

.�1 C j �1; : : : ; �n C j �n/ 7�! .�1; : : : ; �n; �1; : : : ; �n/:

Under this identi�cation, we get an injective R-homomorphism

ˆWEndH.Hn/ �! EndC.C

2n/;

which in terms of matrices is given by

A 7�! ˆ.A/ D�U � xVV xU

�;

where A D U C jV 2 Mn.H/, with U; V 2 Mn.C/. The image of the group Spnunder ˆ is the group Sp2n;C \U2n, with

Sp2n;C ´ fA 2 M2n.C/I tAJA D J g; (12.28)

where

J D�0 �InIn 0

�:

Thus Spn has a natural complex (faithful) representation �W Spn ! GLC.C2n/.

Denoting the canonical basis of C2n by fe1; : : : ; en; e�1; : : : ; e�ng, under the

action of the maximal torus T the representation � splits into the sum of 2n irre-ducible representations in the spaces

Cei ; Ce�i ; i D 1; : : : ; n;

with corresponding weights ˙ Oxi . Since only one of those weights is dominant,namely Ox1, � is irreducible with dominant weight Ox1 D ˇ1, hence � is the funda-mental representation �1. With all the previous examples in mind, we now considerthe representation

ƒ2�W Spn �! GLC.ƒ2.C2n//:


Under the action of the maximal torus, it splits into the direct sum of�2n2

�irreducible

representations in the spaces

Cei ^ ej ; Ce�i ^ e�j ; 1 � i < j � n;

Cei ^ e�j ; 1 � i; j � n;

with corresponding weights

˙. Oxi C Oxj /; 1 � i < j � n;

˙. Oxi � Oxj /; 1 � i � j � n:

Thus the dominant weights occurring in the decomposition are

• .1; 1; 0; : : : ; 0/, with corresponding weight vector e1 ^ e2,

• .0; : : : ; 0/, with corresponding weight vectors ei ^ e�i , i D 1; : : : ; n.

It is easy to see that the weight vector e1 ^ e2 is maximal.Let v be a weight vector corresponding to .0; : : : ; 0/,

v DnX

iD1�iei ^ e�i ; �i 2 C:

Now note that the root vectorMr;rC1 � iNr;rC1 corresponding to the simple roots�r D Oxr � OxrC1, r D 1; : : : ; n � 1, acts on C

2n by the map

x D .x1; : : : ; xn; x�1; : : : ; x�n/ 7�! 2xrC1er � 2x�re�.rC1/;

whereas the root vector Pnn � iQnn corresponding to the simple root �n D 2 Oxnacts on C

2n by the map

x D .x1; : : : ; xn; x�1; : : : ; x�n/ 7�! �4x�nen:

So, one has

.ƒ2�/�.Mr;rC1 � iNr;rC1/ � v D 2.�rC1 � �r/er ^ e�.rC1/; r D 1; : : : ; n� 1;

.ƒ2�/�.Pnn � iQnn/ � v D 0:

Thus, in order for v to be a maximal vector, one must have �rC1 D �r , for anyr D 1; : : : ; n� 1.

Hence, up to a scalar, there exists exactly one maximal vector corresponding to.0; : : : ; 0/, namely the symplectic 2-form

! DnX

iD1ei ^ e�i :


Therefore, the representation ƒ2� splits into two irreducible components: the �rstone with weight Ox1 C Ox2 D ˇ2, in the space

ƒ2ıC2n ´ U.spn;C/ � .e1 ^ e2/;

is the fundamental representation �2, whereas the second is the trivial representa-tion.

Note that the dimension of �2 is�2n2

��1. To get a description of the fundamental

representations �k , k D 3; : : : ; n, observe that the vector

e1 ^ � � � ^ ek 2 ƒk.C2n/

is a maximal vector with corresponding weight ˇk . Hence the vector space of thefundamental representation is the cyclic module

ƒkı C2n ´ U.spn;C/ � .e1 ^ � � � ^ ek/:

As in the case k D 2, the representations ƒk�, k D 3; : : : ; n are not irreducible.

Proposition 12.72. The representations ƒk�, k D 2; : : : ; n decompose under theaction of the symplectic group as

ƒk.C2n/ D ƒkı C2n ˚ .ƒk�2

ı C2n ^ !/˚ .ƒk�4

ı C2n ^ !2/˚ � � � :

This decomposition is often called “Lepage decomposition” following [Lep46].For a proof, see [Bou75], p. 202. Note that

ƒk.C2n/ D ƒkı C2n ˚ .ƒk�2.C2n/ ^ !/;

hence the dimension of the fundamental representation �k is

nk D�2n

k

��2n

k � 2

�:

As we did in the examples above, we will prove here the decomposition bydetermining explicitly the maximal vectors.

Consider the following basis of ƒk.C2n/: for any triple .A; B; C / of disjointsubsets of f1; : : : ; ng such that

CardAC CardB C 2CardC D k;

we set

wA;B;C D ea1^ � � � ^ eap ^ e�b1

^ � � � ^ e�bq^ ec1

^ e�c1^ � � � ^ ecr ^ e�cr ;

where a1 < � � � < ap (resp. b1 < � � � < bq , c1 < � � � < cr ) are the elements of A(resp. B , C ).


Under the action of the maximal torus T , the representation ƒk� splits into thedirect sum of

�2nk

�irreducible representations in the spaces CwA;B;C , with corre-

sponding weightsOxa1

C � � � C Oxap � Oxb1� � � � � Oxbq

:

By Proposition 12.71, the dominant weights are

Ox1 C � � � C Oxk�2l ; l D 0; : : : ; Œk=2� (0 if k is even and l D k=2),

with corresponding weight vectors

wA;;;C ; A D f1; : : : ; k � 2lg .A D ;; if k is even and l D k=2/;

C � fk � 2l C 1; : : : ; ng;CardC D l:

For any given l D 0; : : : ; Œk=2�, let us consider the existence of a maximal vectorwith weight

Ox1 C � � � C Oxk�2l (0 if k is even and l D k=2).

Any such vector v has the form

v DX

C

�CwA;;;C ; �C 2 C;

A D f1; : : : ; k � 2lg .A D ;; if k is even and l D k=2),

where the sum is taken over the distinct subsets C of fk � 2l C 1; : : : ; ng withl elements. One has (possibly up to a sign which has no importance for the proof)

.ƒ2�/�.Mr;rC1 � iNr;rC1/ � v D 0;

for 1 � r � k � 2l;

.ƒ2�/�.Mr;rC1 � iNr;rC1/ � v D � 2X

C;r2C;.rC1/…C�CwA[frg;frC1g;Cnfrg

C 2X

C;r…C;.rC1/2C�CwA[frg;frC1g;CnfrC1g;

for k � 2l C 1 � r � n� 1; and

.ƒ2�/�.Pnn � iQnn/ � v D 0:

Thus, in order for v to be a maximal vector, one must have �C 0[frg D �C 0[frC1g forany subset C 0 of fk � 2l C 1; : : : ; ngnfr; r C 1g with l � 1 elements,r D k � 2l C 1; : : : ; n � 1. Hence, for any subset C of fk � 2l C 1; : : : ; ngwith l elements, one has

�C D �fk�2lC1;:::;k�lg:


Indeed, consider such a subset C D fc1; : : : ; clg (c1 < � � � < cl ), and let j be thelowest integer such that cj 6D k � 2l C j . Then

fk � 2l C j; k � 2l C j C 1; : : : ; cj � 1g \ C D ;:

SettingC 0 D Cnfcj g, we get from the previous result, considering r D k � 2l C j ,k � 2l C j C 1; : : : ; cj � 1,

�C D �C 0[fcj g D �C 0[fcj �1g D � � � D �C 0[fk�2lCj g:

Considering now the set C 0 [ fk � 2l C j g instead of C , one deduces by inductionthat �C D �fk�2lC1;:::;k�lg.

Hence there is, up to a scalar, exactly one maximal vector for the weightOx1 C � � � C Oxk�2l , which can be written as

e1 ^ � � � ^ ek�2l ^ !l :

The corresponding cyclic module is

U.spn;C/ � .e1 ^ � � � ^ ek�2l ^ !l/:

But the symplectic form ! is invariant under the action of Spn, thus

U.spn;C/ � .e1 ^ � � � ^ ek�2l ^ !l / D .U.spn;C/ � .e1 ^ � � � ^ ek�2l // ^ !l

D ƒk�2lı C

2n ^ !l :

Therefore, since corresponding to the weights Ox1 C � � � C Oxk�2l , l D 0; : : : ; Œk=2�,there are exactly Œk=2�C 1 linearly independent maximal vectors, one gets the fol-lowing decomposition of ƒk.C2n/ into irreducible modules:

ƒk.C2n/ D ƒkı C2n ˚ .ƒk�2

ı C2n ^ !/˚ .ƒk�4

ı C2n ^ !2/˚ � � � :

Chapter 13

Symmetric space structure of model spaces

A classical reference for an introduction to symmetric spaces is [Bes87], Chapter 7.By de�nition, a Riemannian manifold .M; gM / is said to be a symmetric space iffor any p in M , there exists an isometry sp of .M; gM / such that sp.p/ D p anddsp D �IdTpM .

Theorem 13.1 (É. Cartan). A Riemannian manifold .M; gM / is a symmetric spaceif and only if there exists a triple .G;H; �/ satisfying the following conditions.

(i) G is a connected Lie group, H is a compact subgroup of G, and � is aninvolutive automorphism of G such that

G�e � H � G� ;

where G� is the group fg 2 GI �.g/ D gg and G�e is the connected compo-nent of the identity in G� .

(ii) There exists a G-invariant metric gG=H on G=H such that .G=H; gG=H / isisometric to .M; gM /.

Notations 13.2. In the following, any symmetric space .M; gM / will be identi�edwith the corresponding homogeneous space .G=H; gG=H / by means of the isometry{W .G=H; gG=H / ! .M; gM /.

The neutral element inG will be denoted by e, the equivalence class inG=H ofany g 2 G by Œg�. The same notation Lg will be used to denote the left action ofg 2 G on G and the induced action on the quotient G=H .

Remark 13.3. Note that the group G acts transitively on M by the isometries{ ı Lg ı {�1, g 2 G, and that H is the isotropy subgroup corresponding to thepoint p ´ {.Œe�/.

Note also that sinceM isG-homogeneous, all the geometric objects onM whichare invariant under isometries are completely determined by their value at p.

Theorem 13.4 ([Hel78]). If in the above theorem the symmetric space .M; gM / issupposed to be compact and simply connected, then there exists a triple .G;H; �/such that G is compact and simply connected, andH D G�e .

370 13. Symmetric space structure of model spaces

From now on, we consider a compact and simply connected symmetric space.M; gM /, together with a triple .G;H; �/ satisfying the conditions of Theorem 13.4.

Let g (resp. h) be the Lie algebra of G (resp. H ). The tangent map of { at thepoint Œe�, gives an isometry

.TŒe�.G=H// ' .g=h; gG=H;Œe�/TŒe�{��! .TpM;gM;p/:

Note that under the identi�cation TŒe�.G=H/ ' g=h, the action of h 2 H ong=h is given by AdG.h/, since for any h 2 H and any X 2 g,

d

dtfLh.Œexp.tX/�/g

ˇtD0 D d

dtfŒh exp.tX/h�1�g

ˇtD0

D d

dtfŒexp.t AdG.h/ �X/�g

ˇtD0:

Now the structure of symmetric space of .M; gM / provides an AdG.H/-invariantsubspace p of g which complements h in g. Indeed, let �� be the tangent map at eof the involutive automorphism � . It is a Lie algebra automorphism of g such that�2� D Idg. SinceH D G�e , one has

h D fX 2 gI ��.X/ D Xg;

thus the decomposition of g into eigenspaces is

g D h ˚ p;

withp D fX 2 gI ��.X/ D �Xg: (13.1)

It follows easily from the condition H D G�e that p is an AdG.H/-invariant sub-space of g: for any X 2 p and any h 2 H

��.AdG.h/.X// D d

dt.��h exp.tX/h�1�/

ˇtD0

D d

dt.h�.exp.tX//h�1/

ˇtD0

D d

dt.h exp.t��.X//h�1/

ˇtD0

D d

dt.h exp.�tX/h�1/

ˇtD0

D �AdG.h/.X/:

Furthermore, we have the following result.

371

Lemma 13.5. The complementing Lie algebras h and p satisfy the commutationrelations

Œh; h� � h; Œh; p� � p; Œp; p� � h:

Proof. For any X and Y in p, one has

�� ŒX; Y � D Œ�� X; �� Y � D ŒX; Y �;

hence the last inclusion. �

We consider on p the AdG.H/-invariant scalar product .�; �/ which makes theisomorphism

ˆW p �! TŒe�.G=H/; X 7�! d

dtfŒexp.tX/�g

ˇˇtD0

; (13.2)

an isometry. Thus AdG.H/ is a compact and connected (by the conditionH D G�e )subgroup of SO.p/.

We �x, once for all, an orthonormal basis fXig, 1 � i � n, of p. The choice ofsuch a basis allows one to identify SO.p/ with SOn, so in what follows AdG.H/ isconsidered as a subgroup of SOn.

Definition 13.6. The symmetric space G=H is said to be irreducible if the repre-sentation AdG WH ! GL.p/ ' GLn is irreducible.

From now on, we assume that our symmetric space .M; gM / is irreducible. Notethat, by the de Rham decomposition theorem (see [Bes87], p. 194, for instance),this is a rather weak condition. In fact, any compact simply connected symmetricspace is the Riemannian product of a �nite number of simply connected compactirreducible symmetric spaces.

Lemma 13.7. AnyAdG.H/-invariant symmetric bilinear form on p is proportionalto .�; �/. In particular, all the AdG.H/-invariant scalar products on p are propor-tional.

Proof. The scalar product .�; �/ allows one to identify any AdG.H/-invariant sym-metric bilinear form on p with an AdG.H/-invariant symmetric endomorphism ofp. This endomorphism is diagonalizable in some orthonormal basis of p. But sinceit is AdG.H/-invariant, any eigenspace is invariant under the action of AdG.H/,hence is equal to p because p is irreducible. Therefore, the endomorphism is ascalar multiple of the identity. �


Note that since the groupG is compact and simply connected,G is semi-simple(cf. Theorem 12.32), so the restriction to p � p of the Killing form B of g sign-changed provides anAdG.H/-invariant scalar product on p. From the above lemma,this scalar product is proportional to .�; �/. Thus we can renormalize the metric gG=Hon G=H (hence the metric gM onM ) in such a way that

.�; �/ D �Bjp: (13.3)

Note also that the Ricci tensor at p, viewed as a symmetric bilinear form on p, isproportional to .�; �/. Hence, because the Ricci tensor is totally determined by itsvalue at p (see the above remark), .M; gM / is an Einstein manifold.

Finally, wemention the following result about the holonomy of symmetric spaces

Theorem 13.8. The holonomy group Hol of any irreducible simply connected sym-metric space G=H is the image of the groupH under the homomorphism

H �! SO.p/ ' SOn; h 7�! AdG.h/jp:

For a proof, see [Bes87], Proposition 10.79.

13.1 Symmetric space structure of spheres

Classically, Sn is the symmetric space SOnC1 = SOn (see for instance [KN69]).

Since SOnC1 is not simply connected, in order to obtain a triple verifying the con-ditions of Theorem 13.4, we consider the two-fold covering �W SpinnC1 ! SOnC1.

ThenG D SpinnC1 acts transitively on Sn by the isometry �.g/, for any g 2 G.

Let fe0; : : : ; eng be the standard basis ofRnC1. We identify Spinn with the subgroup

H of SpinnC1, also denoted by Spinn, de�ned by

H D fv1 � � �v2kI vi 2 spanfe1; : : : ; eng; k vi kD 1; 1 � i � 2kg:

It is easy to see thatH is the isotropy subgroup of the point p D e0, for the transitiveaction of SpinnC1 on S

n. Hence the map

{W SpinnC1=Spinn �! Sn; Œg� 7�! �.g/.p/;

is a diffeomorphism and it becomes an isometry if SpinnC1=Spinn is endowed withthe metric {�can (note that this metric is G-invariant, since G acts on S

n by isome-tries).

13.1. Symmetric space structure of spheres 373

Now consider the involutive automorphism of SpinnC1

� W SpinnC1 �! SpinnC1; �. / ´ e0 � � e0�1 D �e0 � � e0:

Observe that for any 2 H , � e0 D e0 � , henceH � G� , and then H � G�e ,sinceH is connected. But it is easy to see that h is the eigenspace of �� correspond-ing to the eigenvalue 1. Indeed, consider the basis fei � ej g, 0 � i < j � n, of g.One easily gets

��.ei � ej / D ei � ej ; 1 � i < j � n; (13.4a)

��.e0 � ei / D �e0 � ei ; 1 � i � n: (13.4b)

Hence X in g veri�es ��.X/ D X if and only if X 2 h.Thus the two connected subgroups of G, H and G�e , have the same Lie alge-

bra, hence H D G�e . So the triple .G D SpinnC1; H D Spinn; �/ veri�es theconditions of Theorem 13.4. By (13.4), it is easy to see that the eigenspace p of ��corresponding to the eigenvalue �1 has fe0 � eig1�i�n as basis. Under the isometry

ˆW p �! Te0Sn ' he0i?; X 7�! d

dtf�.exp.tX// � e0gjtD0 D ��.X/.e0/;

the vector e0 � ei is mapped onto 2ei , hence setting

Xi ´ 1

2e0 � ei ; 1 � i � n;

we get an orthonormal basis .Xi / of p.Under the identi�cations H ' Spinn and SO.p/ ' SOn (the last being given

by the choice of the above basis), the homomorphism AdG jH is nothing else but thecovering �W Spinn ! SOn. Hence the representation AdG WH ! GL.p/ ' GLnis irreducible, since it is the standard representation of the group Spinn on Rn, seeSection 1.2.2.2. By applying Theorem 13.8, this in particular implies

Remark 13.9. For any n � 2, Hol.Sn; can/ D SOn.

The Killing form B of the group SpinnC1 is given by

B.X; Y / D .n � 1/ tr.��.X/ � ��.Y //; X; Y 2 g:

By Lemma 13.7, its restriction to p sign-changed has to be proportional to the scalarproduct .�; �/ on p. Considering any vector of the above orthonormal basis fXig, onegets

�B.Xi ; Xi / D 2.n � 1/;so in order to have .�; �/ D �Bjp, we have to normalize the canonical metric on thesphere by considering the metric 2.n� 1/can.


13.2 Symmetric space structure of the complex projectivespace

We denote by fe0; : : : ; emg, the canonical basis of CmC1. Recall that the canoni-

cal metric on complex projective space is de�ned by requiring the Hopf �brationS2mC1 ! CPm to be a Riemannian submersion, cf. [BGM71]. Let G be the group

SUmC1 D fA 2 MmC1.C/I t xAA D Id; detCA D 1g:

The group G acts transitively on the sphere S2mC1, and since SUmC1 � SO2mC2,

it acts by isometries. This action factors through the quotient, hence the groupSUmC1 acts transitively on CPm, and by de�nition of the canonical metric, it actsby isometries. The isotropy subgroup H of the point p ´ Œe0� for this transitiveaction of SUmC1 on CPm is given by

S.U1 � Um/ ńA 2 SUmC1I A D

�ei� 0

0 B

�; � 2 R; B 2 Um

o:

Hence the map

{W SUmC1 =S.U1 � Um/ �! CPm; Œg� 7�! g.p/ D Œg.e0/�;

is a diffeomorphism and it becomes an isometry if SUmC1 =S.U1�Um/ is endowedwith the G-invariant metric gG=H D {�can. Now consider the map

� W SUmC1 �! SUmC1; �.A/ D SAS�1;

where S is the .mC 1/ � .mC 1/-matrix

�1 0

0 �Im

�:

It is easy to see that � is an involutive automorphism of SUmC1 such thatG� D S.U1 � Um/, hence G�e D S.U1 � Um/, since S.U1 � Um/ is connected.Hence, the triple .G D SUmC1; H D S.U1 � Um/; �/ veri�es the conditions ofTheorem 13.4.

The eigenspace p of �� corresponding to the eigenvalue �1 is precisely

p DnXx D

�0 �t Nxx 0

�I x 2 Mm;1.C/ ' C

mo:

Since p is endowed with the scalar product .�; �/ that makes the isomorphism ˆ anisometry, it is easy to see from the de�nition of the canonical metric on CPm that

.Xx ; Xx0/ D Rehx; x0i; x; x0 2 Cm; (13.5)

13.3. Symmetric space structure of the quaternionic projective space 375

where h�; �i is the usual Hermitian product on Cm. Hence the map

Cm ' R

2m �! p; x 7�! Xx ;

is an isometry.Identifying p with Cm by this isometry, the adjoint representation of the group

H on p is

AdG

�ei� 0

0 B

�� x D e�i�Bx: (13.6)

This representation is irreducible since its restriction to the subgroupnA 2 H I A D

�1 0

0 B

�; B 2 SUm

o;

is the standard representation of the group SUm, see Paragraph 12.4.1. So the sym-metric space .SUmC1 =S.U1 � Um/; gG=H / is irreducible.

Remark 13.10. The image of the group H under the homomorphism

H �! SO.p/ ' SO2m; h 7�! AdG.h/jp;

is the group Um, hence from Theorem 13.8 it follows that Hol.CPm; can/ D Um.

The Killing form B of the group SUmC1 is given by

B.X; Y / D 2.mC 1/Re.Tr.XY //; X; Y 2 g:

From Lemma 13.7, its restriction to p sign-changed has to be proportional to thescalar product .�; �/ on p. By (13.5), for any x 2 C

m, one has

�B.Xx; Xx/ D 4.mC 1/kxk2 D 4.mC 1/.Xx ; Xx/;

so in order to have .�; �/ D �Bjp, we need to normalize the canonical metric on thecomplex projective space by considering gG=H D 4.mC 1/can.

13.3 Symmetric space structure of the quaternionicprojective space

We consider HmC1 as a right vector space over H and denote by fe0; : : : ; emg,

the canonical basis of HmC1. The quaternionic projective space HPm is the set of

equivalence classes of .m C 1/-tuples of quaternions x D .q0; : : : ; qm/, two such.m C 1/-tuples x and x0 being equivalent if there exists a non-zero quaternion qsuch that x0 D xq.


The canonical metric on HPm is de�ned by requiring the Hopf �bration

S4mC3 �! HPm

to be a Riemannian submersion, cf. [BGM71]. Let G be the group

SpmC1 D fA 2 MmC1.H/I t xAA D I g:

The group G acts transitively on the sphere S4mC3 � H

mC1, and it acts by isome-tries since SpmC1 � SO4mC4. This action factors through the quotient, hence thegroup SpmC1 acts transitively on HPm, and by de�nition of the canonical metric,it acts by isometries. The isotropy subgroup H at the point p ´ Œe0� for this tran-sitive action of SpmC1 on HPm is the group identi�ed with Sp1 � Spm:

H DnA 2 SpmC1I A D

�q 0

0 B

�; q 2 Sp1; B 2 Spm

o:

Hence the map

{W SpmC1 = Sp1 � Spm �! HPm; Œg� 7�! g.p/ D Œg.e0/�;

is a diffeomorphism and it becomes an isometry if SpmC1 = Sp1 � Spm is endowedwith the G-invariant metric gG=H D {�can.

Now consider the map

� W SpmC1 �! SpmC1; �.A/ ´ SAS�1;

where S is the .mC 1/ � .mC 1/ matrix

�1 0

0 �Im

�:

It is straightforward to see that � is an involutive automorphism of SpmC1 such thatG� D Sp1 � Spm, hence G

�e D Sp1 � Spm, since Sp1 � Spm is connected. So the

triple .G D SpmC1; H D Sp1 � Spm; �/ satis�es the conditions of Theorem 13.4.The eigenspace p of �� corresponding to the eigenvalue �1 is

p DnXx D

�0 �t Nxx 0

�I x 2 Mm;1.H/ ' H

mo:

Since p is endowed with the scalar product .�; �/ which makes the isomorphism ˆ

an isometry, from the de�nition of the canonical metric on HPm it follows that

.Xx; Xx0/ D Rehx; x0i; x; x0 2 Hm; (13.7)

where h�; �i is the symplectic product on Hm. Hence the map

Hm ' R

4m �! p; x 7�! Xx;

13.3. Symmetric space structure of the quaternionic projective space 377

is an isometry. Identifying p with Hm by this isometry, the adjoint representation

of the groupH on p is then

AdG

�q 0

0 B

�� x D Bx Nq:

This representation is irreducible since its restriction to the subgroupnA 2 H I A D

�1 0

0 B

�; B 2 Spm

o;

is the standard representation of the group Spm, see Section 12.4.3. We concludethat the symmetric space .SpmC1 = Sp1 � Spm; gG=H / is irreducible.

Remark 13.11. The image of the group H under the homomorphism

H �! SO.p/ ' SO4m; h 7�! AdG.h/jp;

is the group Sp1 � Spm de�ned in (7.2) (this could be indeed a de�nition of thisgroup), hence from Theorem 13.8, Hol.HPm; can/ D Sp1 � Spm.

The Killing form B of the group SpmC1 is given by

B.X; Y / D 4.mC 2/Re.Tr.XY //; X; Y 2 g:

By Lemma 13.7, its restriction to p sign-changed has to be proportional to the scalarproduct .�; �/ on p. By (13.7), one has for any x 2 H

m,

�B.Xx; Xx/ D 8.mC 2/hx; xi D 8.mC 2/.Xx; Xx/;

so in order to have .�; �/ D �Bjp, we have to normalize the canonical metric on thequaternionic projective space by considering gG=H D 8.mC 2/can.

Chapter 14

Riemannian geometry of model spaces

The results of this chapter hold for any symmetric space, but we continue to considera compact, simply connected, n-dimensional irreducible symmetric space.M D G=H; gG=H / as in the previous chapter, with corresponding triple .G;H; �/verifying the conditions of Theorem 13.4. As before, we �x once for all an or-thonormal basis fXig, 1 � i � n, of p, which allows us to identify p with Rn, andwe denote by ˆ the isometry

ˆW p �! TŒe�G=H; X 7�! d

dt.Œexp.tX/�/

ˇtD0:

In the following, for simplicity we denote by ˛ the homomorphism

H �! SOn; h 7�! AdG.h/jp:

The manifold M D G=H has a G-invariant orientation. Indeed, consider then-form

ˆ.X1/� ^ � � � ^ˆ.Xn/�;

on TŒe�G=H , and extend it to any point Œg� ofG=H by applying the diffeomorphismLg given by the left action of some representative g of Œg� on G=H . Note that, thisde�nition does not depend on the choice of the representative since ˛.H/ � SOn.

Proposition 14.1. Let PSOnM be the bundle of positive orthonormal frames ofM .

Let � WG ! G=H be the canonical principal bundle over G=H with structuralgroupH . Consider the associated SOn-principal bundle given byG �˛ SOn. Thenthe principal bundles PSOn

M and G �˛ SOn are isomorphic.Proof. Let Œg� be some element inG=H and let bŒg� be a positive orthonormal frameat Œg�, that is, an isometry R

n ! TŒg�.G=H/ preserving the orientations. Let g bea representative of Œg�, and denote by Lg� the tangent map at the point p ´ Œe�

of Lg . Consider the map ug W Rn ! R

n ' p, de�ned by

ug D ˆ�1 ı Lg��1 ı bŒg�: (14.1)

Since ug is an isometry preserving the orientation of Rn, ug belongs to SOn.

Furthermore for any h in H , it is easy to verify that ugh D ˛.h�1/ug . Hence

380 14. Riemannian geometry of model spaces

the element Œg; ug � in the �bre of G �˛ SOn at Œg� depends only on the equivalenceclass Œg� of g. The map

PSOnM �! G �˛ SOn; bŒg� 7�! Œg; ug �;

gives the claimed SOn-isomorphism between PSOnM and G �˛ SOn. (The inverse

map is given by Œg; u� 7! Lg� ıˆ ı u, where .g; u/ 2 G � SOn is a representativeof Œg; u�). �

Consider the vector bundle associated with the principal bundle � WG ! G=H

by the linear representation ˛WH ! SOn � GLn, that is G �˛ Rn.

Proposition 14.2. The vector bundles TM and G �˛ Rn are isomorphic.

Proof. The tangent bundle TM is the vector bundle associated to PSOnM by thestandard linear representation � of SOn, that isPSOnM ��R

n. Let Œg� be an elementin G=H and ŒbŒg�; x� an element in the �bre of PSOn

M �� Rn over Œg�, where bŒg�is a positive orthonormal frame at Œg� and x 2 R

n. It is easy to see that the elementŒg; ug .x/� of the �bre of G �˛ R

n at Œg�, where ug is the isometry of Rn de�ned

in (14.1), depends only on the equivalence class ŒbŒg�; x�, and that the map

PSOnM �� R

n �! G �˛ Rn; ŒbŒg�; x� 7�! Œg; ug.x/�;

is the claimed isomorphism between the two vectors bundles. (The inverse mapis given by Œg; x� 7! ŒLg� ı ˆ; x�, where .g; x/ 2 G � R

n is a representative ofŒg; x�.) �

Corollary 14.3. Let C1H .G;R

n/ be the space of H -equivariant smooth functionsG ! R

n, that is, the set of functions f WG ! Rn satisfying the condition

f .gh/ D ˛.h�1/ � f .g/; g 2 G; h 2 H:

The space �.TM/ of smooth sections of the tangent bundle ofM is isomorphic toC1H .G;R

n/.

Proof. According to Proposition 14.2, a smooth vector �eld onM , that is, a smoothsection of the bundle TM can be seen as a section of the vector bundleG�˛R

n. Theproof is a classical result of the theory of associated bundles. Let f be a function inC1H .G;R

n/. Take Œg� 2 G=H and denote by g a representative of the class Œg�. Sincef is H -equivariant, the element Œg; f .g/� of the �bre of G �˛ R

n at Œg� dependsonly on the equivalence class Œg� of g, and the map

Xf WG=H �! G �˛ Rn; Œg� 7�! Œg; f .g/�;

is a smooth section of the bundle G �˛ Rn.

14.1. The Levi-Civita connection 381

Conversely, let X be a smooth section of the bundle G �˛ Rn. For any g in G,

there exists a unique xg in Rn such that X.Œg�/ D Œg; xg �. Note that for any h 2 H ,

xgh D ˛.h�1/ � xg . Hence the map

fX WG �! Rn; g 7�! xg ; (14.2)

is H -equivariant. Using local trivializations of the bundle, it follows that fX issmooth. The map X 7! fX is the claimed isomorphism with inverse f 7! Xf .Note that for any tangent vector �eld X , with the identi�cation R

n ' p, the relationbetween X and theH -equivariant function fX is given by

XŒg� D Lg� ıˆ.fX .g//; g 2 G (14.3)


Remark 14.4. Similarly, .p; q/-tensor �elds on M (resp. p-forms on M ), can beidenti�ed withH -equivariant functions G ! T.p;q/Rn (resp. G ! ƒpRn).

14.1 The Levi-Civita connection

Let g D h ˚ p be the canonical decomposition of g induced by the structureof symmetric space of M . Since p is an AdG.H/-invariant subspace of g, theG-invariant horizontal distribution H on G given by

Hg ´ TeLg.p/; g 2 G;

is a principal connection on the canonical bundle � WG ! G=H . Let ! be thecorresponding connection 1-form. Recall that, by de�nition, denoting by prh theprojection g ! h, for any vector �eld X on G one has

!.X/.g/ D prh.TeLg�1.Xg//:

It is a classical result in the theory of connections that this principal connection!induces a linear connection r on the associated bundleG�˛R

n ' TM , connectionwhich is de�ned as follows. Let Œg� be an element of G=H and denote by g arepresentative of the class Œg�. If Y is a section of the bundle G �˛ R

n ' TM , thenby Corollary 14.3 we can identify Y with theH -equivariant function fY WG ! R

n,de�ned by YŒg� D Œg; fY .g/� (see (14.2)). If X is a vector �eld on M , we denoteby XH the horizontal lift of X . Then

.rXY /.Œg�/ D Œg; .XH :fY /.g/�:

Note that if � is a local section of the principal bundle G ! G=H , then

XH :fY .�.Œg�// D .X:��fY /Œg� � ˛�.��!.X// � fY .�.Œg�//:

This implies that the linear connection r is metric, since ˛WH ! SOn.


Proposition 14.5. The connection r is G-invariant, that is, for any g in G and forany tangent vector �elds X and Y ,

Lg�.rXY / D rLg�XLg�Y:

Proof. TheG-invariance of the principal connectionH implies that for any g0 2 G,

.Lg�0X/H D Lg�

0.XH /:

On the other hand, using (14.3), we get for any tangent vector �eld Z that

ZŒg� D d

dtfŒg exp.tfZ.g//�g

ˇtD0;

hence, for any smooth function f WG=H ! R,�Lg�

0Z�Œg�:f D ZŒg�1

0 g�.f ı Lg0/

D d

dtff .Œg exp.tf

Z.g�10 g//�/g

ˇtD0:

Therefore,

.Lg�0Z/Œg� D d

dtfŒg exp.tf

Lg

�0Z.g//�g

ˇtD0

D d

dtfŒg exp

�tfZ.g�10 g//�g

ˇtD0:

Using the “uniqueness” of integral curves, and differentiating at t D 0, this impliesthat for any tangent vector �eld Z

fL

g�0Z.g/ D f

Z.g�10 g/; g 2 G:

So �nally

rL

g�0XLg�

0Y Œg� D Œg; Lg�

0.XH / � f

Lg

�0Y.g/�

D Œg; XHg�1

0g

� fL

g�0Y

ı Lg0�

D Œg; XHg�1

0 g� fY�

D Œg; frXY.g�10 g/�

D Lg�0.rXY /Œg�: �


Definition 14.6. Let � WU � M ! G be a local section of the canonical bundle.G; �; G=H/ and b be the local frame U � M ! PSOn

M de�ned by

bŒg� D Œ�.Œg�/; e�:

Recall that, by Proposition 14.1,

b.Œg�/ D . xX1.Œg�/; : : : ; xXn.Œg�//;

wherexXi .Œg�/ D L�.Œg�/� ıˆ.Xi /:

We will say for short that such a local frame is induced by the local section � .

Proposition 14.7. Let Œg0� be the class of g0 2 G and let � WU � M ! G be alocal section of the canonical bundle .G; �; G=H/ such that �.Œg0�/ D g0. In thelocal frame . xX1.Œg�/; : : : ; xXn.Œg�// induced by the section � , one has

r xXi

xXj .Œg0�/ D Œg0; . zXi/g0

� f xXj�; (14.4)

where zXi are the left-invariant vector �elds on G de�ned by the Xi 2 p. Further-more,

. zXi /g0� f xXj

D d

dtf˛.exp.�tXi/g�1

0 �.Œg0 exp.tXi/�// �Xj gˇtD0: (14.5)

Proof. First, it is easy to see that the horizontal lift of the vector �eld xXi is de�nedby

xXHi .g/ D zXi .�.Œg�//;

hence (14.4), since xXHi .g0/ D zXi .g0/.On the other hand, by (14.3), the H -equivariant function f xXi

W��1.U / ! Rn

corresponding to the local vector �eld xXi veri�es

xXi .Œg�/ D L�.Œg�/� ıˆ.f xXi.�.Œg�///;

whencef xXi

.�.Œg�// D Xi :

Noting that �.Œg�/�1g 2 H , we get

f xXi.g/ D f xXi

.�.Œg�/�.Œg�/�1g/

D ˛.g�1�.Œg�//.Xi /;(14.6)

which implies (14.5). �


Proposition 14.8. The linear connection r is the Levi-Civita connection onM D G=H .

Proof. We already noted that r is a metric connection, so we only have to showthat r is torsion free. Since M is G-homogeneous and r is G-invariant, we onlyhave to verify this property at the point p ´ Œe�. Let . xXi / be the orthonormal basisof Tp.G=H/ given by

xXi D ˆ.Xi /; (14.7)

where fXig is the orthonormal basis of p we already �xed.

Let us compute the value of Tp. xXi ; xXj /. For this we introduce a local orthonor-mal frame bWU � M ! PSOnM , de�ned on an open neighborhood U of p, suchthat b.p/ D . xX1; : : : ; xXn/. Since T is a tensor, the value of Tp. xXi ; xXj / does notdepend on the choice of b, so we are free to consider the local frame induced by thefollowing local section � of the canonical bundle .G; �; G=H/. Since the map

p �! G=H;

nX

iD1xiXi 7�! Œexp.x1X1/ : : : exp.xnXn/�;

is a local diffeomorphism from a neighborhood of 0 in p onto a neighborhood U ofŒe� in G=H , we can consider the local section � WU ! G given by

�.Œexp.x1X1/ : : : exp.xnXn/�/ D exp.x1X1/ : : : exp.xnXn/: (14.8)

We �rst prove that for the local frame f xX1; : : : ; xXng induced by � ,

.r xXi

xXj /.p/ D 0:

Indeed, using (14.4) and (14.5), one gets since �.Œexp.tXi/�/ D exp.tXi/,

. xXHi � f xXj/.e/ D d

dt¹f xXj

.exp.tXi //ºˇtD0

D d

dtf˛.exp.�tXi/�.Œexp.tXi/�//.Xj /g

ˇtD0

D 0:

This implies that .r xXi

xXj /.p/ D .r xXj

xXi /.p/ D 0. Note also that wemay conclude.rg/p D 0, hence rg D 0 using homogeneity, so r is a metric connection.


We now compute Œ xXi ; xXj �.p/. Since the local �ows 'i;t of the vector �elds xXiare de�ned by 'i;t .Œg�/ D Œ�.Œg�/ exp.tXi/�, we get by de�nition of � ,

Œ xXi ; xXj �.p/ D � d

dt

d

dsf'i;t ı 'j;s ı 'i;�t .Œe�/g

ˇtD0sD0

;

D � d

dt

d

dsfŒexp.�tXi/ exp.sXj / exp.tXi/�g

ˇtD0sD0

;

D � d

dt

d

dsfŒexp.sAdexp.�tXi / �Xj /�g

ˇtD0sD0

;

D � d

dt

d

dsfŒexp.s.Adexp.�tXi / �Xj /p/�g

ˇtD0sD0

;

D �ˆ� ddt

f.Adexp.�tXi / �Xj /pgˇtD0

�;

D �ˆ�� ddt

fAdexp.�tXi / �Xj gˇtD0

�

p

�;

D ˆ.ŒXi ; Xj �p/:

So �nallyTp. xXi ; xXj / D �Œ xXi ; xXj �.p/ D �ˆ.ŒXi ; Xj �p/ D 0;

since Œp; p� � h, by Lemma 13.5. �

Wenow examine the curvature ofM . SinceM isG-homogeneous, the curvatureR is completely determined by its value at p D Œe�.

Proposition 14.9. Let f xXig be the orthonormal basis ofTp.G=H/ de�ned in .14.7/.One has

Rp. xXi ; xXj / xXk D �ˆ.ŒŒXi ; Xj �; Xk �/:Proof. SinceR is a tensor, we can choose the same local frame as in the computationof the torsion at p. Since Œ xXi ; xXj �.p/ D 0, we get

Rp. xXi ; xXj / xXk D .r xXir xXj

� r xXjr xXi

/. xXk/.p/:Now, with the notations of the above proof, we have by (14.4)

.r xXir xXj

/. xXk/.p/ D Œe; . zXi/e � . xXHj � f xXk/�:

But, by de�nition of the local section � , see (14.8), we have

. zXi /e � . xXHj :f xXk/ D d

dtf. xXHj /exp.tXi /:f xXk

gˇtD0;

D d

dtf. zXj /exp.tXi /:f xXk

gˇtD0;

D . zXi /e � . zXj :f xXk/:


Hence

Rp.Xi ; Xj /Xk D Œe; Œ zXi ; zXj �e � f xXk�:

Now since Œ zXi ; zXj �.e/ D BŒXi ; Xj �.e/, and ŒXi ; Xj � 2 h, we get by (14.6)

Œ zXi ; zXj �e � f xXkD d

dtf˛.exp.�t ŒXi ; Xj �//.Xk/g

ˇtD0;

D d

dtfAdG.exp.�t ŒXi ; Xj �//.Xk/g

ˇtD0;

D �ŒŒXi ; Xj �; Xk�:

The proof is complete. �

Using this proposition, we compute the value at p of the Ricci tensor.

Proposition 14.10. The Ricci tensor is given by

Ricp. xXi ; xXj / D �12B.Xi ; Xj /;

where B is the Killing form of g.

Proof. We follow [Bes87]. Let .�; �/g be any AdG.H/-invariant extension of thescalar product .�; �/ to g such that .h; p/g D 0. For instance, consider the Killingform sign-changed on h, �Bh, and de�ne

.�; �/g D

8ˆ<ˆ:

.�; �/ on p � p;

�Bh on h � h;

0 otherwise:

Let fEag be a basis of h such that fXi ; Eag is an orthonormal basis of .g; .�; �/g/.Then

B.Xi ; Xj / D tr.ad.Xi / ı ad.Xj //;

DX

k

.ŒXi ; ŒXj ; Xk��; Xk/g CX

a

.ŒXi ; ŒXj ; Ea��; Ea/g:


From the condition Œp; p� � h, we get

X

k

.ŒXi ; ŒXj ; Xk ��; Xk/g DX

k;a

.ŒXj ; Xk�; Ea/g.ŒXi ; Ea�; Xk/g

DX

a

.ŒXj ;X

k

.ŒXi ; Ea�; Xk/gXk �; Ea/g

DX

a

��Xj ; ŒXi ; Ea�

�; Ea

�g

DX

a

.ŒXi ; ŒXj ; Ea��; Ea/g;

where the last equality is obtained using the Jacobi identity and the AdG.H/-in-variance of .�; �/. Hence,

B.Xi ; Xj / D 2X

k

.ŒXi ; ŒXj ; Xk��; Xk/g:

From Proposition 14.9, �nally we obtain

B.Xi ; Xj / D 2X

k

.ŒXi ; ŒXj ; Xk��; Xk/g

D 2X

k

.ŒŒXk ; Xj �; Xi �; Xk/g

D �2X

k

.Rp. xXk ; xXj / xXi ; xXk/

D �2Ricp. xXj ; xXi /

D �2Ricp. xXi ; xXj /: �

Remark 14.11. Recall that for the symmetric spaces we consider, the scalar prod-uct .�; �/ on p is (eventually after a renormalization) �Bjp, (see (13.3)). Hence werecover the fact that those symmetric spaces are Einstein manifolds. Moreover, thescalar curvature Scal is given by

Scal D 1

2dimM: (14.9)


14.2 Spin structures

Theorem 14.12 ([CG88]). Let M D G=H be a compact, simply connected irre-ducible symmetric space, with corresponding triple .G;H; �/ satisfying the condi-tions of Theorem 13.4. Then,M has a spin structure .necessarily unique, sinceMis simply connected/ if and only if the homomorphism

˛WH �! SOn; h 7�! AdG.h/jp;

lifts to a homomorphism Q WH ! Spinn such that the diagram

Spinn

�

��H

Q==④④④④④④④④④④④④

˛// SOn

commutes. In this case, the spin structure is G-invariant.

Proof. The condition is suf�cient. Suppose ˛ lifts to Q WH ! Spinn. Let PSpinnM

be the principal bundle over M with structural group Spinn, associated with theprincipal bundle .G; �; G=H/ by the homomorphism Q , that is

PSpinnM D G � Q Spinn;

and let �M be the map

�M W PSpinnM D G � Q Spinn �! G �˛ SOn ' PSOnM;

Œg; u� 7�! Œg; �.u/�:

It is easy to see that .PSpinnM; �M / is a spin structure onM . Furthermore, this spin

structure is G-invariant since the left action of the group G on PSpinnM given by

g0 � Œg; u� D Œg0g; u� ;

and the right action of the group Spinn on PSpinnM given by

Œg; u� � u0 D Œg; uu0� ;

clearly commute.The condition is necessary. Here is a sketch of the proof. Suppose there exists

a spin structure .PSpinnM; �M / onM D G=H . First, as G is assumed to be simply

connected, the monodromy principle allows us to lift the action ofG onPSOnM to a

G-action on PSpinnM , which commutes with the right action of Spinn on PSpinn

M .

14.2. Spin structures 389

The induced action ofH onPSpinnM stabilizes the �bre .PSpinn

M/Œe� ofPSpinnM

at the point Œe�. Let QbŒe� be a �xed element in .PSpinnM/Œe�. For any h 2 H , denote

by h � QbŒe� the action of h on QbŒe�. Then de�ne Q.h/ 2 Spinn by the relation

h � QbŒe� D QbŒe� � Q .h/:

It can be checked that Q is a homomorphism H ! Spinn which is a lift of ˛, andthat (see [CG88]):

PSpinnM ' G � Q Spinn: �

Examples 14.13. i) For n � 2 the sphere .Sn; can/ has a unique spin structure.Recall that, up to the identi�cation H ' Spinn, the homomorphism ˛ is the two-fold covering �, so ˛ lifts trivially to Q D Id. Note that since Spinn is identi�edwith H , the spin bundle PSpinn

Sn is isomorphic to the canonical bundle

.SpinnC1; �; SpinnC1=Spinn/

via the map

PSpinnSn ´ SpinnC1 �H Spinn �! SpinnC1; Œg; h� 7�! gh:

ii) The quaternionic projective space .HPm; can/ has a unique spin structure.Recall that H is isomorphic to the group Sp1 � Spm, hence is simply connected.It then follows from the classical monodromy principle that the homomorphism˛WH ! SOn lifts to a unique homomorphism Q WH ! Spinn.

iii) The complex projective space .CPm; can/ has a (unique) spin structure ifand only if m is odd. Indeed, by the monodromy principle, ˛ has a lift if and onlyif ˛� .…1.H// is trivial. Note that the map

R � SUm �! H D S.U1 � Um/; .�; B/ 7�! ei� 0

0 e� i�m B

!;

is a covering ofH with kernel isomorphic toZ. Since R�SUm is simply connected,one has…1.H/ ' Z and any non-trivial element of…1.H/ has the form

k W Œ0; 1� �! H; t 7�! k.t / D

0B@e2ik�t 0 0

0 e�2ik�t 0

0 0 Im�1

1CA ; k 2 Z

�:


Consider the natural basis feig1�i�m of Cm and denote by f�ig1�i�2m the basis of

R2m de�ned by �2k�1 D ek , �2k D iek , 1 � k � m. From (13.6), denoting byu�i�j .�/ the rotation with angle � in the plane generated by �i and �j , it is easy tosee that

˛. k.t // D u�1�2.�4k�t/ ı u�3�4

.�2k�t/ ı � � � ı u�2m�1�2m.�2k�t/:

Now ˛� .…1.H// is trivial if and only if the lift e k starting at 1 in Spin2m of any˛. k/ satis�es e k.1/ D 1. But

e k.t / D .cos.2k�t/ � sin.2k�t/�1 � �2/mY

jD2.cos.k�t/ � sin.k�t/�2j�1 � �2j /;

so e k.1/ D .�1/k.m�1/. Hence e k.1/ D 1, for any k 2 Z� if and only if m is odd.

14.3 Spinor bundles on symmetric spaces

Proposition 14.14. Let �nW Spinn ! GLC.†n/, be the spinor representation. Con-sider the vector bundle associated to .G; �; G=H/ by the representation

e�n ´ �n ı Q ;

that is G �e�n†n. Then the spinor bundle †M is isomorphic to G �e�n

†n.

Proof. The proof is analogous to that of Proposition 14.2. By de�nition, the spinorbundle †M is the associated bundle PSpinn

M ��n †n. Let Œg� be an element inG=H and Œ QbŒg�; � an element in the �bre of PSpinn

M ��n †n over Œg�, where QbŒg�is an element of the �bre of PSpinn

M over Œg� and 2 †n. But QbŒg� has the formŒg; u�, where g is a representative of Œg� and u 2 Spinn. Now, it is straightforwardthat the element Œg; �n.u/ � � of the �bre of G �e�n

†n at Œg� depends only on theequivalence class ŒŒg; u�; �, and that the map

PSpinnM ��n †n �! G �e�n

†n; ŒŒg; u�; � 7�! Œg; �n.u/ � �;

is the claimed isomorphism between the two vector bundles. (The inverse map isgiven by Œg; � 7! ŒŒg; e�; �, g 2 G, and 2 †n). �

Corollary 14.15. Let C1H .G;†n/ be the space of H -equivariant smooth functions

G ! †n, that is, the set of functions f WG ! †n satisfying the condition

f .gh/ D e�n.h�1/ � f .g/; g 2 G; h 2 H:

The space �.†M/ of smooth sections of the bundle†M is isomorphic to the spaceC1H .G;†n/.

14.3. Spinor bundles on symmetric spaces 391

Proof. The proof is similar to that of Proposition 14.3. TheH -equivariant functionf‰ corresponding to a spinor �eld ‰ is de�ned by

‰.Œg�/ D Œg; f‰.g/�; (14.10)

where g is some representative of Œg�. �

Remark 14.16. Let X be a vector �eld on M and ‰ a spinor �eld on M . Iden-tify X with the H -equivariant function fX WG ! R

n, see (14.2), and ‰ with theH -equivariant function f‰ . Then the Clifford multiplication X �‰ has the expres-sion

.X � ‰/.Œg�/ D Œg; fX .g/ � f‰.g/�; (14.11)

where g is some representative of Œg�.

Finally, we can retrieve, in that context, the following result.

Proposition 14.17 ([Gut86]). The .complex/ spinor bundle of the sphere .Sn; can/is trivial.

Proof. This is a consequence of the description of (complex) spinor representations.Identifying Spinn with the subgroup H of SpinnC1, one gets two equivalent repre-sentations of Spinn:

� �2m and �2mC1jSpin2m, if n is even, n D 2m ;

� �2mC1 and �C2.mC1/jSpin2mC1

, if n is odd, n D 2mC 1.

Let J be the C-isomorphism†2m ! †2mC1 (resp.†2mC1 ! †C2.mC1/), such that

J ı �2m.h/ D �2mC1jSpin2m.h/ ı J; h 2 Spin2m;

(resp. J ı �2mC1.h/ D �C2.mC1/jSpin2mC1

.h/ ı J , h 2 Spin2mC1). Recall (seeExample 14.13 i) above) that the principal spinor bundle PSpinn

Sn is isomorphic to

the canonical bundle .SpinnC1; �;Sn/. It is then clear that the map

SpinnC1 ��n †n �! Sn �†n

given by

Œg; � 7�!

8<:.�.g/; J�1 ı �2mC1.g/ ı J. //; if n D 2m;

.�.g/; J�1 ı �C2.mC1/.g/ ı J. //; if n D 2mC 1,

is well de�ned and de�nes an isomorphism between the two vector bundles. �


14.4 The Dirac operator on symmetric spaces

The canonical principal connection ! on � WG ! G=H (see Section 14.1) inducesthe covariant derivative r on the associated bundle G �e�n

†n ' †M .Let Œg� be an element of G=H and g a representative of the class Œg�. Let X be

a vector �eld on M and ‰ be a section of the bundle G �e�n†n ' †M . From

Corollary 14.15, we identify ‰ with the H -equivariant function f‰WG ! †n,such that‰.Œg�/ D Œg; f‰.g/�; see (14.10). Denote byXH the horizontal lift of thevector �eld X . Then, by de�nition

.rX‰/.Œg�/ D Œg; .XH :f‰/.g/�: (14.12)

Proposition 14.18. The linear connectionr is the spinorial Levi-Civita connectionon †M .

Proof. Let � be a local section of the bundle .G; �; G=H/. It is easy to verify that

XH :f‰.�Œg�/ D .X:��f‰�Œg�

� Q�.��!.X// � f‰.�Œg�/

D .X:��f‰/Œg� � ��1� ı ˛�

��!.X// � f‰.�Œg�/:

But we saw in Section 14.1 that ˛��!.X/

�is the local expression of the Levi-

Civita connection form in the local frame bundle . xX1; : : : ; xXn/ induced by the sec-tion � , that is

˛�.��!.X// D 1

2

X

i;j

hrX xXi ; xXj iXi ^Xj ;

and to complete the proof it remains to apply Theorem 2.7. �

Remark 14.19. It should be pointed out that the G-invariance of the principal con-nectionH implies theG-invariance of the spinorial Levi-Civita connection: for anyg in G, any vector �eld X , and any spinor �eld ‰, one has

Lg.rX‰/ D rLg�XLg‰;

where Lg denotes the natural left action of G on �.†M/ given by

.Lg0‰/.Œg�/ ´ Œg; f‰.g0

�1g/�:

Nowusing (14.12) and (14.11), we get the local expression of the Dirac operator.Let Œg0� be the class of a g0 in G, and � WU � M ! G a local section of the

canonical bundle .G; �; G=H/, satisfying �.Œg0�/ D g0. Denote by f xXig the localorthonormal frame induced by � ,

xXi .Œg�/ D .L�.Œg�//�.Xi /:

14.4. The Dirac operator on symmetric spaces 393

We already noted in Section 14.1 that theH -equivariant function

f xXiW��1.U / �! R

n

corresponding to the local vector �eld xXi is de�ned by

f xXi.g/ D ˛.g�1�.Œg�//.Xi /;

so that f xXi.g0/ D Xi .

We also noted that the horizontal lift of the vector �eld xXi is de�ned byxXHi .g/ D zXi .�.Œg�//;

where zXi is the left-invariant vector �eld on G corresponding to Xi 2 p, hencexXHi .g0/ D zXi .g0/.Now, for ‰ a spinor �eld on M , let f‰ be the corresponding H -equivariant

function G ! †n. One has

D‰.Œg0�/ Dhg0;

nX

iD1Xi � . zXi :f‰/g0

i;

Hence, we get the following result.

Proposition 14.20. Via the identi�cation of the space of spinor �elds �.†M/ withthe space C1

H .G;†n/ given in Corollary 14.15, the Dirac operator D on M DG=H is the differential operator

DW C1H .G;†n/ �! C1

H .G;†n/;

‰ 7�! D‰;

de�ned by

D‰.g/ DnX

iD1Xi � . zXi :‰/g ; g 2 G; (14.13)

where fXig, i D 1; : : : ; n, is an orthonormal basis of p, and where zXi is the left-invariant vector �eld corresponding to Xi .

Remark 14.21. Denote by Lg the natural left action of G on C1H .G;†n/, that is

.Lg0�‰/.g/ ´ ‰.g0

�1g/; g0 2 G;‰ 2 C1H .G;†n/:

By theG-invariance of the spinorial Levi-Civita connection, cf. Remark 14.19 (andit is clear from (14.13), since the zXi are left-invariant vector �elds), it follows thatthe Dirac operator D is equivariant for this action, that is

Lg ı D D D ı Lg ; g 2 G:

Chapter 15

Explicit computations of the Dirac spectrum

15.1 The general procedure

We use harmonic analysis methods to determine the spectrum of the Dirac operatoronmodel spaces. The seminal work on the subject is due to R. Parthasarathy [Par71],who studied discrete series representations of a noncompact semisimple Lie groupby considering kernels of twisted Dirac operators. Later, M. Cahen and S. Guttstarted a systematic study of the Dirac operator on compact symmetric spaces,cf. [CG88] (see also [CFG89]).

In the previous chapter, we identi�ed the space of spinor �elds on G=H withthe space of H -equivariant functions C1

H .G;†n/. This is a Banach space for thesup-norm topology (the space †n being endowed with the usual (Spinn-invariant)Hermitian scalar product h�; �i). The left action of G on C1

H .G;†n/ induces a topo-logical representation which is said to be induced by the representation

e�nWH �! GLC.†n/

and is classically denoted by IndGH .e�n/.In order to obtain a continuous unitary representation in a Hilbert space, we

consider the Hermitian scalar product on C1H .G;†n/ given by

.‰;‰0/ ´Z

G

h‰.g/; ‰0.g/idg;

where dg is the Haar measure onG. The natural completion of C1H .G;†n/ for this

Hermitian scalar product is theHilbert spaceL2H .G;†n/ ofH -equivariantL2-func-tions G ! †n.

The left action of G on C1H .G;†n/ induces a continuous unitary representation

on the Hilbert space L2H .G;†n/. Being unitary, this representation is reducible,hence can be decomposed into irreducible components which turn out to be �nite-dimensional.

Definition 15.1. In the following, yG is the (countable) set of equivalence classes ofunitary irreducible �nite-dimensional complex representations of G. (Actually, thecondition to be �nite dimensional is redundant, cf. Remark 12.14). Any represen-tative of a in yG is denoted by .� ; V /.

396 15. Explicit computations of the Dirac spectrum

Let us �rst examine how an irreducible representation .� ; V / of G may becontained in the induced representation IndGH .e�n/.

It is well known that every irreducible representation .� ; V / ofG is containedin the regular left representation �reg in the space C0.G;C/ � L2.G;C/, with multi-plicity dimV . Indeed, as explained in the proof of the Peter–Weyl theorem 12.10,any non-trivial vector F 2 V �

D Hom.V ;C/ de�nes a G-equivariant injectivehomomorphism V ! C0.G;C/, de�ned by

v 7�! .g 7�! F.� .g�1/ � v//:

The following result can be considered as a generalization.

Proposition 15.2 (Frobenius reciprocity). For any 2 yG, let HomH .V ; †n/ bethe space of H -equivariant homomorphisms V ! †n, that is

HomH .V ; †n/ D fA 2 Hom.V ; †n/I A ı � .h/ D e�n.h/ ıA; h 2 H g:

The representation .� ; V / is contained in the representation

.IndGH .e�n/; C0H .G;†n//

if and only if HomH .V ; †n/ ¤ f0g. In this case, the multiplicity is

dim.HomH .V ; †n//:

In other words, denoting by ResGH .� / the restriction of .� ; V / to the sub-groupH , and counting multiplicities, we have

mult.� ; IndGH .e�n// D mult.ResGH .� /; e�n/:

Proof. Let A be a non-trivial vector in HomH .V ; †n/. For any v 2 V , it is easyto check that the function

G �! †n; g 7�! A.� .g�1/ � v/

isH -equivariant and continuous (and even smooth). Thus there exists a homomor-phism

FAWV �! C0H .G;†n/; v 7�! .g 7! A�� .g

�1/ � v�/;

which turns out to be G-equivariant, since for any g in G, FA ı � .g/ D Lg ı FA.Hence, by the Schur lemma, FA is injective (otherwise KerFA would be a non-trivialG-invariant subspace of V ). So .IndGH .e�n/; C 0H .G;†n// contains the irreducibleG-space FA.V /.

15.1. The general procedure 397

Conversely, suppose .IndGH .e�n/; C 0H .G;†n// contains a representation equiva-lent to .� ; V /: there exists a G-equivariant injective homomorphism

F WV ,�! C 0H .G;†n/:

It is easy to verify that the map

AWV �! †n; v 7�! F.v/.e/

is a non-trivial vector in HomH .V ; †n/ and that, with the previous notations,FA D F . Thus the map A 7! FA de�nes an isomorphism between HomH .V ; †n/and HomG.V ; C 0H .G;†n//. It follows that the multiplicity of .� ; V / in.IndGH .e�n/; C0H .G;†n// is dim.HomH .V ; †n//. �

Corollary 15.3. For any in yG, the -isotypical part of C0H .G;†n/ is isomorphicto V ˝ HomH .V ; †n/.

Proof. With the notations of the previous result, the map

V ˝ HomH .V ; †n/ �! C0H .G;†n/; v ˝ A 7�! FA.v/

is a G-equivariant isomorphism into the “locally �nite part” of C0H .G;†n/, thatis the G-invariant subspace of C0H .G;†n/ spanned by all the �nite-dimensionalG-spaces contained in C0H .G;†n/. �

Identifying the “locally �nite part” of C0H .G;†n/ with

M

2 yG

V ˝ HomH .V ; †n/;

a generalization of the Peter–Weyl theorem (cf. [BtD85], Theorem 5.7, Chapter III),yields

Theorem 15.4. The locally �nite part

M

2 yG

V ˝ HomH .V ; †n/

of C0H .G;†n/ is dense (for the sup-norm topology) in C0H .G;†n/. The unitaryrepresentation L2H .G;†n/ splits into the Hilbert sum

L2H .G;†n/ DM

2 yG

V ˝ HomH .V ; †n/:


Proof. The Peter–Weyl theorem implies, after tensoring with the �nite-dimensionalHermitian space †n, that the direct sum

M

2 yG

V ˝ Hom.V ; †n/

is bothL2-orthogonal (see also the orthogonality relations from Theorem 12.3) anddense inside C0.G;†n/, both for the sup norm and for the L2 norm.

Consider the linear map

PH W C0.G;†n/ �! C0.G;†n/; .PHf /.g/ ´Z

H

e�n.h/f .gh/dh:

It is immediate to see that this map takes values in the space of H -equivariantfunctions C0H .G;†n/, is continuous with respect to the C0 topology, and also withrespect to theL2 topology. Moreover, PH acts as the identity on C0H .G;†n/, and isself-adjoint with respect to theL2 inner product. It follows thatPH is the orthogonalprojector onto L2H .G;†n/.

Let v ˝ A 2 V ˝ Hom.V ; †n/ ,! C0.G;†n/. Then from the de�nition,

PH .v ˝ A/ D v ˝Z

H

e�n.h/ ı A ı � .h�1/dh 2 V ˝ HomH .V ; †n/:

Thus, if PH; WHom.V ; †n/ ! HomH .V ; †n/ denotes the orthogonal projec-tor, we have that the restriction of PH to V ˝ Hom.V ; †n/ equals 1 ˝ PH; .We have thus constructed a self-adjoint projector preserving each term of theorthogonal Hilbert sum

M

2 yG

V ˝ Hom.V ; †n/ D L2.G;†n/:

It follows thatL2H .G;†n/ (the image of this projector) decomposes into the orthog-onal Hilbert sum of the images

M

2 yG

V ˝ HomH .V ; †n/;

as claimed. �

This result may be also re-formulated as the decomposition of the Hilbert spaceL2H .G;†n/, viewed as a C0.G/-module, by means of the idempotent elementsof the convolution algebra C0.G/ de�ned by irreducible characters, cf. [BtD85],Theorem (5.10), Chapter III.


The extension of the Dirac operator to L2H .G;†n/, being G-equivariant(cf. Remark 14.21), it leaves invariant the -isotypical parts V ˝ HomH .V ; †n/(which are in the domain ofD, since they consist of C1-functions). More precisely,we have the following result.

Proposition 15.5. The Dirac operator leaves invariant the space

V ˝ HomH .V ; †n/

andDjV ˝HomH .V ;†n/ D Id ˝ D ;

where D is the operator

D W HomH .V ; †n/ �! HomH .V ; †n/;

A 7�! �nX

iD1Xi � .A ı � �.Xi //;

fXig being some orthonormal basis of p.

Proof. Let v ˝ A be an element in V ˝ HomH .V ; †n/. From (14.13), one has

D.v ˝ A/ DnX

iD1Xi � . zXi � .v ˝ A//:

But

zXi � .v ˝ A/.g/ D d

dtfA.� .exp.�tXi/g�1/ � v/g

ˇtD0;

D A�� ddt

f� .exp.�tXi//gˇtD0 ı � .g�1/

�� v�

D �.A ı � �.Xi //.� .g�1/ � v/:

Hence

D.v ˝ A/ D v ˝�

�nX

iD1Xi � .A ı � �.Xi //

�: �

Note that sinceD is formally self-adjoint, the restriction ofD to any non-trivialspace V ˝ HomH .V ; †n/ is a Hermitian operator on the �nite-dimensionalHermitian space

.V ˝ HomH .V ; †n/; .�; �/jV ˝HomH .V ;†n//;


hence, is diagonalizable with real eigenvalues. By Theorem 15.4 and the aboveresult, one concludes that

Spec.D/ D[

2 yG

Spec.DjV ˝HomH .V ;†n// D[

2 yG

Spec.D /:

(This implies the well-known property that D has a discrete and real spectrum).But because of the symmetric space structure, one can proceed further.

Proposition 15.6. The spectrum of D is symmetric with respect to the origin.Furthermore it is completely determined by the spectrum of its square D2.

Proof. Let� be an eigenvalue ofD and let‰ be a non-trivial function in C1H .G;†n/

such that D‰ D �‰. Consider the function ��‰, where � is the involutive auto-morphism of G de�ning the symmetric structure. Since

�.exp.tXi // D exp.t��.Xi // D exp.�tXi/;cf. (13.1), for any g 2 G one gets

zXi .��‰/.g/ D d

dtf‰.�.g exp.tXi///g

ˇtD0

D d

dtf‰.�.g/�.exp.tXi///g

ˇtD0

D d

dtf‰.�.g/ exp.�tXi//g

ˇtD0

D � zXi .‰/.�.g//:Hence,

D.��‰/.g/ D �D‰.�.g// D ��‰.�.g// D ��‰.g/;

so the spectrum of D is symmetric with respect to the origin.The operator D2 is also G-equivariant, hence the spaces V ˝ HomH .V ; †n/

is invariant, and by Proposition 15.5

D2jV ˝HomH .V ;†n/ D Id ˝ D2 : (15.1)

Thus

Spec.D2/ D[

2 yG

Spec.D2jV ˝HomH .V ;†n// D

[

2 yG

Spec.D 2/:

As noted before, there exists an orthonormal basis of V ˝ HomH .V ; †n/ whichdiagonalizesDjV ˝HomH .V ;†n/; but it also diagonalizesD

2jV ˝HomH .V ;†n/. Thisleads to the conclusion that

Spec.D/ D[

2 yG

f˙p�I� 2 Spec.D

2/g: �


Now, the following formula for the square ofD holds. This is a particular caseof the Parthasarathy formula (cf. Proposition 3.1 in [Par71]).

Proposition 15.7 (Parthasarathy formula). For any A in HomH .V ; †n/,

D2 A D A ı C.� /C Scal

8A;

where C.� / is the Casimir operator of the representation � .

Proof. We follow S. Sulanke [Sul79]. We �rst recall the expression of the Casimiroperator. Let fEag be a basis of h such that fXi ; Eag is an orthonormal basis of.g;�B/, B being the Killing form of g. The Casimir operator of a representation.�; V / is the endomorphism of V

C.�/ ´ �X

i

��.Xi / ı ��.Xi / �X

a

��.Ea/ ı ��.Ea/:

Now, for A in HomH .V ; †n/,

D 2A D

X

i;j

Xi �Xj � .A ı ��.Xj / ı ��.Xi //

D 1

2

X

i;j

Xi �Xj � .A ı ��.Xj / ı ��.Xi //

C 1

2

X

i;j

Xj � Xi � .A ı ��.Xi / ı ��.Xj //

D �12

X

i;j

Xi � Xj � .A ı ��.ŒXi ; Xj �//

�X

i

A ı ��.Xi / ı ��.Xi /

D �12

X

i;j

Xi � Xj � .A ı ��.ŒXi ; Xj �//

C A ı C.� /

CX

a

A ı ��.Ea/ ı ��.Ea/:


But

A ı ��.ŒXi ; Xj �/ D e�n�.ŒXi ; Xj �/ ı A;

since ŒXi ; Xj � 2 h and A is H -equivariant. Moreover, recall that

e�n� D �n� ı Q�;

Q� D ��1 ı ˛�;

and

˛�.X/ D ad.X/jp; X 2 h:

Hence,

˛�.ŒXi ; Xj �/ D �12

X

k;l

B.ŒŒXi ; Xj �; Xk�; Xl/Xk ^Xl

and

Q�.ŒXi ; Xj �/ D �14

X

k;l

B.ŒŒXi ; Xj �; Xk�; Xl /Xk �Xl :

Thus

X

i;j

Xi � Xj � e�n�.ŒXi ; Xj �/ D �14

X

i;j;k;l

B.ŒŒXi ; Xj �; Xk �; Xl/Xi �Xj � Xk �Xl :

On the other hand,

X

a

A ı ��.Ea/ ı ��.Ea/ DX

a

e�n�.Ea/ ı e�n�.Ea/ ıA:

But

˛�.Ea/ D �12

X

i;j

B.ŒEa; Xi �; Xj /Xi ^Xj

and

Q�.Ea/ D �14

X

i;j

B.ŒEa; Xi �; Xj /Xi � Xj :


Consequently,X

a

e�n�.Ea/ ı e�n�.Ea/

D 1

16

X

a

X

i;j;k;l

B.ŒEa; Xi �; Xj /B.ŒEa; Xk �; Xl/Xi �Xj �Xk � Xl

D � 1

16

X

i;j;k;l

B�X

a

B.ŒXi ; Ea�; Xj /ŒEa; Xk�; Xl�Xi � Xj �Xk �Xl

D 1

16

X

i;j;k;l

B�X

a

B.Ea; ŒXi ; Xj �/ŒEa; Xk�; Xl�Xi �Xj � Xk �Xl

D � 1

16

X

i;j;k;l

B.ŒŒXi ; Xj �; Xk�; Xl/Xi �Xj �Xk �Xl ;

the third equality being obtained using the Ad.G/-invariance of the Killing form.Thus

D2 A D A ı C.� /C

� 116

X

i;j;k;l

B.ŒŒXi ; Xj �; Xk�; Xl /Xi �Xj �Xk �Xl�

ı A:

Now, we can avoid super�uous computations by observing that according to Propo-sition 14.9, the term inside the parentheses in the above equality is equal to

1

16

X

i;j;k;l

Rp. xXi ; xXj ; xXk; xXl/ xXi � xXj � xXk � xXl ;

where xXi D ˆ.Xi /, ˆ being the isometry between .p;�Bjp/ and .TpM;gM;p/de�ned in (13.2). By the proof of the Schrödinger–Lichnerowicz formula, cf. (2.49),the above expression is equal to Scal

8Id. �

It is also well known that the Casimir operator of a representation .�; V / ofG commutes with any element of �.g/. Hence, by the Schur lemma, the Casimiroperator of an irreducible representation .� ; V / is a scalar multiple of the identity

C.� / D c Id;

the constant c depending only on the equivalence class of � . Since Scal D12dimM , cf. (14.9), we can re-formulate the above proposition as

Proposition 15.8. For any A 2 HomH .V ; †n/, one has

D2 A D

�c C dimM

16

�A;

where c is the eigenvalue of the Casimir operator of � .


In order to conclude, we only have to determine those of the HomH .V ; †n/which are non-trivial.

Proposition 15.9. Let

.e�n; †n/ DpM

iD1.e�n;i ; †n;i /

be the decomposition into irreducible components of the representation

e�nWH ! GLn;C.†n/:

For any i D 1; : : : ; p, letmult.e�n;i ;Res

HG .� //

be the multiplicity of the irreducible representation e�n;i in the restriction ResHG .� /

of the representation � to H . Then

dim.HomH .V ; †n// DpX

iD1mult.e�n;i ;Res

HG .� //:

Proof. Consider

.ResHG .� /; V / DqM

iD1.ResHG .� /i ; V ;i/;

the irreducible decomposition of the representation .ResHG .� /; V /. For any i D1; : : : ; q, one identi�es HomH .V ;i ; †n/with a subspace of HomH .V ; †n/ by con-sidering the injective homomorphism

HomH .V ;i ; †n/ �! HomH .V ; †n/;

Ai 7�! A �

8<:Ai on V ;i ,

0 on V ;j , j 6D i .

Under this identi�cation, one obtains the decomposition

HomH .V ; †n/ DqM

jD1HomH .V ;j ; †n/:

Thus,

HomH .V ; †n/ DqM

jD1

� pM

iD1HomH .V ;j ; †n;i/

�;


and

dim.HomH .V ; †n// DpX

iD1

� qX

jD1dim.HomH .V ;j ; †n;i//

�:

But by the Schur lemma, dim.HomH .V ;j ; †n;i// D 1 if V ;j and †n;i are equiva-lent, and zero otherwise, so, in any case,

dim.HomH .V ;j ; †n;i// D dim.HomH .†n;i ; V ;j //:

We conclude that

qX

jD1dim.HomH .V ;j ; †n;i // D

qX

jD1dim.HomH .†n;i ; V ;j //;

is the multiplicity of e�n;i in ResHG .� /. �

We can summarize all the previous results as

Theorem 15.10. Let .M; gM / be a spin compact, simply connected, n-dimensionalirreducible symmetric space and .G;H; �/ the corresponding accurate triple:M D G=H , with G compact and simply connected, and � WG ! G is an invo-lution such thatH is the identity component of the subgroup of �xed elements of � .

Denote by

†n DpM

iD1†n;i

the decomposition of the spinor space †n into irreducible H -modules, under theaction of the groupH induced by the spin structure.

The spectrum of the Dirac operator is

Spec.D/ Dn

˙rc C n

16I

2 yGW there exists i 2 f1; : : : ; pgsuch that mult.V jH ; †n;i/ � 1o;

where yG is the set of equivalence classes of irreducible unitary complex representa-tions ofG, c the eigenvalue of the Casimir operator of any representative .� ; V /of 2 yG, and mult.V jH ; †n;i/ is the multiplicity of the irreducible H -module†n;i in theH -module V .


Remark 15.11. From (15.1) and Proposition 15.8, it is clear that the multiplicity ofan eigenvalue of D2 of the form c C n

16is at least equal to

dim.V / dim.HomH .V ; †n//I

it may be greater, since two distinct elements and 0 in yG may verify c D c 0

(cf. for instance the example of complex projective spaces below).

Thus, the spectrum can be in principle determined by means of the followingsteps.

(1) Determine the sets yG and yH : equivalence classes of irreducible representa-tions can be described in terms of dominant weights relative to the choice ofa maximal torus.

(2) Determine the “branching rules” giving the decomposition of an irreducibleG-representation into irreducible H -representations. This is a classical prob-lem in representation theory, but far from being obvious in general: only afew results are known for classical groups.

(3) Decompose the representation

e�nWH �! GLC.†n/;

into irreducibleH -representations. This can be done by determining the max-imal vectors (cf. Theorem 12.58) of the representation, or by using the fol-lowing result of R. Parthasarathy: an H -dominant weight ˇH occurs in thedecomposition if and only if there exists an element w in the Weyl group ofG such that ˇH D w � ıG � ıH , where ıG (resp. ıH ) is the half-sum of thepositive roots of G (resp. H ). Furthermore, it occurs with multiplicity one(cf. Lemma 2.2. in [Par71]).

(4) Using Proposition (15.9), determine the ’s in yG such that HomH .V ; †n/ ¤f0g.

(5) For those , compute the Casimir eigenvalue c . By Freudenthal’s formula,c can be computed from the dominant weight ˇ :

c D hˇ ; ˇ C 2ıGi;

where h�; �i is the natural extension of the Killing form of g sign-changed tothe space of weights i t�.

15.2. Spectrum of the Dirac operator on spheres 407

15.2 Spectrum of the Dirac operator on spheres

Using the above method, S. Sulanke computed in [Sul79] the spectrum of the Diracoperator on the sphere .Sn; can/, n � 2. The two cases n even and n odd have to beconsidered separately.

First, let us consider the even case, n D 2m. Relative to the standard maximaltorus, dominant weights of the group Spin2mC1 are of the form .k1; k2; : : : ; km/,where the ki are all simultaneously integers or half-integers satisfying the conditionk1 � k2 � � � � � km�1 � km � 0, cf. Section 12.4.2.

Under the identi�cation H D Spin2m, the representation e�2m is the spin rep-resentation (cf. Example (14.13, i)), which is known to split into two irreduciblecomponents �˙

2m. Using the branching rules of the group Spin2mC1 and its sub-group Spin2m, given in [Žel73], Theorem 2, Chapter XVIII, (see also [Bra97b]and [Bra99]) one shows that the restriction to H of an irreducible representationof Spin2mC1 contains one of the representations �˙

2m if and only if its dominantweight has the form

1

2.2k C 1; 1; : : : ; 1/; k 2 N:

In this case, each representation occurs with multiplicity one, hence

dim.HomH .V ; †2m// D 2:

Denoting by k the element of yG with dominant weight 12.2kC1; 1; : : : ; 1/, k 2 N,

the eigenvalues of D2 are given by

c kC n

16D 1

2.2m� 1/.mC k/2:

Furthermore, since c k¤ c k0 if k ¤ k0, the multiplicity of 1

2.2m�1/ .m C k/2 is2 dim.V k

/, cf. Remark 15.11, and theWeyl dimension formula (cf. Theorem 12.67)gives the value of dim.V k

/:

dim.V k/ D 2m

�2mC k � 1

k

�:

Let us now consider the odd case, n D 2m � 1, m � 1. Relative to the stan-dard maximal torus, the dominant weights of the group Spin2m are of the form.k1; k2; : : : ; km/, where the ki are all simultaneously integers or half-integers veri-fying the condition k1 � k2 � � � � � km�1 � jkmj, cf. Section 12.4.2.

Under the identi�cation H D Spin2m�1, the representation A�2m�1 is the spinrepresentation, which is irreducible. Using the branching rules of the group Spin2mand its subgroup Spin2m�1, given in [Žel73], Theorem 3, Chapter XVIII, one can


show that the restriction to H of an irreducible representation of Spin2m containsthe representation �2m�1 if and only if its dominant weight has the form

1

2.2k C 1; 1; : : : ; 1/ or

1

2.2k C 1; 1; : : : ;�1/; k 2 N:

Furthermore, the multiplicity of �2m�1 is equal to one in each case. Denoting by ˙k

the element in yG corresponding to the dominant weight 12.2k C 1; 1; : : : ;˙1/,

the eigenvalues of D2 are given by

c

Ck

C n

16D c �

kC n

16D 1

4.m� 1/

�2m� 1

2C k

�2:

Since c ˙

k

¤ c ˙

k0, if k ¤ k0, the multiplicity of the eigenvalue 1

4.m�1/ .2m�12

C k/2

is

dim.V

Ck

/C dim.V �k/ D 2m

�2mC k � 2

k

�:

Thus, thanks to the symmetry of the spectrum of the Dirac operator, we deduce thefollowing theorem.

Theorem 15.12 ([Sul79]). The eigenvalues of the Dirac operator on .Sn; can/, are

˙�n2

C k�; k 2 N;

with multiplicity

2Œn2 �

�nC k � 1

k

�:

Note that the lowest eigenvalue of D2 is n2

4D n

n�1Scal4.

There are other approaches to compute the spectrum and determine multiplici-ties. One is due to C. Bär, cf. [Bär91] and [Bär94]. It is based on the fact that thespinor bundle †S

n (which is known to be trivializable, cf. Proposition 14.17), canbe trivialized by Killing spinors for the constant � D 1

2as well as � D �1

2. This is

shown by proving that the curvature of the modi�ed connection de�ned by

zrX‰ ´ rX‰ � �X � ‰;

vanishes at every point.

15.3. Spectrum of the Dirac operator on the complex projective space 409

Let .‰j / be a trivialization of the bundle†Sn byKilling spinors with constant�.

From the Schrödinger–Lichnerowicz-type formula

.D C �/2 D zr� zr C 1

4.n� 1/2;

one deduces, considering an orthogonal L2 eigenbasis f0 � 1; f1; f2; : : : of theLaplace operator, that the functions fi‰j form an orthogonal L2 eigenbasis of theoperator .D C �/2. Using the explicit expression of the eigenvalues and multiplic-ities of the Laplace operator on S

n, the eigenvalues and multiplicities of the Diracoperator on S

n are then derived from those of the operator .D C �/2. See [Bär91]or [Bär94] for details.

Another approach is due to A. Trautman [Tra93] and [Tra95]. It is based on thetheory of spin (pin) structures on hypersurfaces (the spin structure on the sphere.Sn; can/ is inherited from the standard spin structure on R

nC1) and on a formularelating the Dirac operator on R

nC1 (restricted to Sn) to the Dirac operator on

Sn, which is analogous to the formula relating the respective Laplace operators.

The explicit determination of the eigenvalues, multiplicities, as well as a descrip-tion of the corresponding eigenspinors is given in [Tra95].

Onemore approach we know is that introduced by R. Camporesi and A. Higuchiin [CH96]. It is based on a separation-of-variables technique in geodesic polarcoordinates which allows to express eigenfunctions onS

n in terms of those onSn�1.

By an induction procedure, the spectrum of the Dirac operator on Sn is derived from

that on S2. We refer to [CH96] for details.

15.3 Spectrum of the Dirac operator on the complexprojective space

The “Harmonic Analysis” method also provides the spectrum of the Dirac oper-ator on the complex projective space .CPmD2qC1; can/, q � 0. It was given byM. Cahen, A. Franc, and S. Gutt in [CFG89] and [CFG94].

The standard maximal torus T ofG D SUmC1 (cf. Section 12.4.1) is a maximalcommon torus of G and H D S.U1 � Um/. Relative to this torus, the dominantweights of G are of the form .k1; : : : ; km/ where the ki are integers verifying thecondition k1 � k2 � � � � � km � 0, whereas the dominant weights of H are ofthe form .`1; : : : ; `m/, where the ì are integers verifying the condition `2 � `3 �� `m � 0 (the dominant weights of H do not have the same expression asin [CFG89] since our choice of the isotropy subgroupH is different).


The decomposition of the representation

e�2mWS.U1 � Um/ �! GLC.†2m/

into irreducible components is given by

†2m DmM

kD0†2m;k ; (15.2)

where †2m;k is the H -subspace corresponding to the dominant weight

ˇk D .q C 1 � k; 1; : : : ; 1„ ƒ‚ …k

; 0; : : : ; 0/; 0 � k � m � 1;

ˇm D .�.q C 1/; 0; : : : ; 0/:

In the basis of†2m introduced in Section 12.4.2, the corresponding maximal vectoris w for ˇ0 and u1 � � �uk � w for ˇk , 1 � k � m.

Remark 15.13. Decomposition (15.2) induces the decomposition of the spinor bun-dle of the spin Kähler manifold CPm into eigenspaces under the action of the Kählerform, cf. Lemma 6.4.

Indeed, the local expression of the Kähler form is invariant under the action ofthe group Um, hence, under the action of the group S.U1�Um/. Thus, viewed as anendomorphism of†2m, it leaves invariant all the subspaces†2m;k andmoreover, bythe Schur lemma, its restriction to any of those subspaces is a scalar multiple of theidentity. Computation of its action on maximal vectors shows that the eigenvaluecorresponding to the subspace †2m;k is i.2k �m/.

The branching rules of the group SUmC1 and its subgroup S.U1�Um/ are givenin [IT78]; they can also be derived from the branching rules given in [CFG89].

Using this result, one shows that an irreducibleG-representation � has the prop-erty that m.� jH ; †2m;k/ ¤ 0, for some integer k, 0 � k � m, if and only if itsdominant weight ˇ has the form

ˇ D .2l C q � k; l; : : : ; l„ ƒ‚ …k�1

; l � 1; : : : ; l � 1/;

where 1 � k � m and l is an integer such that l � maxf1; k � qg, or

ˇ D .2l C q C 1 � k; l C 1; : : : ; l C 1„ ƒ‚ …k

; l; : : : ; l/;

where 0 � k � m� 1 and where l is an integer such that l � maxf0; k � qg. Thusfrom Theorem 15.10, it can be deduced

15.3. Spectrum of the Dirac operator on the complex projective space 411

Theorem 15.14 ([CFG89, SS93, CFG94]). The Dirac operator on .CPm; can/,m odd, has the eigenvalues ˙2

p�l;k , ˙2

p�l;k , where

�l;k D l2 C 1

2l.3m � 2k � 1/C 1

2.m � k/.m � 1/

D�l C m � 1

2

�.l Cm � k/;

with k 2 f1; : : : ; mg, and l 2 N such that l � maxf1; k � m�12

g, and

�l;k D l2 C 1

2l.3m � 2k C 1/C 1

2.m � k/.mC 1/

D�l C mC 1

2

�.l Cm � k/;

with k 2 f0; : : : ; m� 1g, and l 2 N such that l � maxf0; k � m�12

g.

Note that for m D 1 one recovers the spectrum of .S2; 14can/ ' .CP1; can/,

and that the lowest eigenvalue of D2 is

4�1;mC1

2D 4�0;m�1

2D .mC 1/2 D mC 1

m

Scal

4:

There are relations between the two series �l;k and �l;k , for instance

�lC1;kC1 D �l;k D �l�kC m�12 ;m�k�1:

For this reason, it seems dif�cult to deduce multiplicities (using Remark 15.11),since distinct elements and 0 in yG may have the same Casimir eigenvalue c .

Using a different approach, a formula for multiplicities was obtained by B. Am-mann and C. Bär (as an example) in [AB98]. Indeed, endowing the sphere S

2mC1

with the Berger metric g` with parameter `, the Hopf �bration

S1 �! S

2mC1 �! CPm;

becomes a Riemannian submersion with totally geodesic �bers of length 2�`.It turns out that, form odd, upon collapsing the �bers of this S

1-principal bundle(that is, making ` tend to zero), eigenvalues of the Dirac operator on .CPm; can/ areexactly the limits of the eigenvalues of the Dirac operator of .S2mC1; g`/ that havea �nite limit.

From the explicit determination of the eigenvalues andmultiplicities of the Diracoperator on .S2mC1; g`/ given by C. Bär in [Bär96], one obtains


Theorem 15.15 ([AB98]). The Dirac operator on .CPm; can/, m odd, has theeigenvalues

˙2r�

a1 C mC 1

2

��a2 C mC 1

2

�;

where a1; a2 2 N�, ja1 � a2j � m�1

2, with multiplicity

.a1 Cm/Š.a2 Cm/Š.a1 C a2 CmC 1/

a1Ša2Š�a1 C mC1

2

� �a2 C mC1

2

�mŠ�a1 � a2 C m�1

2

�Š�a2 � a1 C m�1

2

�Š:

15.4 Spectrum of the Dirac operator on the quaternionicprojective space

The spectrum of theDirac operator on the quaternionic projective space .HPm; can/,m � 1, was computed in [Mil92] using the above “Harmonic Analysis” method.

The standard maximal torus T of G D SpmC1 (cf. Section 12.4.3) is a maximalcommon torus of G and H D Sp1 � Spm. Relative to this torus, the dominantweights of G are of the form .k1; : : : ; kmC1/, where the ki are integers verifyingthe condition k1 � k2 � � � � � kmC1 � 0, whereas the dominant weights of H areof the form .`1; : : : ; `mC1/, where the ì are integers verifying the condition `1 � 0

and `2 � `3 � � � � � `mC1 � 0. The decomposition of the representation

e�4mW Sp1 � Spm 7�! GLC.†4m/;

into irreducible components is given by

†4m DmM

kD0†4m;k ; (15.3)

where †4m;k is the H -subspace corresponding to the dominant weight

ˇk D .k; 1; : : : ; 1; 0; : : : ; 0„ ƒ‚ …k

/:

Remark 15.16. Decomposition (15.3) was previously stated, in a different context,in [BS83] and [Wan89]. As in the Kähler case, it corresponds to the decompositionof the spinor bundle of the spin quaternion-Kähler manifold HPm into eigenspacesunder the action of the fundamental 4-form, cf. Section 7.2.

15.4. Spectrum of the Dirac operator on the quaternionic projective space 413

Indeed, the local expression of the fundamental 4-form is invariant under theaction of the group Sp1 � Spm, hence, under the action of the group Sp1 � Spm. Thus,viewed as an endomorphism of†4m, it leaves invariant all the subspaces†4m;k andmoreover, by the Schur lemma, its restriction to any of those subspaces is a scalarmultiple of the identity.

Computation of its action on maximal vectors, shows that the eigenvalue corre-sponding to the subspace †4m;k is 6m � 4k.k C 2/ (see [HM95b] for details).

The branching rules of the group SpmC1 and its subgroup Sp1 � Spm are givenin [Lep71] and [Tsu81]. Using the result of [Tsu81], one shows that an irreducibleG-representation � is such that m.� jH ; †4m;k/ ¤ 0, for some integer k,0 � k � m, if and only if its dominant weight ˇ has the form

ˇ D .p C q � 1; q; 1; : : : ; 1; 0; : : : ; 0„ ƒ‚ …p

/;

where 1 � p � m � 1, and q 2 N�, or

ˇ D .p C q C 1; q; 1; : : : ; 1; 0; : : : ; 0„ ƒ‚ …p

/;

where 0 � p � m � 1, and q 2 N if p D m � 1, q 2 N� otherwise. Thus, from

Theorem 15.10, one can deduce

Theorem 15.17 ([Mil92]). The Dirac operator on .HPm; can/ has the eigenvalues˙p�p;q , ˙p

�p;q , where

�p;q D .2mC p C 2q/2 � .p C 2/2

D 4.mC p C q C 1/.mC q � 1/;

where 1 � p � m � 1 and q 2 N�,

�p;q D .2mC p C 2q C 1/2 � .p � 1/2

D 4.mC p C q/.mC q C 1/;

where 1 � p � m; and q 2 N if p D m, q 2 N� otherwise.

Note that for m D 1, one recovers the spectrum of .S4; 14can/ ' .HP1; can/,

and that the lowest eigenvalue of D2 is

�1;1 D 4m.mC 3/ D mC 3

mC 2

Scal

4:


There are relations between the two series �l;k and �l;k ; for instance we have�p;q D �pC3;q�2. For this reason, it seems dif�cult to deduce multiplicities withthis method. Up to our knowledge, there is no known formula giving multiplicities.

15.5 Other examples of spectra

There is a generalization of the above method to spin compact homogeneous spacesdue to C. Bär, cf. [Bär91] and [Bär92a].

All results about spin structures and spinor bundles are mainly the same as inChapter 14, but the Dirac operator has a more complicated expression cf. [Bär92a],Theorem 1), because of the expression of the Levi-Civita connection (cf. [KN69]Theorem 3.3, Chapter X).

All the harmonic analysis results of Chapter 15 remain obviously true, but the ex-pression ofD in Proposition 15.5 has to be changed to (cf. Proposition 1, [Bär92a])

D .A/ D �nX

iD1Xi � .A ı � �.Xi //

C� nX

iD1ˇiXi C

X

1�i<j<k�n˛ijkXi �Xj �Xk

�� A;

where

˛ijk D 1

4.hŒXi ; Xj �p; Xki C hŒXj ; Xk �p; Xii C hŒXk ; Xi �p; Xj i/

and

ˇi D 1

2

nX

jD1hŒXj ; Xi �p; Xj i:

The main difference with respect to the “symmetric” case, is that we cannotproceed further here, since there is no result analogous to Theorem 15.10. With thehelp of the above formula, C. Bär computed the spectrum of the Dirac operator onodd-dimensional spheres with Berger metrics [Bär96].

Actually, there are only few examples where Dirac spectra can be computedexplicitly. Examples, known to us, of Dirac spectra on closed Riemannian manifoldsare listed in Table 6.

15.5. Other examples of spectra 415

Table 6

Spin manifold References Remarks

�at tori Rn=� [Fri84] homogeneous space

sphere .Sn; can/ [Sul79], [Bär94], [Tra93], irreducible

[Tra95], [CH96] symmetric space

spherical space form Sn=� [Bär94] homogeneous space

sphere S2mC1 [Hit74], form D 1 homogeneous space

with Berger metric [Bär96], for generalm

3-dimensional lens space S3=.Z=kZ/ [Bär92a] homogeneous space

with the Berger metric

3-dimensional (�at) [Pfä00] homogeneous space

Bieberbach manifolds

some n-dimensional (�at) [MP06] homogeneous space

Bieberbach manifolds

n-dimensional Heisenberg [AB98] homogeneous space

manifolds

SU2=Q8, Q8 �nite group of quaternions, endowed [Gin08] homogeneous space

with a three-parameter family of homogeneous

metrics, and for the four different spin structures

�nPSL2.R/ [SS87] homogeneous space

� co-compact Fuchsian group

simply connected compact [Feg87] symmetric

Lie groupG with canonical metric space

induced by the Killing form sign-changed

complex projective space [CFG89],[CFG94] Kähler

.CP2mC1; can/ [SS93] irreducible

symmetric space

quaternionic projective space [Bun91b], form D 2 quaternion-Kähler

.HPm; can/ [Mil92], for generalm irreducible

symmetric space

real Grassmannian Gr2.R2m/ [Str80b], form D 3 Kähler

with metric induced by [Str80a], for generalm irreducible

the Killing form sign-changed symmetric space

real Grassmannian Gr2p.R2m/ [See97] irreducible

with metric induced by symmetric space,

the Killing form sign-changed Kähler if and

only if p D 1

complex Grassmannian Gr2.CmC2/ [Mil98] Kähler and

(m even), with metric induced by quaternion-Kähler

the Killing form sign-changed irreducible

symmetric space

G2=SO4 , with metric induced by [See99] quaternion-Kähler

the Killing form sign-changed irreducible

symmetric space


The method described above has also been used to compute the Dirac spectrumof the compacti�ed Minkowski space S

1� S3, with the metric r2canS1 �R2canS3 ,

where r; R are two real parameters, cf. [OV93]. This is the only example known tous in the pseudo-Riemannian case.

As noted before, only few results are known, since even the decomposition ofthe spinor representation under the action of the subgroupH is far from being easyin general (see for instance the partial results obtained in [Kli07] for Grassmannmanifolds).

However, the �rst eigenvalue may be explicitly obtained by using analogousmethods.

Theorem 15.18 ([Mil05]). Let G=H be a compact, simply connected, n-dimen-sional irreducible symmetric space withG compact and simply connected, endowedwith the metric induced by the Killing form of G sign-changed. Assume that G andH have the same rank and that G=H has a spin structure. Let ˇk , k D 1; : : : ; p,be the H -dominant weights occurring in the decomposition into irreducible com-ponents of the spin representation under the action of H . Then the square of the�rst eigenvalue of the Dirac operator is

2 min1�k�p

kˇkk2 C n=8;

where k�k is the norm associated with the scalar product h�; �i induced by the Killingform of G sign-changed.

The proof is based on a lemma of Parthasarathy given in [Par71] and alreadymentioned above. The formula has been developed further, involving only dataof a root system of G, in order to obtain a more ef�cient expression for explicitcomputations.

Theorem 15.19 ([Mil06]). Under the same assumptions as in the preceding theo-rem, let ˆ be the set of non-zero roots of G with respect to the maximal torus T ,common to G and H . Let ˆC

G be the set of positive roots of G and ˆCK , the set

of positive roots of K, with respect to a �xed lexicographic ordering in ˆ. Let ıG(resp. ıK) be the half-sum of the positive roots of G .resp. K/. Then the square ofthe �rst eigenvalue of the Dirac operator is given by

2kıG � ıKk2 C 4X

�2ƒh�; ıKi C n

8;

where ƒ is the setƒ ´ f� 2 ˆC

G I h�; ıKi < 0g;the scalar product h�; �i on weights being induced by the Killing form of G sign-changed.


As an example, the formula is used in [Mil06] to obtain the list of the �rsteigenvalues of the Dirac operator for the spin compact irreducible symmetric spacesendowed with a quaternion-Kähler structure; see Table 7.

Table 7

G=K Square of the �rst eigenvalue of D

HPn D SpmC1=.Spm � Sp1/mC 3

mC 2

m

2D mC 3

mC 2

Scal

4

Gr2.CmC2/ D SUmC2=S.Um � U2/mC 4

mC 2

m

2D mC 4

mC 2

Scal

4

(m even)

eGr4.RmC4/ D SpinmC4=SpinmSpin4

m2 C 6m� 4m.mC 2/

m

2D m2 C 6m� 4

m.mC 2/

Scal

4

(m even)

G2=SO4

3

2D 3

2

Scal

4

E6=.SU6SU2/41

6D 41

30

Scal

4

E7=.Spin12SU2/95

9D 95

72

Scal

4

E8=.E7SU2/269

15D 269

210

Scal

4

We end by noting that harmonic analysis methods can also be used in thenon-compact case. Here again, we may restrict to simply connected non-compactirreducible symmetric spaces. Since their sectional curvature is non-positive,cf. [Hel78], they are diffeomorphic to R

n by the Hadamard–Cartan theorem, henceposses a unique spin structure.

The standard example of non-compact simply connected irreducible space is thereal hyperbolic space

Hn.R/ D Spin1;n=Spinn;

with the metric induced by the Killing form of Spin1;n.Its Dirac spectrum has been computed by U. Bunke [Bun91a] using harmonic

analysis methods. We will not describe this method here, because harmonic analy-sis on non-compact homogeneous spaces is far more complicated than the compactcase. But the approach is mainly the same as in the compact case. It is based on the


decomposition of the unitary representation L2HDSpinn.G D Spin1;n; †n/, but in-

stead of the result given in Theorem 15.4, here the representation has to be decom-posed into a “direct integral” of irreducible ones, because the set of equivalenceclasses of irreducible G-representations (which need not to be �nite dimensional)does not need to be countable.

As in the compact case, the computation of the spectrum of the Dirac operatorD is derived from the study of the restriction of its square D2 to the componentsof the above decomposition. Here again a key role is played by the Parthasarathyformula, cf. Proposition 15.7, which brings one back to study the action of theCasimir operator of Spin1;n on the components.

We refer to [Bun91a] for details. Note however that there is an incorrect state-ment concerning the eigenvalue 0 (the correct statement using harmonic analysismay be found in [CP99]).

There are other approaches to compute the spectrum of the Dirac operator.Camporesi and Higuchi [CH96] obtain the spectrum by a separation-of-variablestechnique in geodesic polar coordinates and con�rm the computation using theabove harmonic analysis method.

Another approach is given by C. Bär in [Bär00], second remark in Paragraph 3:it is based on the fact that

yHn.R/ ´ .Hn.R/ with a point removed/;

is isometric to the warped product

Sn�1 � .0;1/;

with the warped product metric

ds2.x; t / D %.t/2canSn�1.x/C dt2; where %.t/ D sinh.t /:

It is then proved that the square of the Dirac operator D2 on yHn.R/ is unitarilyequivalent to

L�L�, where the sum is taken over all eigenvalues � of the Dirac

operator on .Sn�1; can/, and where L� is the operator on L2..0;1/;C; dt/ givenby

L� D � d2

dt2C � cosh.t /C �2

sinh2.t /:

This leads to the computation of the spectrum of D2 on yHn.R/, and the spectrumof D2 on Hn is then derived using a “decomposition principle” which allows tocompare the spectra on these two manifolds.


The study of the spectrum of the Dirac operator on noncompact symmetricspaces has been considerably improved by two papers, one by S. Goette and U. Sem-melmann, and one by R. Camporesi and E. Pedon. Here again, we only men-tion those results since they require some non-trivial knowledge in “noncompact”harmonic analysis.

Indeed, unlike the compact case, the spectrum of the L2-closed extension ofthe Dirac operator to L2.†M/ (which is self-adjoint since any symmetric space isa complete manifold) is the union of the point spectrum Specp.D/, de�ned as theset of complex numbers � for which D � �Id is not injective, and the continuousspectrum Specc.D/, de�ned as the set of complex numbers � for which D � �Idhas a discontinuous inverse with dense domain.

In [GS02], S. Goette and U. Semmelmann, have determined the point spectrumof the Dirac operator on all noncompact symmetric spaces. Their result can be statedas follows.

Theorem 15.20 ([GS02]). The point spectrum of D on an irreducible symmetricspace G=K of noncompact type is nonempty if and only if G=K is isometric to

SUp;q =S.Up � Uq/

with p C q odd, and in this case Specp.D/ D f0g.

Referring to this result, R. Camporesi and E. Pedon proved in [CP01] thefollowing theorem.

Theorem 15.21 ([CP01]). For a noncompact Riemannian symmetric space G=Kof rank one, the continuous spectrum Specc.D/ is equal to R except if G=K is thecomplex hyperbolic spaceHn.C/ with n even, in which case

Specc.D/ D�

� 1;�12

i[h12;C1

�:

Note that the exceptional case, the complex hyperbolic space

Hn.C/ D SU.1; n/=S.U1 � Un/

with n even, is precisely the one for which the point spectrum is nonempty.Let us mention here that partial results in the study of the spectrum of Hn.C/

and Hn.H/ were obtained in [Sei92] and [Bai97]. Note also that results on eigen-values and eigenspaces of certain twisted Dirac operators over the symmetric spacesH 2n.R/ andHn.C/ can be found in [GV94].

Bibliography

—A —

[Ada69] J. F. Adams, Lectures on Lie groups.Mathematics Lecture Note Series.W. A. Benjamin, New York and Amsterdam, 1969.MR 0252560 Zbl 0206.31604 305, 340, 341

[Ale68] D. V. Alekseevskii, Riemannian spaces with exceptional holonomygroups. Funkcional. Anal. i Priložen 2 (1968), 1–10. English transl.:Functional Anal. Appl. 2 (1968), 97–105.MR 0231313 Zbl 0175.18902 178, 207, 208

[AGI98] B. Alexandrov, G. Grantcharov, and S. Ivanov, An estimate for the �rsteigenvalue of the Dirac operator on compact Riemannian spin manifoldadmitting parallel one-form. J. Geom. Phys. 28 (1998), 263–270.MR 1658743 Zbl 0934.58026 142, 146

[AM95] D. V. Alekseevskii and S. Marchiafava, Transformations of a quater-nionic Kähler manifold. C. R. Acad. Sci. Paris Sér. I 320 (1995),703–708.MR 1323942 Zbl 0839.53026 279

[AB98] B. Ammann and C. Bär, The Dirac operator on nilmanifolds and col-lapsing circle bundles. Ann. Global Anal. Geom. 16 (1998), 221–253.MR 1626659 Zbl 0911.58037 411, 412, 415

[ADH09a] B. Ammann, M. Dahl, and E. Humbert, Surgery and harmonic spinors.Adv. Math. 220 (2009), 523–539.MR 2466425 Zbl 1159.53021 129, 158

[ADH09b] B. Ammann, M. Dahl, and E. Humbert, Surgery and the spinorial � -in-variant. Comm. Partial Differential Equations 34 (2009), 1147–1179.MR 2581968 Zbl 1183.53041 129, 158

http://www.ams.org/mathscinet-getitem?mr=0252560

http://zbmath.org/?q=an:0206.31604













422 Bibliography

[AH05] B. Ammann and E. Humbert, Positive mass theorem for the Yamabeproblem on spin manifolds. Geom. Funct. Anal. 15 (2005), 567–576.MR 2221143 Zbl 1087.53045 155, 156

[AH08] B. Ammann and E. Humbert, The spinorial � -invariant and 0-dimen-sional surgery. J. Reine Angew. Math. 624 (2008), 27–50.MR 2456623 Zbl 1158.53040 129, 158

[AHM06] B. Ammann, E. Humbert, and B. Morel, Mass endomorphism andspinorial Yamabe type problems on conformally �at manifolds. Comm.Anal. Geom. 14 (2006), 163–182.MR 2230574 Zbl 1126.53024 158

[AMM13] B. Ammann, A. Moroianu, and S. Moroianu, The Cauchy problems forEinstein metrics and parallel spinors. Comm. Math. Phys. 320 (2013),173–198.MR 3046994 Zbl 1271.53048 242

[Apo96] V. Apostolov, private communication, 1996.261

[ADM11] V. Apostolov, T. Draghici, and A. Moroianu, A splitting theorem forKähler manifolds with constant eigenvalues of the Ricci tensor. Inter-nat. J. Math. 12 (2011), 769–789.MR 1850671 Zbl 1111.53303 277

[ABS64] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules. Topology 3

(1964), suppl. 1, 3–38.MR 0167985 Zbl 0146.19001 204, 283

[AHS78] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-di-mensional Riemannian geometry. Proc. Roy. Soc. London 362 (1978),425–461.MR 0506229 Zbl 0389.53011 265

—B —

[Bai97] P. D. Baier,Über den Dirac Operator auf Mannigfaltigkeiten mit Zylin-derenden. Diplomarbeit. Universität Freiburg, Freiburg, 1997.419















Bibliography 423

[BB12] W. Ballmann and C. Bär, Boundary value problems for elliptic dif-ferential operators of �rst order. In Surveys in differential geometry.Vol. XVII. In memory of C. C. Hsiung. Lectures from the Journal ofDifferential Geometry (JDG) International Symposium on Geometryand Topology held at Lehigh University, Bethlehem, PA, May 2010.Edited by H. D. Cao and S. T. Yau. Surveys in Differential Geome-try 17, 2012, International Press, Boston, MA, 1–78.MR 3076058 MR 3087284 (collection) Zbl 1239.00059 (collection)152

[Bär91] C. Bär, Das Spektrum von Dirac-Operatoren. Dissertation, RheinischeFriedrich-Wilhelms-Universität Bonn, Bonn, 1990. Bonner Mathema-tische Schriften 217. Universität Bonn, Mathematisches Institut, Bonn,1991.MR 1178937 Zbl 0748.53022 408, 409, 414

[Bär92a] C. Bär, The Dirac operator on homogeneous spaces and its spectrum on3-dimensional lens spaces. Arch. Math. (Basel) 59 (1992), 65–79.MR 1166019 Zbl 0786.53030 414, 415

[Bär92b] C. Bär, Lower eigenvalue estimates for Dirac operators.Math. Ann. 293(1992), 39–46.MR 1162671 Zbl 0741.58046 138, 140

[Bär93] C. Bär, Real Killing spinors and holonomy. Comm. Math. Phys. 154(1993), 509–521.MR 1224089 Zbl 0778.53037 146, 236, 238, 239, 242

[Bär94] C. Bär, The Dirac operator on space forms of positive curvature.J. Math. Soc. Japan 48 (1994), 69–83.MR 1361548 Zbl 0848.58046 408, 409, 415

[Bär96] C. Bär, Metrics with harmonic spinors. Geom. Funct. Anal. 6 (1996),899–942.MR 1421872 Zbl 0867.53037 70, 411, 414, 415

[Bär98] C. Bär, Extrinsic bounds for eigenvalues of the Dirac operator. Ann.Global Anal. Geom. 16 (1998), 573–596.MR 1651379 Zbl 0921.58065 149, 153


















424 Bibliography

[Bär99] C. Bär, Zero sets of solutions to semilinear elliptic systems of �rst order.Invent. Math. 138 (1999), 183–202.MR 1714341 Zbl 0942.35064 147

[Bär00] C. Bär, The Dirac operator on hyperbolic manifolds of �nite volume.J. Differential Geometry 54 (2000), 439–488.MR 1823312 Zbl 1030.58021 95, 418

[Bär09] C. Bär, Spectral bounds for Dirac operators on open manifolds. Ann.Global Anal. Geom. 36 (2009), 67–79.MR 2520031 Zbl 1193.53119 159

[BD04] C. Bär and M. Dahl, The �rst Dirac eigenvalues on manifolds with pos-itive scalar curvature. Proc. Amer. Math. Soc. 132 (2004), 3337–3344.MR 2073310 Zbl 1053.58012 143

[BGM05] C. Bär, P. Gauduchon, andA. Moroianu, Generalized cylinders in semi-Riemannian and spin geometry. Math. Z. 249 (2005), 545–580.MR 2121740 Zbl 1068.53030 242, 243

[BS83] R. Barker and S. M. Salamon, Analysis on a generalized Heisenberggroup. J. London Math. Soc. (2) 28 (1983), 184–192.MR 0703476 Zbl 0522.22008 180, 412

[BC05] R. Bartnik and P. T. Chrusciel, Boundary value problems for Dirac-typeequations. J. Reine Angew. Math. 579 (2005), 13–73.MR 2124018 Zbl 1174.58305 152

[Bau81] H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudo-Riemannsche Mannigfaltigkeiten. Teubner-Texte zur Mathematik 41.BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1981.MR 0701244 Zbl 0519.53054 69

[BFGK91] H. Baum, Th. Friedrich, R. Grunewald, and I. Kath, Twistor andKillingspinors on Riemannian manifolds. Teubner-Texte zur Mathematik 124.Teubner Verlag, Stuttgart etc., 1991.MR 1164864 Zbl 0734.53003 133, 149, 238, 270

[Ber79] L. Bérard Bergery, Variétés quaternioniennes. Notes d’une conférenceà la table ronde Variétés d’Einstein. Espalion, 1979.265, 268



















Bibliography 425

[Ber66] M. Berger, Remarques sur les groupes d’holonomie des variétés rie-manniennes. C. R. Acad. Sci. Paris Sér. A 262 (1966), 1316–1318.MR 0200860 Zbl 0151.28301 178, 266

[BGM71] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d’une variété rie-mannienne. Lecture Notes in Mathematics 194. Springer Verlag, Berlinetc., 1971.MR 0282313 Zbl 0223.53034 374, 376

[BGV92] N. Berline, E. Getzler, and N. Vergne, Heat kernels and Dirac opera-tors. Grundlehren der Mathematischen Wissenschaften 298. SpringerVerlag, Berlin etc., 1992.MR 1215720 Zbl 0744.58001 1, 96

[Bes87] A. Besse, Einstein manifolds. Ergebnisse der Mathematik und ihrerGrenzgebiete (3) 10. Springer Verlag, Berlin etc., 1987.MR 0867684 Zbl 0613.53001 138, 176, 178, 179, 187, 202, 246, 282,299, 369, 371, 372, 386

[Bon67] E. Bonan, Sur les G-structures de type quaternionien. Cahiers Topolo-gie Géom. Différentielle 9 (1967), 389–461.MR 0233302 Zbl 0171.20802 178

[Bou75] N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII. Groupeset algèbres de Lie. Chapitre VII. Sous-algèbres de Cartan, élémentsréguliers. Chapitre VIII. Algèbres de Lie semi-simples déployés. Actu-alités Scienti�ques et Industrielles 1364. Hermann, Paris, 1975. Englishtrasl. by A. Pressley: Elements of mathematics. Lie groups and Lie al-gebras. Chapters 7–9. Springer Verlag, Berlin, 2005.MR 0453824 Zbl 0329.17002 366

[Bou05] J. P. Bourguignon, A brief introduction to Riemannian and spino-rial geometries. In J.-P. Bourguignon, Th. Branson, A. Chamseddine,O. Hijazi and R. J. Stanton (eds.), Dirac operators: yesterday and to-day. Proceedings of the Summer School and Workshop held in Beirut,August 27–September 7, 2001. International Press, Somerville, MA,2005, 45–69.MR 2205366 MR 2205362 (collection) Zbl 1099.53026Zbl 1089.53003 (collection) 39

















426 Bibliography

[BG92] J. P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Diracet variations de métriques. Comm. Math. Phys. 144 (1992), no. 3,581–599.MR 1158762 Zbl 0755.53009 148, 242

[BG99] C. Boyer and K. Galicki, 3-Sasakian manifolds. In Surveys in differ-ential geometry: essays on Einstein manifolds. Lectures on geometryand topology, sponsored by Lehigh University’s Journal of DifferentialGeometry. Edited by C. LeBrun and M. Wang Surveys in DifferentialGeometry VI. International Press, Cambridge, MA, 1999, 123–184.MR 1798609 MR 1798603 (collection) Zbl 1008.53047Zbl 961.00021 (collection) 265, 280, 282, 299

[Bra96] Th. Branson, Nonlinear phenomena in the spectral theory of geometriclinear differential operators. In W. Arveson, Th. Branson and I. Segal(eds.), Quantization, nonlinear partial differential equations, and op-erator algebra. Proceedings of the 1994 John von Neumann Sympo-sium on Quantization and Nonlinear Wave Equations held at the Mas-sachusetts Institute of Technology, Cambridge, Massachusetts, June7–11, 1994. American Mathematical Society, Providence, R.I., 1996,27–65.MR 1392983 MR 1392981 (collection) Zbl 0857.58042Zbl 0840.00054 (collection) 56

[Bra97] Th. Branson, Stein–Weiss operators and ellipticity. J. Funct. Anal. 151(1997), 334–383.MR 1491546 Zbl 0904.58054 56, 132

[Bra97b] Th. Branson, Spectral theory of invariant operators, sharp inequalities,and representation theory. In J. Slovák (ed.), Proceedings of the 16th

Winter School on geometry and physics, Srní, Czech Republic, January13–20, 1996. Supplemento ai Rend. Circ. Mat. Palermo (2) 46 (1997),29–54.MR 1469020 Zbl 0884.58100 Zbl 0866.00050 (collection) 407

[Bra98] Th. Branson, Second order conformal covariants. Proc. Amer MathSoc. 126 (1998), 1031–1042.MR 1422849 Zbl 0890.47030 176

[Bra99] Th. Branson, Spectra of self-gradients on spheres. J. Lie Theory 9

(1999), 491–506.MR 1718236 Zbl 1012.22026 407




















Bibliography 427

[Bra05] Th. Branson, Conformal structure and spin geometry. In J.-P. Bour-guignon, Th. Branson, A. Chamseddine, O. Hijazi and R. J. Stanton(eds.), Dirac operators: yesterday and today. Proceedings of the Sum-mer School and Workshop held in Beirut, August 27–September 7,2001. International Press, Somerville, MA, 2005, 163–191.MR 2205371 MR 2205362 (collection) Zbl 1109.53051Zbl 1089.53003 (collection) 176

[BH97] Th. Branson and O. Hijazi, Vanishing theorems and eigenvalueestimates in Riemannian spin geometry. Internat. J. Math. 8 (1997),921–934.MR 1482970 Zbl 0897.58005 140

[BH00] Th. Branson and O. Hijazi, Improved forms of some vanishing the-orems in Riemannian spin geometry. Internat. J. Math. 11 (2000),291–304.MR 1769610 Zbl 1110.53305 140

[BtD85] T. Bröker and T. tom Dieck, Representations of compact Lie groups.Graduate Texts in Mathematics 98, Springer Verlag, Berlin etc., 1985.MR 0781344 Zbl 0581.22009 305, 312, 313, 315, 340, 341, 397, 398

[Bry87] R. L. Bryant, Metrics with exceptional holonomy. Ann. Math. (2) 126(1987), 525–576.MR 0916718 Zbl 0637.53042 227, 233, 236

[Buc98] V. Buchholz,DieDirac und Twistorgleichung in der konformenGeome-trie. Diplomarbeit. Humboldt-Universität zu Berlin, Berlin, 1998.254

[Bun91a] U. Bunke, The spectrum of the Dirac operator on the hyperbolic space.Math. Nachr. 153 (1991), 179–190.MR 1131941 Zbl 0803.58056 417, 418

[Bun91b] U. Bunke, Upper bounds of small eigenvalues of the Dirac operator andisometric immersions. Ann. Global Anal. Geom. 9 (1991), 109–116.MR 1136121 Zbl 0698.53025 415

[Bur93] J. Bureš, Dirac operator on hypersurfaces. Comment. Math. Univ. Car-olin. 34 (1993), 313–322.MR 1241739 Zbl 0781.53031 149



















428 Bibliography

—C —

[CFG89] M. Cahen, A. Franc, and S. Gutt, Spectrum of the Dirac operator oncomplex projective space P2q�1.C/. Lett. Math. Phys. 18 (1989),165–176.MR 1010996 Zbl 0706.58010 395, 409, 410, 411, 415

[CFG94] M. Cahen, A. Franc, and S. Gutt, Erratum to “Spectrum of the Diracoperator on complex projective space P2q�1.C/.” Lett. Math. Phys. 32(1994), 365–368.MR 1310300 409, 411, 415

[CG88] M. Cahen and S. Gutt, Spin structures on compact simply connectedRiemannian symmetric spaces. Simon Stevin 62 (1988), 209–242.MR 0976428 Zbl 0677.53057 388, 389, 395

[CH96] R. Camporesi and A. Higuchi, On the eigenfunctions of the Dirac op-erator on spheres and real hyperbolic spaces. J. Geom. Phys. 20 (1996),1–18.MR 1407401 Zbl 0865.53044 409, 415, 418

[CP99] R. Camporesi and E. Pedon, Harmonic analysis for spinors on real hy-perbolic spaces. Colloq. Math. 87 (2001), 245–286.MR 1814669 Zbl 0968.22009 418

[CP01] R. Camporesi and E. Pedon, The continuous spectrum of the Dirac op-erator on noncompact Riemannian symmetric spaces of rank one. Proc.Amer. Math. Soc. 130 (2002), 507–516.MR 1862131 Zbl 1003.43007 419

[CHZ12] D. Chen, O. Hijazi, and X. Zhang, The Dirac–Witten operator onpseudo-Riemannian manifolds. Math. Z. 271 (2012), 357–372.MR 2917148 Zbl 1248.53039 151, 159

[Che62] C. Chevalley, Algebraic theory of spinors. Columbia Univ. Press,New York, 1962.MR 0060497 Zbl 0057.25901 13

[Con11] D. Conti, Embedding into manifolds with torsion.Math. Z. 268 (2011),725–751.MR 2818726 Zbl 1232.53044 243


















Bibliography 429

[CS06] D. Conti and S. Salamon, Reduced holonomy, hypersurfaces and exten-sions. Int. J. Geom. Methods Mod. Phys. 3 (2006), 899–912.MR 2264396 Zbl 1116.53031 243

[CS07] D. Conti and S. Salamon, Generalized Killing spinors in dimension 5.Trans. Amer. Math. Soc. 359 (2007), 5319–5343.MR 2327032 Zbl 1130.53033 243

—D—

[Die75] J. Dieudonné, Eléments d’analyse. Tome V. Chapitre XXI. Cahiers Sci-enti�ques. Fasc. XXXVIII. Gauthier-Villars, Paris, 1975.MR 0467781 Zbl 0326.22001 244, 305, 321, 339, 341

—E —

[Esc92] J. F. Escobar, Conformal deformation of a Riemannian metric to ascalar �at metric with constant mean curvature on the boundary. Ann.of Math. (2) 136 (1992), 1–50.MR 1173925 Zbl 0766.53033 153

— F —

[Feg87] H. Fegan, The spectrum of the Dirac operator on a simply connectedcompact Lie group. Simon Stevin 61 (1987), 97–108.MR 0914927 Zbl 0631.58032 56, 132, 415

[Fri80] Th. Friedrich, Der erste Eigenwert des Dirac-Operators einerkompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalar-krümmung. Math. Nachr. 97 (1980), 117–146.MR 0600828 Zbl 0462.53027 129, 265

[Fri81] Th. Friedrich, A remark on the �rst eigenvalue of the Dirac operator on4-dimensional manifolds. Math. Nachr. 102 (1981), 53–56.MR 0642141 Zbl 0481.53039 239

[Fri84] Th. Friedrich, Zur Abhängigkeit des Dirac-Operators von der Spin-Struktur. Colloq. Math. 48 (1984), 57–62.MR 0750754 Zbl 0542.53026 415

















430 Bibliography

[Fri93] Th. Friedrich, The classi�cation of 4-dimensional Kähler manifoldswith small eigenvalue of the Dirac operator. Math. Ann. 295 (1993),565–574.MR 1204838 Zbl 0798.53065 275

[Fri00] Th. Friedrich, Dirac-Operatoren in der Riemannschen Geometrie. Miteinem Ausblick auf die Seiberg–Witten-Theorie. Advanced Lecturesin Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 1997. Englishtransl. by A. Nestke: Dirac operators in Riemannian geometry. Gradu-ate Studies in Mathematics 25. American Mathematical Society, Prov-idence, RI, 2000.MR 1777332 MR 1476425 (transl.) Zbl 0887.58060Zbl 0949.58032 (transl.) 1, 158

[Fri98] Th. Friedrich, On the spinor representation of surfaces in Euclidean3-space. J. Geom. Phys. 28 (1998), 143–157.MR 1653146 Zbl 0966.53042 158, 242

[FK01] Th. Friedrich and E.-C. Kim, Some remarks on theHijazi inequality andgeneralizations of the Killing equation for spinors. J. Geom. Phys. 37(2001), 1–14.MR 1806438 Zbl 0979.53052 159

[FK08] Th. Friedrich and E.-C. Kim, Eigenvalues estimates for the Dirac op-erator in terms of Codazzi tensors. Bull. Korean Math. Soc. 45 (2008),no. 2, 365–373.MR 2419083 Zbl 1218.53049 159

[FK02a] Th. Friedrich and K.-D. Kirchberg, Eigenvalue estimates of the Diracoperator depending on the Ricci tensor. Math. Ann. 324 (2002),799–816.MR 1942250 Zbl 1033.53040 159

[FK02b] Th. Friedrich and K.-D. Kirchberg, Eigenvalue estimates for the Diracoperator depending on the Weyl tensor. J. Geom. Phys. 41 (2002),196–207.MR 1877926 Zbl 1007.53039 159

[FT00] Th. Friedrich and A. Trautman, Spin spaces, Lipschitz groups, andspinor bundles. Ann. Global Anal. Geom. 18 (2000), Special issue inmemory of A. Gray (1939–1998), 221–240.MR 1795095 Zbl 0964.15033 49



















Bibliography 431

[FH91] W. Fulton and J. Harris, Representation theory. A �rst course.GraduateTexts in Mathematics 129, Springer Verlag, New York etc., 1991.MR 1153249 Zbl 0744.22001 305

—G—

[GV94] E. Galina and J. Vargas, Eigenvalues and eigenspaces for the twistedDirac operator over SU.n; 1/ and Spin.2n; 1/. Trans. Amer. Math.Soc. 345 (1994), 97–113.MR 1189792 Zbl 0826.22008 419

[Gal79] S. Gallot, Equations différentielles caractéristiques de la sphère. Ann.Sci. École Norm. Sup. (4) 12 (1979), 235–267.MR 0543217 Zbl 0412.58009 237, 299

[GHL87] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry.Universi-text, Springer Verlag, Berlin etc., 1987.MR 0909697 Zbl 0636.53001 315

[GM75] S. Gallot and Y. Meyer, Opérateur de courbure et laplacien des formesdifférentielles d’une variété riemannienne. J. Math. Pures Appl. (9) 54(1975), 259–284.MR 0454884 Zbl 0316.53036 130

[Gau84] P. Gauduchon, La 1-forme de torsion d’une variété hermitienne com-pacte. Math. Ann. 267 (1984), 495–518.MR 0742896 Zbl 0523.53059 262

[Gau95b] P. Gauduchon, Structures deWeyl–Einstein, espaces de twisteurs et var-iétés de type S1 � S3. J. Reine Angew. Math. 469 (1995), 1–50.MR 1363825 Zbl 0858.53039 262

[Gau95a] P. Gauduchon, L’opérateur de Penrose kählérien et les inégalités deKirchberg. Preprint 1995.56, 161, 173, 254

[GMS11] P. Gauduchon, A. Moroianu, and U. Semmelmann, Almost complexstructures on quaternion-Kähler manifolds and inner symmetric spaces.Invent. Math. 184 (2011), 389–403.MR 2793859 Zbl 1225.53028 4

















432 Bibliography

[Gin04] N. Ginoux, Dirac operators on Lagrangian submanifolds. J. Geom.Phys. 52 (2004), no. 4, 480–498.MR 2098837 Zbl 1077.53026 158

[Gin08] N. Ginoux, The spectrum of the Dirac operator on SU2=Q8.Manuscripta math. 125 (2008), no. 3, 383–409.MR 2373068 Zbl 1242.58003 415

[Gin09] N. Ginoux, The Dirac spectrum. Lecture Notes in Mathematics 1976.Springer Verlag, Berlin, 2009. MR 2509837 Zbl 1186.58020 1, 129,146

[GH08] N. Ginoux and G. Habib, Remarques sur les spineurs de Killingtransversaux. C. R. Math. Acad. Sci. Paris 346 (2008), 657–659.MR 2423273 Zbl 1152.53035 148

[GH10] N. Ginoux, and G. Habib, A spectral estimate for the Dirac operator onRiemannian �ows. Cent. Eur. J. Math. 8 (2010), 950–965.MR 2727442 Zbl 1221.53054 147, 148

[GM02] N. Ginoux and B. Morel, On eigenvalue estimates for the submanifoldDirac operator. Internat. J. Math. 13 (2002), 533–548.MR 1914563 Zbl 1049.58028 153, 159

[God82] R. Godement, Introduction à la théorie des groupes de Lie. Tome II.Publications Mathématiques de l’Université Paris VII 12. Universitéde Paris VII, U.E.R. de Mathématiques, Paris, 1982.MR 0679860 Zbl 0533.22001 28

[GS02] S. Goette and U. Semmelmann, The point spectrum of the Dirac opera-tor on noncompact symmetric spaces. Proc. Am. Math. Soc. 130 (2002),915–923. MR 1866049 Zbl 0995.58005 419

[GW98] R. Goodman and N. R. Wallach, Representations and invariants of theclassical groups. Encyclopedia of Mathematics and its Applications 68.Cambridge University Press, Cambridge, 1998.MR 1606831 Zbl 0948.22001 305, 341

[Gra76] A. Gray, The structure of nearly Kähler manifolds. Math. Ann. 223(1976), 233–248.MR 0417965 Zbl 0345.53019 231, 236

[GL80b] M. Gromov and H. B. Jr. Lawson, Spin and scalar curvature in the pres-ence of a fundamental group I. Ann. Math. 111 (1980), 209–230.MR 0569070 Zbl 0445.53025 129























Bibliography 433

[GL80a] M. Gromov and H. B. Jr. Lawson, The classi�cation of simply-connected manifolds of positive scalar curvature. Ann. Math. (2) 111(1980), 423–434.MR 0577131 Zbl 0463.53025 129

[GL83] M. Gromov and H. B. Jr. Lawson, Positive scalar curvature and theDirac operator on complete Riemannian manifolds. Inst. Hautes ÉtudesSci. Publ. Math. 58 (1983), 295–408.MR 720933 Zbl 0538.53047 129

[Gro06] N. Große, On a conformal invariant of the Dirac operator on noncom-pact manifolds. Ann. Global Anal. Geom. 30 (2006), 407–416.MR 2265322 Zbl 1104.53046 158, 159

[Gro11] N. Große, The Hijazi inequality on conformally parabolic manifolds.Diff. Geom. Appl. 29 (2011), 838–849.MR 1235.53054 Zbl 2846280 158, 159

[Gro12] N. Große, Solutions of the equation of a spinorial Yamabe-type problemon manifolds of bounded geometry. Comm. Partial Differential Equa-tions 37 (2012), 58–76.MR 2864806 Zbl 1239.53068 158

[GMP12] C. Guillarmou, S. Moroianu, and J. Park, Bergman and Calderón pro-jectors for Dirac operators. J. Geom. Anal. 24 (2014), 298–336MR 3145926 72, 73

[GL98] M. Gursky and C. LeBrun, Yamabe invariants and Spinc structures.Geom. Funct. Anal. 8 (1998), 965–977.MR 1664788 Zbl 0931.53019 158

[Gut86] S. Gutt, Killing spinors on spheres and projective spaces. In Spinorsin physics and geometry. Proceedings of the First Conference held inTrieste, September 11–13, 1986. Edited by A. Trautman and G. Furlan.World Scienti�c, Singapore, 1988, 238–248.MR 0992419 MR 0992405 (collection) Zbl 0947.53501 (collection)391







http://www.ams.org/mathscinet-getitem?mr=1235.53054

http://zbmath.org/?q=an:2846280









434 Bibliography

—H—

[Hab07] G. Habib, Energy momentum tensor on foliations. J. Geom. Phys. 57(2007), 2234–2248.MR 2360240 Zbl 1136.53042 148, 159

[HR09] G. Habib and K. Richardson, A brief note on the spectrum of the basicDirac operator. Bull. London Math. Soc. 41 (2009), 683–690.MR 2521364 Zbl 1175.53037 148, 159

[Hel78] S. Helgason, Differential geometry, Lie groups and symmetric spaces.Pure and Applied mathematics 80. Academic Press, New York etc.,1978.MR 0514561 Zbl 0451.53038 369, 417

[HM99] M. Herzlich and A. Moroianu, Generalized Killing spinors and con-formal eigenvalue estimates for Spinc manifolds. Ann. Global Anal.Geom. 17 (1999), 341–370.MR 1705917 Zbl 0988.53020 159

[Hij86b] O. Hijazi, A conformal lower bound for the smallest eigenvalue of theDirac operator and Killing spinors. Comm. Math. Phys. 104 (1986),151–162.MR 0834486 Zbl 0593.58040 69, 137, 138, 140

[Hij86a] O. Hijazi, Caractérisation de la sphère par les premières valeurs propresde l’opérateur de Dirac en dimensions 3, 4, 7 et 8. C. R. Acad. Sci. ParisSér. I Math. 307 (1986), 417–419.MR 0862207 Zbl 0606.53024 239

[Hij91] O. Hijazi, Première valeur propre de l’opérateur de Dirac et nombre deYamabe. C. R. Acad. Sci. Paris Sér. IMath. 313 (1991), 865–868.MR 1138566 Zbl 0738.53030 139, 140

[Hij94] O. Hijazi, Eigenvalues of the Dirac operator on compact Kähler mani-folds. Comm. Math. Phys. 160 (1994), 563–579.MR 1266064 Zbl 0794.53042 142, 161, 162, 173

[Hij95] O. Hijazi, Lower bounds for the eigenvalues of the Dirac operator.J. Geom. Phys. 16 (1995), 27–38.MR 1325161 Zbl 0823.34074 147, 148, 159



















Bibliography 435

[HL88] O. Hijazi and A. Lichnerowicz, Spineurs harmoniques, spineurs-twisteurs et géométrie conforme. C. R. Acad. Sci. Paris Sér. IMath. 307(1988), 833–838.MR 0978250 Zbl 0659.53016 140

[HM95a] O. Hijazi and J.-L. Milhorat, Décomposition du �bré des spineurs d’unevariété spin Kähler-quaternionienne sous l’action de la 4-forme fonda-mentale. J. Geom. Phys. 15 (1995), 320–332.MR 1322424 Zbl 0821.53012 180, 181

[HM95b] O. Hijazi and J.-L. Milhorat, Minoration des valeurs propres del’opérateur de Dirac sur les variétés spin Kähler-quaternioniennes.J. Math. Pures Appl. 15 (1995), 387–414.MR 1354336 Zbl 0859.58029 142, 175, 181, 187, 413

[HM97] O. Hijazi and J.-L. Milhorat, Twistor operators and eigenvalues of theDirac operator on compact quaternion-Kähler spin manifolds. Ann.Global Anal. Geom. 2 (1997), 117–131.MR 1448719 Zbl 0876.53021 142, 175, 176

[HMR02] O. Hijazi, S. Montiel, and A. Roldán, Eigenvalue boundary problemsfor the Dirac operator. Comm. Math. Phys. 231 (2002), 375–390.MR 1946443 Zbl 1018.58020 151

[HMR03] O. Hijazi, S. Montiel, and A. Roldán, Dirac operators on hypersur-faces of manifolds with negative scalar curvature. Ann. Global Anal.Geom. 23 (2003), 247–264.MR 1966847 Zbl 1032.53040 154

[HMU06] O. Hijazi, S. Montiel, and F. Urbano, Spinc geometry of Kähler mani-folds and the Hodge Laplacian on minimal Lagrangian submanifolds.Math. Z. 253 (2006), 821–853.MR 2221101 Zbl 1101.53029 158

[HMZ01a] O. Hijazi, S. Montiel, and X. Zhang, Dirac operator on embedded hy-persurfaces. Math. Res. Lett. 8 (2001), 195–208.MR 1825270 Zbl 0988.53019 149, 152, 153, 159

[HMZ01b] O. Hijazi, S. Montiel, and X. Zhang, Eigenvalues of the Dirac operatoron manifolds with boundary. Comm. Math. Phys. 221 (2001), 255–265.MR 1845323 Zbl 0997.58015 149, 152, 159



















436 Bibliography

[HMZ02] O. Hijazi, S. Montiel, and X. Zhang, Conformal lower bounds for theDirac operator of embedded hypersurfaces. Asian J. Math. 6 (2002),23–36.MR 1902645 Zbl 1016.58013 154, 159

[HZ01a] O. Hijazi and X. Zhang, Lower bounds for the eigenvalues of theDirac operator. I. The hypersurface Dirac operator. Ann. Global Anal.Geom. 19 (2001), 355–376.MR 1842575 Zbl 0989.53030 153, 159

[HZ01b] O. Hijazi and X. Zhang, Lower bounds for the eigenvalues of theDirac operator. II. The submanifold Dirac operator. Ann. Global Anal.Geom. 20 (2001), 163–181.MR 1857176 Zbl 1019.53019 153, 159

[HZ03] O. Hijazi and X. Zhang, The Dirac–Witten operator on spacelike hyper-surfaces. Comm. Anal. Geom. 11 (2003), no. 4, 737–750.MR 1081.58023 Zbl 2015174 151, 159

[Hit74] N. Hitchin, Harmonic spinors. Advances in Math. 14 (1974), 1–55.MR 0358873 Zbl 0284.58016 5, 69, 70, 129, 137, 167, 204, 215, 251,415

[Hit01] N. Hitchin, Stable forms and special metrics. In Global differential ge-ometry: the mathematical legacy of Alfred Gray. Papers from the Inter-national Congress on Differential Geometry held in memory of Profes-sor A. Gray in Bilbao, September 18–23, 2000. Edited byM. Fernándezand J. A. Wolf. Contemporary Mathematics 288. American Mathemat-ical Society, Providence, RI, 2001, 70–89.MR MR1871001 MR 1870993 (collection) Zbl 1004.53034Zbl 0980.00033 (collection) 243

[Hom06] Y. Homma, Estimating the eigenvalues on quaternionic Kähler mani-folds. Internat. J. Math. 17 (2006), 665–691.MR 2246886 Zbl 1096.58011 176

[Hor87] L. Hörmander, The analysis of linear partial differential operators.III. Pseudodifferential operators. Grundlehren der MathematischenWissenschaften, 274. Springer, Berlin, 1985.MR 0781536 Zbl 0601.35001 99

[Hum72] J. E. Humphreys, Introduction to Lie algebras and representation the-ory. Graduate Texts in Mathematics 9. Springer Verlag, Berlin etc.,1972.MR 0323842 Zbl 0254.17004 305, 331, 336, 339, 341







http://www.ams.org/mathscinet-getitem?mr=1081.58023

http://zbmath.org/?q=an:2015174



http://www.ams.org/mathscinet-getitem?mr=MR1871001










Bibliography 437

— I—

[IT78] A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian onSn and P n.C/. Osaka J. Math. 15 (1978), 515–546.MR 0510492 Zbl 0392.53033 410

[Ish73] S. Ishihara, Quaternion Kählerian manifolds and �bred Riemannianspaces with Sasakian 3-structure. Kodai Math. Sem. Rep. 25 (1973),321–329.MR MR0324592 Zbl 0267.53023 267, 299

[Ish74] S. Ishihara, Quaternion Kählerian manifolds. J. Differential Geometry 9(1974), 483–500.MR 0348687 Zbl 0297.53014 178, 187, 267

[IK72] S. Ishihara and M. Konishi, Fibred Riemannian spaces with Sasakian3-structure.Differential Geometry, in Honor of Kentaro Yano. Edited byS. Kobayashi, M. Obata, and T. Takahashi. Kinokuniya. Tokyo, 1972,179–194.MR 0326621 MR 0324589 (collection) Zbl 0252.53041Zbl 0242.00016 (collection) 267

—K—

[Käh62] E. Kähler, Der innere Differentialkalkül. Rend. Mat. e Appl. (5) 21

(1962), 425–523.MR 0150681 Zbl 0127.31404 13

[Kim02] E. C. Kim, A local existence theorem for the Einstein–Dirac equation.J. Geom. Phys. 44 (2002), 376–405.MR 1969788 Zbl 1074.53038 242

[Kir86] K.-D. Kirchberg, An estimation for the �rst eigenvalue of the Dirac op-erator on closed Kähler manifolds of positive scalar curvature. Ann.Global Anal. Geom. 3 (1986), 291–325.MR 0910548 Zbl 0629.53058 142, 161, 162, 167, 172

[Kir88] K.-D. Kirchberg, Compact six-dimensional Kähler spin manifolds ofpositive scalar curvature with the smallest possible �rst eigenvalue ofthe Dirac operator. Math. Ann. 282 (1988), 157–176.MR 0960839 Zbl 0628.53061 162



http://www.ams.org/mathscinet-getitem?mr=MR0324592
















438 Bibliography

[Kir90] K.-D. Kirchberg, The �rst eigenvalue of the Dirac operator on Kählermanifolds. J. Geom. Phys. 7 (1990), 449–468.MR 1131907 Zbl 0734.53050 142, 161, 162, 173, 175

[KS95] K.-D. Kirchberg and U. Semmelmann, Complex contact structures andthe �rst eigenvalue of the Dirac operator on Kähler manifolds. Geom.Funct. Anal. 5 (1995), 604–618.MR 1339819 Zbl 0855.53026 272, 274

[Kli07] F. Klinker, The decomposition of the spinor bundle of Grassmann man-ifolds. J. Math. Phys. 48 (2007), Article id. 113511, 26 pp.MR 2370254 Zbl 1152.14305 416

[Kna01] A. W. Knapp, Representation theory of semisimple groups. Anoverview based on examples. PrincetonMathematical Series 36. Prince-ton University Press, Princeton, NJ, 1986.MR 0855239 Zbl 0604.22001 305

[Kob59] S. Kobayashi, Remarks on complex contact manifolds. Proc. Amer.Math. Soc. 10 (1959), 164–167.MR 0111061 Zbl 0090.38502 272

[Kob61] S. Kobayashi, On compact Kähler manifolds with positive de�nite Riccitensor. Ann. of Math. (2) 74 (1961), 570–574.MR 0133086 Zbl 0107.16002 276

[KN63] S. Kobayashi and K. Nomizu, Foundations of differential geometry.Vol. I. Interscience Publishers, a division of John Wiley & Sons, NewYork etc., 1963.MR 0152974 Zbl 0119.37502 42, 218

[KN69] S. Kobayashi and K. Nomizu, Foundations of differential geometry.Vol. II. Interscience Publishers, a division of John Wiley & Sons, NewYork etc., 1969.MR Zbl 0175.48504 225, 259, 372, 414

[Koi81] N. Koiso, Hypersurfaces of Einstein manifolds. Ann. Sci. École Norm.Sup. (4) 14 (1981), 433–443.MR 0654205 Zbl 0493.53041 244















http://www.ams.org/mathscinet-getitem?mr=




Bibliography 439

[Kon75] M. Konishi, On manifolds with Sasakian 3-structure over quaternionKähler manifolds. Kodai Math. Sem. Rep. (1975), 194–200.MR 0377782 Zbl 0308.53035 267, 280

[Kra66] V. Y. Kraines, Topology of quaternionic manifolds. Trans. Amer. Math.Soc. 122 (1966), 357–367.MR 0192513 Zbl 0148.16101 178

[KSW98a] W. Kramer, U. Semmelmann, and G. Weingart, The �rst eigenvalue ofthe Dirac operator on quaternionic Kähler manifolds. Comm. Math.Phys. 199 (1998), 327–349.MR 1666875 Zbl 0941.53032 142, 176, 279, 297, 298, 299

[KSW98b] W. Kramer, U. Semmelmann, and G. Weingart, Quaternionic Killingspinors. Ann. Global Anal. Geom. 16 (1998), 63–87.MR 1616598 Zbl 0912.53033 142, 176

[KSW99] W. Kramer, U. Semmelmann, and G. Weingart, Eigenvalue estimatesfor the Dirac operator on quaternionic Kähler manifolds. Math. Z. 230(1999), 727–751.MR 1686563 Zbl 0926.53018 142, 176, 181, 208

— L —

[LR10] M.-A. Lawn and J. Roth, Isometric immersions of hypersurfaces in 4-di-mensional manifolds via spinors. Differential Geom. Appl. 28 (2010),205–219.MR 2594463 Zbl 1209.53035 158, 242

[LR11] M.-A. Lawn and J. Roth, Spinorial characterizations of surfaces into3-dimensional pseudo-Riemannian space forms. Math. Phys. Anal.Geom. 14 (2011), 185–195.MR 2824602 Zbl 1242.53019 158

[LM89] H. B. Jr. Lawson andM.-L. Michelson, Spin geometry. PrincetonMath-ematical Series 38. Princeton University Press, Princeton, NJ, 1989.MR 1031992 Zbl 0688.57001 1, 14, 24, 96, 198, 203, 204, 238, 240

[LeB95] C. LeBrun, Fano manifolds, contact structures, and quaternionic geom-etry. Internat. J. Math. 6 (1995), 419–437.MR 1327157 Zbl 0835.53055 272



















440 Bibliography

[LeB97] C. LeBrun, Yamabe constants and the perturbed Seiberg–Witten equa-tions. Comm. Anal. Geom. 5 (1997), 535–553.MR 1487727 Zbl 0901.53028 158

[LS94] C. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds. Invent. Math. 118 (1994), 109–132.MR 1288469 Zbl 0815.53078 178, 179

[LW99] C. LeBrun andM. Wang (eds.), Surveys in differential geometry: essayson Einstein manifolds. Lectures on geometry and topology, sponsoredby Lehigh University’s Journal of Differential Geometry. Surveys inDifferential Geometry VI. International Press, Cambridge, MA, 1999.MR 1798603 Zbl 0961.00021 446

[LP87] J. Lee and T. Parker, The Yamabe problem. Bull. Amer. Math.Soc. (N.S.) 17 (1987), 37–91.MR 0888880 Zbl 0633.53062 148

[Lep46] Th. N. Lepage, Sur certaines congruences des formes alternées. Bull.Soc. Roy. Sci. Liège 15 (1946), 21–31.MR 0020072 Zbl 0060.04202 366

[Lep71] J. Lepowsky, Multiplicity formulas for certain semisimple Lie groups.Bull. Amer. Math. Soc. 77 (1971), 601–605.MR 0301142 Zbl 0228.17003 413

[Lic58] A. Lichnerowicz, Géométrie des groupes de transformations. Travauxet Recherches Mathématiques III. Dunod, Paris, 1958.MR 0124009 Zbl 0096.16001 130

[Lic63] A. Lichnerowicz, Spineurs harmoniques. C. R. Acad. Sci. Paris 257,(1963), 7–9.MR 0156292 Zbl 0136.18401 127

[Lic87] A. Lichnerowicz, Spin manifolds, Killing spinors and universality ofthe Hijazi inequality. Lett. Math. Phys. 3 (1987), 331–344.MR 0895296 Zbl 0624.53034 140

[Lic88] A. Lichnerowicz, Les spineurs-twisteurs sur une variété spinoriellecompacte. C. R. Acad. Sci. Paris Sér. IMath. 306 (1988), 381–385.MR 0934624 Zbl 0641.53014 72





















Bibliography 441

[Lic89] A. Lichnerowicz, On the twistor-spinors. Lett. Math. Phys. 18 (1989),333–345.MR 1028200 Zbl 0685.53017 140

[Lic90] A. Lichnerowicz, La première valeur propre de l’opérateur de Diracpour une variété kählérienne et son cas limite. C. R. Acad. Sci. ParisSér. I Math. 311 (1990), 717–722.MR 1081632 Zbl 0713.53040 162, 173, 265

[Lot86] J. Lott, Eigenvalue bounds for the Dirac operator. Paci�c J. Math. 125(1986), 117–126.MR 0860754 Zbl 0605.58044 140

—M—

[Mae08] D. Maerten, Optimal eigenvalue estimate for the Dirac–Witten opera-tor on bounded domains with smooth boundary. Lett. Math. Phys. 86(2008), 1–18.MR 2460723 Zbl 1179.53056 151, 159

[McI91] B. McInnes, Methods of holonomy theory for Ricci-�at Riemannianmanifolds. J. Math. Phys. 32 (1991), 888–896.MR 1097773 Zbl 0737.53040 216

[Mel05] R. B. Melrose, Lectures on pseudodifferential operators. Preprint 2005.http://math.mit.edu/zrbm/Lecture_notes.html 99

[MP06] R. J. Miatello and R. A. Podestá, The spectrum of twisted Dirac oper-ators on compact �at manifolds. Trans. Amer. Math. Soc. 358 (2006),4569–4603.MR 2231389 Zbl 1094.58014 415

[Mic80] M.-L. Michelsohn, Clifford and spinor cohomology of Kähler mani-folds. Amer. J. Math. 102 (1980), 295–313.MR 0595007 Zbl 0462.53035 167

[Mil92] J.-L. Milhorat, Spectre de l’opérateur de Dirac sur les espaces projectifsquaternioniens. C. R. Acad. Sci. Paris Sér. IMath. 314 (1992), 69–72.MR 1149642 Zbl 0753.47039 412, 413, 415

[Mil98] J.-L. Milhorat, Spectrum of the Dirac operator on Gr2.CmC2/. J. Math.Phys. 39 (1998), 594–609.MR 1489638 Zbl 0919.58062 415











http://math.mit.edu/~rbm/Lecture_notes.html









442 Bibliography

[Mil99] J.-L. Milhorat, Eigenvalues of the Dirac operator on compactquaternion-Kähler manifolds. Rapport de Recherche 99/10-3. Univer-sité de Nantes, Nantes, 1999.181

[Mil05] J.-L. Milhorat, The �rst eigenvalue of the Dirac operator on compactspin symmetric spaces. Comm. Math. Phys. 259 (2005), 71–78.MR 2169968 Zbl 1079.53068 416

[Mil06] J.-L. Milhorat, A formula for the �rst eigenvalue of the Dirac operatoron compact spin symmetric spaces. J. Math. Phys. 47 (2006), Articleid. 043503, 12 pp.MR 2226340 Zbl 1111.58023 416, 417

[Mon99] S. Montiel, Unicity of constant mean curvature hypersurfaces in someRiemannian manifolds. Indiana Univ. Math. J. 48 (1999), 711–748.MR 1722814 Zbl 0973.53048 154

[Mon05] S. Montiel, Dirac operators and hypersurfaces. In J.-P. Bourguignon,Th. Branson, A. Chamseddine, O. Hijazi andR. J. Stanton (eds.),Diracoperators: yesterday and today. Proceedings of the Summer School andWorkshop held in Beirut, August 27–September 7, 2001. InternationalPress, Somerville, MA, 2005, 271–281.MR 2205378 MR 2205362 (collection) Zbl 1109.58030Zbl 1089.53003 (collection) 149

[Mor03] B. Morel, The energy-momentum tensor as a second fundamental form.Preprint 2003.arXiv:math.DG/0302205 242

[Mor05] B. Morel, Surfaces in S3 and H

3 via spinors. In Actes de Séminaire deThéorie Spectrale et Géométrie. Vol. 23. Année 2004–2005. Séminairede Théorie Spectrale et Géométrie 23. Université de Grenoble I, InstitutFourier, Saint-Martin-d’Hères, 2005, 131–144.MR 2270227 MR 2270221 (collection) Zbl 1106.53004Zbl 1089.35003 (collection) 158, 242

[Mor95] A. Moroianu, La première valeur propre de l’opérateur de Dirac surles variétés kählériennes compactes. Comm. Math. Phys. 169 (1995),373–384.MR 1329200 Zbl 0832.53054 265, 275, 279











http://arxiv.org/abs/math.DG/0302205







Bibliography 443

[Mor96] A. Moroianu, Opérateur de Dirac et submersions riemanniennes.Ph.D. thesis. École Polytechnique, Paris, 1996.279, 283, 286

[Mor97a] A. Moroianu, On Kirchberg’s inequality for compact Kähler mani-folds of even complex dimension. Ann. Global Anal. Geom. 15 (1997),235–242.MR 1456510 Zbl 0890.53058 265, 276

[Mor97b] A. Moroianu, Parallel and Killing spinors on Spinc manifolds. Comm.Math. Phys. 187 (1997), 417–428.MR 1463835 Zbl 0888.53035 215, 239

[Mor98a] A. Moroianu, Géométrie spinorielle et groupes d’holonomie. CoursPeccot. Collège de France, Paris, 1998.254

[Mor98b] A. Moroianu, Spinc manifolds and complex contact structures. Comm.Math. Phys. 193 (1998), 661–673.MR 1624855 Zbl 0908.53024 268, 269, 272, 274, 275

[Mor99] A. Moroianu, Kähler manifolds with small eigenvalues of the Dirac op-erator and a conjecture of Lichnerowicz. Ann. Inst. Fourier (Greno-ble) 49 (1999), 1637–1659.MR 1723829 Zbl 0946.53040 265, 276

[Mor00] A. Moroianu, On the in�nitesimal isometries of manifolds with Killingspinors. J. Geom. Phys. 35 (2000), 63–74.MR 1767942 Zbl 0982.53042 136

[Mor08] S. Moroianu, Weyl laws on open manifolds. Math. Ann. 340 (2008),1–21.MR 2349766 Zbl 1131.58019 95

[MO04] A. Moroianu and L. Ornea, Eigenvalue estimates for the Dirac oper-ator and harmonic 1-forms of constant length. C. R. Math. Acad. Sci.Paris 338 (2004), 561–564.MR 2057030 Zbl 1049.58029 142, 143

[MO08] A. Moroianu and L. Ornea, Conformally Einstein products and nearlyKähler manifolds. Ann. Global Anal. Geom. 33 (2008), 11–18.MR 2369184 Zbl 1139.53016 237

















444 Bibliography

[MS96] A. Moroianu and U. Semmelmann, Kählerian Killing spinors, com-plex contact structures and twistor spaces. C. R. Acad. Sci. ParisSér. I Math. 323 (1996), 57–61.MR 1401629 Zbl 0853.53048 272

[MS00] A. Moroianu and U. Semmelmann, Parallel spinors and holonomygroups. J. Math. Phys. 41 (2000), 2395–2402.MR 1751897 Zbl 0990.53051 215, 219

—N—

[Nak10] R. Nakad, Lower bounds for the eigenvalues of the Dirac operator onSpinc manifolds. J. Geom. Phys. 60 (2010), 1634–1642.MR 2661160 Zbl 1195.53063 158, 159

[Nak11a] R. Nakad, The energy-momentum tensor on Spinc manifolds. Int.J. Geom. Methods Mod. Phys. 8 (2011), 345–365.MR 2793055 Zbl 1221.53082 158, 159

[Nak11b] R. Nakad, The Hijazi inequalities on complete Riemannian Spinc man-ifolds. Adv. Math. Phys. 2011 (2011), 1–14.MR 2781175 Zbl 1218.81050 158, 159

[NR12] R. Nakad and J. Roth, Hypersurfaces of Spinc manifolds and Lawsoncorrespondence. Ann. Global Anal. Geom. 42 (2012), 421–442.MR 2972623 Zbl 1258.53048 158

—O —

[OV93] C. Ohn and A.-J. Vanderwinden, The Dirac operator on compacti�edMinkowski space. Acad. Roy. Belg. Bull. Cl. Sci. (6) 4 (1993), 255–268.MR 1285871 Zbl 0812.53054 416

[O’N66] B. O’Neill, The fundamental equations of a submersion. MichiganMath. J. 13 (1966), 459–469.MR 0200865 Zbl 0145.18602 236

[O’N83] B. O’Neill, Semi-Riemannian geometry.With applications to relativity.Pure and Applied Mathematics 103. Academic Press, New York etc.,1983.MR 0719023 Zbl 0531.53051 76



















Bibliography 445

— P —

[PT82] T. Parker and C. H. Taubes, On Witten’s proof of the positive energytheorem. Comm. Math. Phys. 84 (1982), 223–238.MR 0661134 Zbl 0528.58040 148

[Par71] R. Parthasarathy, Dirac operator and the discrete series. Ann. ofMath. (2) 96 (1971), 1–30. MR 0318398 Zbl 0249.22003 395, 401,406, 416

[Pen76] R. Penrose, Nonlinear gravitons and curved twistor theory. In The riddleof gravitation – on the occasion of the 60th birthday of Peter Bergmann.Proceedings of a Conference held at Syracuse University, Syracuse,N.Y., March 20–21, 1975. General Relativity and Gravitation 7 (1976).Plenum Publishing Corp., New York, 1976, 31–52.MR 0439004 MR 0401033 (collection) Zbl 0354.53025 265

[PM73] R. Penrose and M. A. H. MacCallum, Twistor theory: an approachto the quantization of �elds and space-time. Phys. Rep. 6C (1973),241–315.MR 0475660 265

[Pfä00] F. Pfäffel, The Dirac spectrum of Bieberbach manifolds. J. Geom.Phys. 35 (2000), 367–385.MR 1780760 Zbl 0984.58017 415

—R —

[Rau05] S. Raulot, Optimal eigenvalues estimate for the Dirac operator on do-mains with boundary. Lett. Math. Phys. 73 (2005), 135–145.MR 2173542 Zbl 1085.53040 152, 159

[Rau06] S. Raulot, The Hijazi inequality on manifolds with boundary. J. Geom.Phys. 56 (2006), 2189–2202.MR 2248870 Zbl 1102.53033 152, 159

[Rot10] J. Roth, Spinorial characterizations of surfaces into three-dimensionalhomogeneous manifolds. J. Geom. Phys. 60 (2010), 1045–1061.MR 2647301 Zbl 1198.53058 158

















446 Bibliography

[Rot11] J. Roth, Isometric immersions into Lorentzian products. Int. J. Geom.Methods Mod. Phys. 8 (2011), 1–22.MR 2847259 Zbl 1245.53046 158

— S —

[Sal82] S. M. Salamon, Quaternionic Kähler manifolds. Invent. Math. 67

(1982), 143–171.MR 0664330 Zbl 0486.53048 178, 180, 207, 208, 265, 268

[Sal99] S. M. Salamon, Quaternion-Kähler geometry. In LeBrun andWang [LW99], Surveys in differential geometry: essays on Ein-stein manifolds. Lectures on geometry and topology, sponsored byLehigh University’s Journal of Differential Geometry. Edited byC. LeBrun and M. Wang. Surveys in Differential Geometry VI. Inter-national Press, Cambridge, MA, 1999, 83–121.MR 1798608 MR 1798603 (collection) Zbl 1003.53039Zbl 0961.00021 (collection) 176, 265, 268

[SS87] J. Seade and B. Steer, A note on the eta function for quotients ofPSL2.R/ by co-compact Fuchsian groups. Topology 26 (1987), 79–91.MR 0880510 Zbl 0664.58042 415

[See97] L. Seeger,Der Dirac-Operator auf kompakten symmetrischen Räumen.Diplomarbeit. Universität Bonn, Bonn, 1997.415

[See99] L. Seeger, The spectrum of the Dirac operator on G2= SO.4/. Ann.Global Anal. Geom. 17 (1999), 385–396.MR 1705919 Zbl 0935.58017 415

[Sei92] S. Seifarth, The discrete spectrum of the Dirac operators on certain sym-metric spaces. Preprint n. 25 of the IAAS, Berlin, 1992.419

[SS93] S. Seifarth and U. Semmelmann, The spectrum of the Dirac operator onthe odd-dimensional complex projective space CP2m�1. Preprint n. 95of the Sonderforschungsbereich 288, Berlin, 1993.http://cds.cern.ch/record/256877/�les/P00020021.pdf 411, 415













http://cds.cern.ch/record/256877/files/P00020021.pdf

Bibliography 447

[SW01] U. Semmelmann andG. Weingart, Vanishing theorems for quaternionicKähler manifolds. In Quaternionic structures in mathematics andphysics. Proceedings of the 2nd Meeting held in Rome, September 6–10,1999. Edited by S. Marchiafava, P. Piccinni and M. Pontecorvo. Uni-versità degli Studi di Roma La Sapienza, Dipartimento di MatematicaGuido Castelnuovo, Rome, 1999, 377–404.MR 1848677 MR 1848652 (collection) Zbl 0987.53018Zbl 0985.53001 (collection)http://www.emis.de/proceedings/QSMP99/semmelmann.pdf 176, 204,207, 208, 209, 210

[SW02] U. Semmelmann andG. Weingart, Vanishing theorems for quaternionicKähler manifolds. J. Reine Angew. Math. 544 (2002), 111–132.MR 1887892 Zbl 1001.53030 176, 204, 207, 208, 209, 210

[SW68] E. Stein and G. Weiss, Generalization of the Cauchy–Riemann equa-tions and representations of the rotation group. Amer. J. Math. 90(1968), 163–196.MR 0223492 Zbl 0157.18303 56, 132

[Sto90] S. Stolz, Simply connected manifolds of positive scalar curvature. Bull.Amer. Math. Soc. 23 (1990), 427–432.MR 1056561 Zbl 0719.53018 129

[Sto92] S. Stolz, Simply connected manifolds of positive scalar curvature. Ann.of Math. (2) 136 (1992), 511–540.MR 1189863 Zbl 0784.53029 129

[Str80b] H. Strese, Über den Dirac-Operator auf Grassmann-Mannig-faltigkeiten. Math. Nachr. 98 (1980), 53–59.MR 0623693 Zbl 0473.53047 415

[Str80a] H. Strese, Spektren symmetrischer Räume. Math. Nachr. 98 (1980),75–82.MR 0623695 Zbl 0473.53048 415

[Sul79] S. Sulanke, Die Berechnung des Spektrums des Quadrates des Dirac-Operators auf der Sphäre. Doktorarbeit. Humboldt-Universität zuBerlin, Berlin, 1979.130, 401, 407, 408, 415





http://www.emis.de/proceedings/QSMP99/semmelmann.pdf













448 Bibliography

—T —

[Tan71] S. Tanno, Killing vectors on contact Riemannian manifolds and �ber-ings related to the Hopf �brations. TôhokuMath. J. 23 (1971), 313–333.MR 0287477 Zbl 0232.53026 268

[Tra93] A. Trautman, Spin structures on hypersurfaces and the spectrum of theDirac operator on spheres. In Spinors, twistors, Clifford algebras andquantum deformations. Proceedings of the Second Max Born Sym-posium, dedicated to Professor Jan Rzewuski on the occasion of his75th birthday, held at Sobótka Castle, September 24–27, 1992. Editedby Z. Oziewicz, B. Jancewicz and A. Borowiec. Fundamental Theoriesof Physics 52. Kluwer Academic Publishers Group, Dordrecht, 1993,25–29.MR 1258344 MR 1258341 (collection) Zbl 0882.58059Zbl 0870.00031 (collection) 149, 409, 415

[Tra95] A. Trautman, The Dirac operator on hypersurfaces. Acta Phys.Polon. B 26 (1995), 1283–1310.MR 1357003 Zbl 0966.58504 409, 415

[Tsu81] C. Tsukamoto, Spectra of Laplace–Beltrami operators on SO.n C2/= SO.2/ � SO.n/ and Sp.nC 1/= Sp.1/ � Sp.n/. Osaka J. Math. 18(1981), 407–426.MR 0628842 Zbl 0477.53048 297, 413

—V —

[vNW84] P. van Nieuwenhuizen and N. P. Warner, Integrability conditionsfor Killing spinors. Comm. Math. Phys. 93 (1984), 1637–1659.MR 0742196 Zbl 0549.53011 239

[Vil70] J. Vilms, Totally geodesic maps. J. Differential Geometry 4 (1970),73–79. MR 0262984 Zbl 0194.52901 280

—W—

[Wan89] M. Y. Wang, Parallel spinors and parallel forms. Ann. Global Anal.Geom. 7 (1989), 59–68.MR 1029845 Zbl 0688.53007 180, 215, 216, 222, 279, 299, 412

















Bibliography 449

[Wan95] M. Y. Wang, On non-simply connected manifolds with non-trivial par-allel spinors. Ann. Global Anal. Geom. 13 (1995), 31–42.MR 1327109 Zbl 0826.53041 215, 216, 217

[Wey46] H. Weyl,Classical groups, their invariants and representations. Prince-ton University Press, Princeton, N.J., 1939.MR 0000255 Zbl 0020.20601 340

[Wit81] E. Witten, A new proof of the positive energy theorem. Comm. Math.Phys. 80 (1981), 381–402.MR 0626707 Zbl 1051.83532 148

[Wit94] E. Witten, Monopoles and four-manifolds. Math. Res. Lett. 1 (1994),769–796.MR 1306021 Zbl 0867.57029 158

[Wol65] J. Wolf, Complex homogeneous contact manifolds and quaternionicsymmetric spaces. J. Math. Mech. 14 (1965), 1033–1047.MR 0185554 Zbl 0141.38202 179

—Z —

[Žel73] D. P. Želobenko, Compact Lie groups and their representations. Trans-lations of Mathematical Monographs 40. American Mathematical So-ciety, Providence, R.I., 1973.MR 0473098 Zbl 0272.22006 305, 313, 341, 407

[Zha98] X. Zhang, Lower bounds for eigenvalues of hypersurface Dirac opera-tors. Math. Res. Lett. 5 (1998), 199–210.MR 1617917 Zbl 0961.58013 153

[Zha99] X. Zhang, A remark on: Lower bounds for eigenvalues of hypersurfaceDirac operators. Math. Res. Lett. 6 (1999), 465–466.MR 1713145 Zbl 0959.58040 153

















Index

3-Sasakian structure, 225, 267

associated vector bundle, 39auxiliary line bundle, 45

canonical Laplacian, 205canonical Weyl structure, 259character, 307class function, 308Clifford algebra, 11Clifford bundle, 40Clifford multiplication, 31Clifford product, 42closed Weyl structure, 50compact operator, 118complex contact structure, 272complex spin group, 26complex volume element, 30conformal complex spin group, 26conformal Dirac operator, 251conformal Laplacian, 138conformal spin group, 22conformal spin Laplacian, 253conformal structure, 49curvature tensor, 43

density bundle, 49determinant line bundle, 46Dirac operator, 62Dirac–Witten operator, 150dominant weight, 337, 338

exact Weyl structure, 50

Faraday form, 50�rst Chern class, 88Fourier transform, 100Fredholm operator, 119

fundamental representation, 338fundamental weight, 338

Gauduchon metric, 262generalized Killing spinor, 242generic 3-forms, 229generic 4-form, 234gradients„ 56

Hermitian product, 33Hodge manifold, 269hyper-Hermitian structure, 260hyper-Kähler manifold, 266hyper-Kähler structure, 226

imaginary Killing spinor, 133index of the Dirac operator, 96integral weight, 318irreducible character, 307

Kähler structure, 161Kählerian Killing spinor, 172Killing spinor, 131, 133

Lee form, 50Levi-Civita connection, 42local gauge, 42

maximal vector, 336

nearly Kähler structure, 229Nijenhuis tensor, 224

open star, 86

parametrix, 112Penrose operator, 60positive quaternion-Kähler manifold,

266

452 Index

positive root, 331projectable spinor, 285projectable vector �eld, 75pseudo-differential operator, 107

quaternion-Kähler manifold, 176, 266quaternionic bundle, 266quaternionic structure, 36

real Clifford algebra, 15real Killing spinor, 133, 236, 239real structure, 36regular element, 340regular Sasakian structure, 267regular vector, 330Ricci tensor, 51Riemannian cone, 224root, 322root system, 330

Sasakian structure, 223scalar curvature, 51Schrödinger–Lichnerowicz formula,

83, 253Schur lemma, 306Schwartz functions, 99simple root, 330simplex, 85singular element, 340singular vector, 330smoothing operator, 110

Sobolev spaces, 116spin curvature operator, 43spin group, 22spin structure, 40, 52Spinc structure, 45, 52spinor bundle, 41spinor �eld, 42spinorial metric, 41standard horizontal vector �eld, 77standard metric, 262standard vertical vector �eld, 77Stein–Weiss operators, 56Stiefel–Whitney class, 90suspension, 60symmetric space, 369

triangulation, 85twistor space, 268twistor-spinor, 133

universal property, 12

weight, 318weight lattice, 318weighted Dirac operator, 263Weyl chamber, 330Weyl group, 319Weyl structure, 49Weyl–Einstein structure, 52

Yamabe operator, 138

Documents

EMS Monographs in Mathematics - 213.230.96.51:8090