Schedule and abstracts of short talks

Schedule

Monday, August 27, 5.30 pm

Session A :

1. Alma L. Albujer : Some rigidity results for complete spacelike hypersurfaces with constant

mean curvature in certain Lorentzian spaces

2. Simone Gutt: Symplectic submanifolds of symplectic manifolds

3. Gabriel Ruiz Hernandez : Surfaces in ℝ4 with constant principal angles with respect to a

plane

Session B:

1. Marian Munteanu : Biharmonic ideal hypersurfaces in Euclidean spaces

2. Jonatan Herrera : On the interplay between Finsler and Lorentzian manifolds: the

Busemann completion

3. Michel Nguiffo Boyom : A characteristic invariant of statistical models

Tuesday, August 28, 11.15 am

Session A:

1. Francisco Palomo : Spacelike surfaces through the lightcone of 4-dimensional Lorentz-

Minkowski spacetime

2. Miguel Domínguez-Vázquez : On the existence of inhomogeneous isoparametric foliations of

higher codimension on complex projective spaces

3. Tommy Murphy : Riemannian foliations of symmetric spaces with complex leaves

Session B:

1. Alicia Prieto-Martin: on η-Einstein para-S manifolds

2. Paul-Andi Nagy: Ricci flat metrics and harmonic morphisms from twists

3. Balázs Csikós : A characterization of harmonic spaces

Wednesday, August 29, 11.15 am

Session A:

1. Miroslava Antic : Affine hypersurfaces with warped product structure

2. Xianfeng Wang : Lagrangian submanifolds in complex projective space with parallel second

fundamental form

3. Lars Schaefer : On nearly pseudo-Kaehler manifolds

Session B:

1. Konstantina Panagiotidou : Results on the Parallelness of the Structure Jacobi Operator of

Real Hypersurfaces in CP2 and CH2

2. Ana Hinic Galic : On the curvatures of subalgebras of nilpotent Lie algebras

3. Sandra Gavino-Fernandez : Quasi-Einstein manifolds in Lorentzian signature

Thursday, August 30, 11.15 am

Session A:

1. Takashi Okayasu : On the construction of complete minimal hypersurfaces in the Euclidean

space

2. Tom Mestdag: The projective Finsler metrizability problem

3. Veronica Martin Molina : A method to obtain Sasaki-Einstein and para-Sasaki-Einstein

metrics from some contact metric (κ,μ)-spaces

Session B:

1. Esteban Calviño-Louzao : Four-dimensional symplectic pairs

2. Hiroshi Matsuzoe : Quasi-Statistical Manifolds and Geometry of Affine Distributions

3. Albert Borbely: On minimal surfaces satisfying the Omori-Yau maximum principle

Abstracts

Alma L. Albujer : Some rigidity results for complete spacelike hypersurfaces with constant mean

curvature in certain Lorentzian spaces spaces

Abstract:

In this communication we present some rigidity results for complete spacelike hypersurfaces with

constant mean curvature in the Lorentzian product space $\mathbbH^n\times\mathbbR_1$. In

order to obtain our results we need to impose some assumptions on the normal hyperbolic angle of

the hypersurface. On the other hand it is well known that a spacelike entire graph in

$\mathbbH^n\times\mathbbR_1$ is not necessarily complete. However, we are able to prove

that this fact is true under our assumptions. As a consequence, we also present non-parametric

versions of our results. Finally, we also give similar results for complete spacelike hypersurfaces with

constant mean curvature in certain Robertson-Walker spacetimes. Our results are a consequence of

the generalized Omori-Yau maximum principle for complete Riemannian manifolds.This

communication is part of a joint work with Fernanda E. C. Camargo and Henrique F. de Lima.

Miroslava Antic : Affine hypersurfaces with warped product structure

Abstract: pdf

Albert Borbely: On minimal surfaces satisfying the Omori-Yau maximum principle

Abstract: Complete minimal immersions satisfying the Omori-Yau maximum principle into R^3 are

considered. Martin and Morales showed that the limit set of a proper minimal immersion into a

convex set can be a small subset of the boundary of the convex set. We will show that the limit set

of a proper minimal immersion satisfying the Omori-Yau maximum principle into a convex set must

be the whole boundary of the convex set. In case of nonproper and nonplanar immersions satisfying

the Omori-Yau maximum principle we prove that the convex hull of the immersion is a half-space or

IR^3.

Michel Nguiffo Boyom: A characteristic invariant of statistical models.

Abstract: One of the relevant tools to study the differential geometry of a statistical model (M,P) for

measurable set are the so-called $\apha$-connections [1], [2]. These connections together with the

Fisher metric (M,g) endow the models with structures of statistical manifold [3]. The aim of the

conference is to discuss two relevant concerns.

(1) Given a straight line D through the LC connection of the Riemannian manifold (M,g) in the space

of linear connections in M, how plenty big is the set of locally flat \alpha$-connections supported by

D? [4]

(2) The second concern is to point out a geometrical invariant which determines (M,P) up to

isomorphism of statistical models.

Some references.

[1] S. Amari and H. Nagaoka. Methods of Information Geometry, Transl Math Monogr. vol 191, Amer

Math Soc-Oxford 2000.

[2] H. Shima. Diff Geom of Hessian manifolds, Word Sci. Publ 2007.

[3] H.Furuhata. Statistical hypersurface in the space of Hessian curvature zero. Diff Geom and its

Appl. 29 (2011) 586-590.

[5] M. Nguiffo Boyom. Cohomology of Koszul-Vinberg algebra. Pacific J. Math vol 225 (2006) 119-153.

Esteban Calviño-Louzao : Four-dimensional symplectic pairs.

Abstract: PDF

Balázs Csikós : A characterization of harmonic spaces

Abstract: We prove that in a complete, connected, and simply connected Riemannian manifold, the

volume of the intersection of two small geodesic balls of equal radii depends only on the distance

between the centers and the common value of the radii if and only if the space is harmonic. (joint

work with Márton Horváth)

Miguel Dominguez-Vazquez : On the existence of inhomogeneous isoparametric foliations of higher

codimension on complex projective spaces

Abstract: pdf

Sandra Gavino-Fernandez : Quasi-Einstein manifolds in Lorentzian signature

Abstract: pdf

Simone Gutt : Symplectic submanifolds of symplectic manifolds

Abstract: Space forms are locally symmetric complete connected Riemannian manifolds of constant

sectional curvature; they are one class of ambient manifolds whose submanifolds have been widely

studied. We define a symplectic analog of space forms as a class of locally symmetric, complete,

connected symplectic manifolds which we completely describe. We prove a symplectic version of the

fundamental theorem of "local submanifold geometry".

Jonatan Herrera : On the interplay between Finsler and Lorentzian manifolds: the Busemann

completion.

Abstract: pdf

Ana Hinic Galic : On the curvatures of subalgebras of nilpotent Lie algebras

Abstract: PDF

Veronica Martin Molina: A method to obtain Sasaki-Einstein and paraSasaki-Einstein metrics from

some contact metric (κ,μ)-spaces

Abstract: pdf

Hiroshi Matsuzoe : Quasi-Statistical Manifolds and Geometry of Affine Distributions

Astract: pdf

Tom Mestdag : The projective Finsler metrizability problem

Abstract: This talk is concerned with the problem of determining whether a projective-equivalence

class of sprays is the geodesic class of a Finsler function. We address both the local and the global

aspects of this problem. We present our results mostly in terms of a multiplier, that is, a type (0,2)

tensor field along the tangent bundle projection. In the course of the analysis we consider several

related issues of interest including the positivity and strong convexity of positively-homogeneous

functions, the relation to the so-called Rapcsak conditions, some peculiarities of the two-dimensional

case, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and

projective classes of sprays. This talk is based on joint work with M. Crampin and D.J. Saunders.

Marian Munteanu : Biharmonic ideal hypersurfaces in Euclidean spaces

Abstract: This is a joint work with Bang Yen Chen. Let $x:M \to E^m$ be an isometric immersion from

a Riemannian n-manifold into a Euclidean m-space. Denote by $\Delta$ and $\vec x$ the Laplace

operator and the position vector of $M$, respectively. Then M is called biharmonic if $\Delta^2 \vec

x=0$. The following Chen's biharmonic Conjecture made in 1991 is well-known and stays open: The

only biharmonic submanifolds of Euclidean spaces are the minimal ones. We prove that the

biharmonic conjecture is true for $\delta(2)$-ideal and $\delta(3)$-ideal hypersurfaces of a Euclidean

space of arbitrary dimension.

Tommy Murphy : Riemannian foliations of symmetric spaces with complex leaves

Abstract: PDF

Paul-Andi Nagy : Ricci flat metrics and harmonic morphisms from twists

Abstract: I will outline the classification of Kaehler metrics admitting homothetic foliations by

complex curves. These arise as deformations that preserve the symplectic structure of the Calabi

metrics on projectivised bundles. Applications to the construction of new Ricci flat almost Kaehler

metrics as well as harmonic morphisms will be presented. This will be based on joint works with

S.G.Chiossi, Bull. London Math. Soc. 44 (2012) 113-124 and arXiv:1102.1995, to appear in Annali

Scuola Normale Pisa.

Takashi Okayasu : On the construction of complete minimal hypersurfaces in the Euclidean space

Abstract: In the famous paper "Minimal cones and the Bernstein problem", Bombieri, De Giorgi and

Giusti constructed the first example of a complete minimal hypersurface in $2m$-dimensional

Euclidean space that has $O(m)\times O(m)$-invariant symmetry. Later in 2005, Alencar et al.

classified those minimal hypersurfaces completely. Both authors defined some vector field whose

integral curve become the generating curve for $O(m)\times O(m)$-invariant minimal hypersurface.

They analysed the vector field carefully to conclude the existence of minimal hypersurfaces.

In our talk, we will give a completely new and direct proof of the existence of $O(m)\times O(m)$-

invariant minimal hypersurfaces. Moreover, we will show that we can construct many complete

minimal hypersurfaces in the Euclidean spaces for every cohomogeneity 2 representation of

orthogonal group. We use a free parameter, instead of the arc length parameter, to represent the

generating curve $(x(t), y(t)$ for minimal hypersurfaces. We reduce the minimal surface equation to

a system of ordinary differential equations of $x$ and $y$. The key point of our proof is that this

system of ODE has a first integral, and by using this first integral we obtain an equivalent system of

ODE whose solutions are easy to find. Our method can be applied to construct complete minimal

hypersufaces in the hyperbolic spaces.

Francisco Palomo : Spacelike surfaces through the lightcone of 4-dimensional Lorentz-Minkowski

spacetime

Abstract: pdf

Konstantina Panagiotidou : Results on the Parallelness of the Structure Jacobi Operator of Real

Hypersurfaces in CP2 and CH^2

Abstract: pdf

Alicia Prieto-Martin : On η-Einstein para-S manifolds

Abstract: pdf

Gabriel Ruiz Hernandez : Surfaces in $\R^4$ with constant principal angles with respect to a plane

Abstract: We study surfaces in $\R^4$ whose tangent spaces have constant principal angles with

respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal

angles.The existence of such surfaces turns out to be equivalent to the existence of a special local

symplectomorphism of $\R^2$. We classify all surfaces with one principal angle equal to $0$ and

observe that they can be constructed as the union of normal holonomy tubes. We also classify the

complete constant angles surfaces in $\R^4$ with respect to a plane. They turn out to be extrinsic

products.

Lars Schaefer : On nearly pseudo-Kaehler manifolds

Abstract: pdf

Xianfeng Wang : Lagrangian submanifolds in complex projective space with parallel second

fundamental form

Abstract: From the Riemannian geometric point of view, one of the most fundamental problems in

the study of Lagrangian submanifolds is the classification of Lagrangian submanifolds with parallel

second fundamental form. In the 1980's, H. Naitoh completely classified the Lagrangian submanifolds

with parallel second fundamental form and without Euclidean factor in complex projective space, by

using the theory of Lie groups and symmetric spaces. He showed that such a submanifold is always

locally symmetric and is one of the symmetric spaces:

$\bf SO(k+1)/\bf SO(k)\,(k\geq 2)$, $\bf SU(k)/\bf SO(k)\,(k\geq 3)$, $\bf SU(k)\,(k\geq 3)$,

$\bf SU(2k)/\bf Sp(k)\,(k\geq 3)$, $\bf E_6/\bf F_4$.

In this talk, we completely classify the Lagrangian submanifolds in complex projective space with

parallel second fundamental form by an elementary geometrical method. We prove that such a

Lagrangian submanifold is either totally geodesic or the Calabi product of a point with a lower

dimensional Lagrangian submanifold with parallel second fundamental form, or the Calabi product of

two lower dimensional Lagrangian submanifolds with parallel second fundamental form, or one of

the standard symmetric spaces: $\bf SU(k)/\bf SO(k)\, (k\geq 3)$, $\bf SU(k)\, (k\geq 3)$, $\bf

SU(2k)/\bf Sp(k)\, (k\geq 3)$, $\bf E_6/\bf F_4$. This was joint work with Prof. F. Dillen, Prof.

H. Li and Prof. L. Vrancken.

Affine hypersurfaces with warped productstructure

The aim of this work is to classify all the strictly locally convex affine hy-persurface Mn+1, n ≥ 2 in Rn+2 such that there exists an affine hypersphereMn in Rn+1 such that Mn+1 = I ×ρ Mn, where I ⊂ R and the function ρdepends only on I, meaning Mn+1 admits a warped product structure. If ∂

∂t

and ∂∂xi

, i = 1, . . . , n span the tangent bundles of I and Mn, respectively, thenthere exist differentiable functions λ1, λ2, µ1 and µ2 such that the differencetensor and the shape operator are of the following form

K(∂

∂t,

∂

∂t) = λ1

∂

∂t, K(

∂

∂t,

∂

∂xi

) = λ2∂

∂xi

,

S∂

∂t= µ1

∂

∂t, S

∂

∂xi

= µ2∂

∂xi

, ∀i = 1, . . . , n.

Conversely, suppose Mn+1 is a locally strongly convex hypersurface of theaffine space Rn+2 such that its tangent bundle is an orthogonal sum, withrespect to the metric h, of two distributions, one-dimensional D1 spanned byunit vector field T and n-dimensional D2 (n ≥ 2), with orthonormal frameX1, X2, . . . , Xn such that

K(T, T ) = λ1T, K(T, X) = λ2X,

ST = µ1T, SX = µ2X, ∀X ∈ D2.

Let γ1, γ2 : R → R be functions such that γ′1 6= 0, γ′1γ′′2 6= γ′′1γ′2. Then Mn+1

an affine hypersphere such that KT = 0 or is affine congruent to one of thefollowing immersions

1. f(t, x1, . . . , xn) = (γ1(t), γ2(t)g2(x1, . . . , xn)), where g2 : Rn → An+1 isa proper affine hypersphere centered at the origin, for γ1, γ2 such thatγ2 6= 0, γ′1γ2−γ1γ

′2 6= 0, and moreover, sgn(γ′1γ2) = sgn(γ′1γ2−γ1γ

′2) =

sgn(γ′1γ′′2 − γ′′1γ′2),

1

Miroslava Antic

2. f(t, x1, . . . , xn) = γ1(t)C(x1, . . . , xn)+γ2(t)en+1, where C : Rn → An+2

is an improper affine sphere given by

C(x1, . . . , xn) = (x1, . . . , xn, p(x1, . . . , xn), 1)

, with the affine normal en+1, for γ1, γ2 such that sgn(γ′1γ′′2−γ′′1 γ′2

γ′1) =

−sgnγ1,

3. f(t, x1, . . . , xn) = C(x1, . . . , xn)+γ2(t)en+1+γ1(t)en+2 where C : Rn →An+2 is previously given improper affine sphere, for γ1, γ2 such thatsgn(γ′1γ

′′2 − γ′′1γ′2) = sgnγ′1.

This is a join work with F. Dillen, K. Schoels and L. Vrancken

2

Four-dimensional symplectic pairsEsteban Calvino-Louzao

University of Santiago de Compostela (Spain)(Joint work with E. Garcıa-Rıo, M.E. Vazquez-Abal and R. Vazquez-Lorenzo)

A symplectic pair on a four-dimensional manifold M is a pair of nontrivial closedtwo-forms (ω, η) of constant and complementary ranks, for which ω (resp., η) re-stricts to the leaves of the kernel foliation of η (resp., ω) as a symplectic form [2].Equivalently a symplectic pair can be defined by two symplectic forms (Ω+,Ω−)satisfying

Ω+ ∧ Ω− = 0, Ω+ ∧ Ω+ = −Ω− ∧ Ω−

Let (M, g) be a Riemannian manifold, then symplectic pairs induce an almost-Kahler and an opposite almost-Kahler structure on (M, g). Some rigidity resultsare well known for symplectic pairs. For instance, in [1] the authors show thatany almost-Kahler and opposite Kahler structure (J, J ′) on a four-dimensional Rie-mannian manifold with J-invariant Ricci operator is locally isometric to the uniqueproper three-symmetric space.

The situation in the indefinite setting is richer than in Riemannian context wheresymplectic pairs can induce almost-Kahler and opposite almost-Kahler structuresor almost-paraKahler and opposite almost-paraKahler ones. The purpose of thistalk is twofold. Firstly, we show an extension of the rigidity result of [1] to theindefinite case and secondly we construct four-dimensional manifolds admittingalmost-paraKahler and opposite paraKahler structures (J, J′) with J-invariant Riccioperator which are not homogeneous. Moreover we show that these structures arenot rigid even in the homogeneous case [3].

References

[1] V. Apostolov, J. Armstrong, T. Draghici, Local rigidity of certain classes of almost Kahler

4-manifolds. Ann. Global Anal. Geom., 21, (2002) 151-176.

[2] G. Bande and D. Kotschick, The geometry of symplectic pairs, Trans. Am. Math. Soc., 358,(2006), 1643–1655.

[3] E. Calvino-Louzao, E. Garcıa-Rıo, M.E. Vazquez-Abal and R. Vazquez-Lorenzo, Local rigid-

ity and nonrigidity of symplectic pairs. Ann. Global Anal. Geom., 41, (2012) 241-252.

1

On the existence of inhomogeneous

isoparametric foliations of higher codimension

on complex projective spaces

Miguel Domınguez-Vazquez

PADGE2012, Leuven 2012

A hypersurface of a Riemannian manifold is called isoparametric if its nearbyparallel hypersurfaces have constant mean curvature. In Euclidean and realhyperbolic spaces these objects are homogeneous as shown by Segre and Cartan.However, in spheres there are inhomogeneous examples, which increases theinterest of this topic (see [3] for a new advance on this problem). Recently,Dıaz-Ramos and the author [1] have constructed examples of inhomogeneousisoparametric hypersurfaces in complex hyperbolic spaces as well.

The foundations of isoparametric submanifolds of higher codimension inspace forms were established by Terng [4]. It turns out that every isopara-metric submanifold extends to a global (singular Riemannian) foliation of thewhole space. Thorbergsson [5] showed that every (irreducible) isoparametric fo-liation of codimension greater than one on a sphere is homogeneous, in contrastto the codimension one case, where inhomogeneous examples are known.

In [2], a complete classification of (irreducible) isoparametric foliations ofcodimension greater than one on complex projective spaces is obtained. Sur-prisingly, there exist inhomogeneous examples. Moreover, we proved that everyirreducible isoparametric foliation on the complex projective n-space is homo-geneous if and only if n + 1 is a prime number.

The aim of the talk will be to explain the results of [2] mentioned above.

References

[1] J. C. Dıaz-Ramos, M. Domınguez-Vazquez, Inhomogeneous isoparametrichypersurfaces in complex hyperbolic spaces, Math. Z., to appear.

[2] M. Domınguez-Vazquez, Isoparametric foliations on complex projectivespaces, arXiv:1204.3428.

[3] R. Miyaoka, Isoparametric hypersurfaces with (g,m) = (6, 2), Ann. ofMath., to appear.

[4] C. L. Terng, Isoparametric submanifolds and their Coxeter groups, J. Dif-ferential Geom. 21 (1985), 79–107.

[5] G. Thorbergsson, Isoparametric foliations and their buildings, Ann. ofMath. (2) 133 (1991), 429–446.

Conference on Pure and Applied Differential GeometryAugust 27–August 30, 2012

KU Leuven, Belgium

Quasi-Einstein manifolds in Lorentzian signatureby S. Gavino-Fernandez

Email: sandra.gavino@usc.es

Faculty of Mathematics, University of Santiago de Compostela, 15782Santiago de Compostela, Spain.

Summary

A pseudo-Riemannian manifold (M, g) of dimension n + 2, n ≥ 1, is quasi-Einstein if there exists a smooth function f : M → R such that

ρ+ Hesf −µdf ⊗ df = λ g, (1)

where ρ and Hesf are the Ricci tensor and the Hessian of f , for some constantsµ, λ ∈ R [1], [2]. If the function f is constant we get the Einstein equation and ifµ = 0 we obtain the gradient Ricci soliton equation [3]. Moreover, the existenceof quasi-Einstein metrics is closely related to the existence of warped productEinstein metrics. Indeed, if M ×f F is Einstein, then φ = −(dimF ) ln f is aquasi-Einstein structure: ρ+ Hesφ− 1

dimF dφ⊗ dφ = λ g.We present a review of some recent results on the geometry of Lorentzian

quasi-Einstein metrics. After emphasizing its relation with gradient Ricci soli-tons and warped product Einstein spaces, we focus on locally conformally flatmetrics. In this case (1) provides enough information on the geometry of (M, g)and the geometry of the level sets of the potential function f to obtain a localclassification of such manifolds. We emphasize that the existence of exampleswhere the level sets of f are degenerate hypersurfaces allow us to construct newLorentzian examples without Riemannian analog. Plane waves play an essentialrole in our analysis.

References

[1] G. Catino, C. Mantegazza, L. Mazzieri and M. Rimoldi, Locally confor-mally flat quasi-Einstein manifolds, J. Reine Angew. Math., to appear.

[2] Jeffrey S. Case, Singularity theorems and the Lorentzian splitting theoremfor the Bakry-Emery-Ricci tensor, J. Geom. Phys. 60 (2010), 477-490.

[3] M. Brozos-Vazquez, E. Garcıa-Rıo and S. Gavino-Fernandez, Locally con-formally flat Lorentzian gradient Ricci solitons, J. Geom. Anal., DOI:10.1007/s12220-011-9283-z.

[4] M. Brozos-Vazquez, E. Garcıa-Rıo and S. Gavino-Fernandez, Locally con-formally flat Lorentzian quasi-Einstein manifolds, arXiv: 1202.1245v1[math.DG].

1

On the interplay between Finsler and Lorentzian

manifolds: the Busemann completion

Jonatan Herrera

Abstract

In this talk, we will describe the notion of Busemann completion on Finslermanifolds, a construction which is motivated by the study of the causal bound-ary in stationary spacetimes. In addition, we will also present the general-ization to Finsler manifolds of two classical constructions of the RiemannianGeometry, the Cauchy completion and the Gromov compactification. Finally,we will emphasize the main differences with the Riemannian case and the re-lations between them.

This talk is based on the following work in collaboration with Jose Luis Flores(U. Malaga) and Miguel Sanchez (U. Granada),

J.L. Flores, J. Herrera, M. Sanchez, Gromov, Cauchy and causal bound-aries for Riemannian, Finslerian and Lorentzian manifolds (2010), toappear in Memoirs of the AMS. Available at arXiv:1011.1154.

1

PADGE 2012Conference on Pure and Applied Differential Geometry,KU Leuven, Belgium,August 27- August 30, 2012

ON THE CURVATURES OF SUBALGEBRAS OF NILPOTENT LIEALGEBRAS

ANA HINIC GALIC

ABSTRACT: A metric Lie algebra g is a Lie algebra equipped with an inner product

〈·, ·〉. Consider a subalgebra h of g, and let ∇h denote the Levi-Civita connection defined

by the restriction of 〈·, ·〉 to h. Similarly, let Rh, Kh, Rich, sh denote the curvature operator,

sectional curvature, Ricci curvature and scalar curvature, respectively, defined by ∇h.

These are the intrinsic curvatures of h. The extrinsic sectional, Ricci and scalar curvatures

on h are obtained by using the Levi-Civita connection ∇ of g.

In this talk we give some relations between intrinsic and extrinsic curvatures when g is

a metric nilpotent Lie algebra. We also present results and examples for various special

cases. This is a joint work with Prof. Dr Grant Cairns, Dr Yuri Nikolayevsky (La Trobe

University, Australia) and Prof. Dr Marcel Nicolau (Universitat Autonoma de Barcelona,

Spain).

Department of Mathematics and Statistics, La Trobe University, Melbourne, Aus-

tralia 3086

E-mail address: A.HinicGalic@latrobe.edu.au

1

A method to obtain Sasaki-Einstein and paraSasaki-Einstein metricsfrom some contact metric (κ, µ)-spaces1

Veronica Martın MolinaUniversidad de Sevilla, Spain. E-mail: veronicamartin@us.es

The contact metric (κ, µ)-spaces are contact metric manifolds (M,φ, ξ, η, g) satisfyingthe following formula for any vector fields X,Y on M :

R(X,Y )ξ = κ (η (Y )X − η (X)Y ) + µ (η (Y )hX − η (X)hY ) ,

for some constants κ and µ, where 2h denotes the Lie derivative of φ in the direction ofthe vector field ξ. This class of Riemannian manifolds was introduced in [1] as a naturalgeneralisation of both the contact metric manifolds such that R(X,Y )ξ = 0 and of theSasakian condition R(X,Y )ξ = η (Y )X − η (X)Y . Despite their technical appearance,there are good reasons for studying then and they have generated an extensive literature.

On the other hand, we have the almost paracontact manifolds, which were defined in[3] as (2n+1)-dimensional smooth manifolds M with a (1, 1)-tensor field ϕ, a vector fieldξ and a 1-form η satisfying that η(ξ) = 1, ϕ2 = I−η⊗ξ and that dimD+ = dimD− = n,where D± is the eigendistribution of ϕ corresponding to the eigenvalue ±1.

Note than when n = 1 these manifolds are Lorentz ones. Paracontact equivalents ofthe notions of contact metric, Sasakian, η-Einstein and the h operator were defined andstudied in [3, 4].

We will see here the method used in [2] to obtain η-Einstein Sasakian or η-EinsteinparaSasakian metrics from non-Sasakian contact metric (κ, µ)-spaces. We will give anexplicit expression for their curvature tensors and the values of κ and µ for which suchmetrics are Sasaki-Einstein and paraSasaki-Einstein.

References

[1] D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifoldssatisfyng a nullity condition. Israel J. Math. 91 (1995), 189–214.

[2] B. Cappelletti Montano, A. Carriazo and V. Martın-Molina, Sasaki-Einstein andparaSasaki-Einstein metrics from (κ, µ)-structures. Submitted. arXiv:1109.6248v1.

[3] S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures onmanifolds. Nagoya Math. J. 99 (1985), 173–187.

[4] S. Zamkovoy, Canonical connections on paracontact manifolds. Ann. Glob. Anal.Geom. 36 (2009), 37–60.

1Joint work with B. Cappelletti Montano (University of Caglieri, ITALY) and A. Carriazo (Universityof Sevilla, SPAIN). A. C. and V. M.-M. are partially supported by the PAI group FQM-327 (Juntade Andalucıa, Spain) and by the MTM2011-22621 grant (MEC, Spain).

Quasi-Statistical Manifolds and Geometry of Affine Distributions

Hiroshi Matsuzoe (Nagoya Institute of Technology)

　Abstract

In this presentation, we study non-integrable generalizations of statistical manifolds andaffine immersions.

The notion of statistical manifold was originally introduced by Lauritzen in statistics, andreformulated by Kurose from the viewpoint of affine differential geometry. Let (M,h) be asemi-Riemannian manifold, and ∇ a torsion-free affine connection on M . The triplet (M,h,∇)is said a statistical manifold if (∇, h) is a (0, 3)-Codazzi pair on M , that is, ∇h is a totallysymmetric (0, 3)-tensor field. It is known that statistical manifolds can be induced from non-degenerate equiaffine immersions in affine hypersurface theory. It is also known that statisticalmanifolds are useful objects in differential geometric theory of statistical inferences, which iscalled information geometry.

A quasi-statistical manifold is a quantum or a non-integrable version of statistical manifold.In information geometry, it is known that affine connections on quantum state spaces admitnon-vanishing torsion in general. Kurose introduced the notion of statistical manifold admittingtorsion to formulate such geometric structures. He also introduced a non-integrable version ofaffine immersions, called affine distributions, and studied fundamental properties of statisticalmanifolds admitting torsion and of affine distributions. Later, the speaker generalized the sta-tistical manifold structure, and introduced the notion of quasi-statistical manifold. Let M be amanifold, h a non-degenerate (0, 2)-tensor field, and ∇ an affine connection. We say that thetriplet (M,∇, h) is a quasi-statistical manifold if the following formula holds:

(∇Xh)(Y,Z) − (∇Y h)(X,Z) = −h(T∇(X,Y ), Z),

where T∇ is the torsion tensor field with respect to ∇. If h is a semi-Riemannian metic, thenthe triplet (M,∇, h) is said a statistical manifold admitting torsion.

In this presentation, we study fundamental properties of quasi-statistical manifolds admittingtorsion and geometry of affine distributions. We then show that a non-degenerate equiaffinedistribution induces a quasi-statistical manifold.

References

[1] S. Amari and H. Nagaoka, Method of Information Geometry, Amer. Math. Soc., OxfordUniversity Press, 2000

[2] T. Kurose, Statistical manifolds admitting torsion, Geometry and Something, FukuokaUniversity, 2007 (in Japanese).

[3] M. Henmi and H. Matsuzoe, Geometry of pre-contrast functions and non-conservativeestimating functions, AIP Conference Proceedings Volume 1340: International Workshopon Complex Structures, Integrability and Vector Fields, Amer. Inst. of Physics, 1340(2011),32–41.

[4] H. Matsuzoe, Statistical manifolds and affine differential geometry Adv. Stud. Pure Math.,57(2010), 303–321.

RIEMANNIAN FOLIATIONS OF KAHLER MANIFOLDS

WITH COMPLEX LEAVES

TOMMY MURPHY

Due to work of Nagy [7] it is known that singular Riemannian folia-tions of compact Kahler manifolds admitting complex leaves are extremelyrigid. Such objects arise, for example, in twistor theory and in the theory ofgroup actions. We obtain classification results for Kahler manifolds, focusingmostly on Hermitian symmetric spaces. For example, the only singular Rie-mannian foliation of complex projective space with complex leaves is givenby the leaves of the twistor fibration π : CP 2n+1 → HPn. We strengthenthis theorem to a local statement for foliations of open subsets of complexprojective space by complex hypersurfaces. Finally we answer a questionof Alfred Gray [1]: when do the tubes around a complex submanifold ofprojective space form a singular Riemannian foliation?

References

[1] Gray, A. The minimal focal distance of a complex hypersurface in complex projectivespace, Conference on Differential Geometry and Topology(Lecce, 1989). Note Mat. 9(1989), suppl., 119-122.

[2] Gray, A. Volumes of tubes about complex submanifolds of complex projective space,Trans. Amer. Math. Soc. 291 (1985), no. 2, 437-449.

[3] Gray, A. Volumes of tubes about Kahler submanifolds expressed in terms of Chernclasses, J. Math. Soc. Japan 36 (1984), no. 1, 23-35.

[4] Murphy, T. Riemannian foliations of projective space with complex leaves, preprint,arXiv:1202.5989.

[5] Murphy, T. Curvature-adapted submanifolds of symmetric spaces, Indiana U. Math.J., to appear.

[6] Murphy, T. Complex Riemannian foliations of Kahler manifolds, preprint.[7] Nagy, P.A. Rigidity of Riemannian foliations with complex leaves on Khler manifolds,

J. Geom. Anal. 13 (2003), no. 4, 659667.

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Francisco J. PalomoSpacelike surfaces through the lightcone of 4-dimensional

Lorentz-Minkowski spacetime

On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowskispace a distinguished smooth function is considered. It is shown how both ex-trinsic and intrinsic geometry of such a surface are codified by this function.The existence of a local maximum is assumed to decide when the spacelikesurface must be totally umbilical, deriving a Liebmann type result. On theother hand, every spacelike surface through the lightcone has a geometrica-lly meaningful lightlike normal vector field. Several sufficient assumptions ona compact spacelike surface with non-degenerate second fundamental formin the direction of that normal vector field are established to get that thesurface is a totally umbilical round sphere. With this aim, we develop anew formula which relates the Gauss curvature of the induced metric andthe Gauss curvature of the second fundamental form. This formula helps toprove that totally umbilical round spheres are the only compact spacelikesurfaces through a lightcone such that the second fundamental form is non-degenerate and its Gauss curvature equals two. Another characterizationsinvolving the first non-trivial eigenvalue of the Laplacian and the area ofthe second fundamental form are also obtained for totally umbilical roundspheres.

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Results on the Parallelness of the Structure Jacobi Operator

of Real Hypersurfaces in CP 2 and CH2

Konstantina PanagiotidouMathematics Division-School of Technology, Aristotle University of Thessaloniki, Greece

e-mail:kapanagi@gen.auth.gr

The following results are based on a joint work with Professor Philippos J. Xenos and are

included in my Ph. D. thesis.

A real hypersurface M in a complex space form Mn(c), is an immersed submanifold and its

dimension is 2n − 1. An almost contact metric structure (φ, ξ, η, g) can be defined on M and it

is induced from the Kaehler metric and the complex structure J of Mn(c). A real hypersurface

is said to be Hopf if the structure vector field ξ is principal, i.e. Aξ = αξ, where A is the shape

operator of M. The classification problem of real hypersurfaces in non-flat complex space form is

of great importance in the area of Differential Geometry. It was initiated by Takagi in 1973 (see

[12], [13]), who classified the homogeneous real hypersurfaces of CPn. Further work on this area

was done by Cecil and Ryan ( see [2]) and finally Kimura in [3] gave the local classification of

Hopf hypersurfaces with constant principal curvatures. In the case of CHn, Berndt in [1] classified

Hopf hypersurfaces with constant principal curvatures.

The structure Jacobi operator of a real hypersurface M is denoted by l and is given by the

relation lX = RξX = R(X, ξ)ξ. It plays an important role in the study of them and many

results concerning different types of parallelness of it such as Lie ξ-parallelness, i.e. Lξl = 0,

Lie parallelness, i.e. LX l = 0, for any vector field X on TM, ξ-parallelness in combination

with other conditions, i.e. ∇ξl = 0, parallelness, i.e. ∇X l = 0, for any vector field X on

TM, semi-parallelness, i.e. R · l = 0 has been proved by many geometers (see [5], [8], [9],

[10], [11]). Motivated by the work that has been done, new results about three dimensional real

hypersurfaces in CP 2 and CH2, when their structure Jacobi operator satisfies certain parallel

conditions, are presented. More precisely, the non-existence of real hypersurfaces in CP 2 or

CH2, whose structure Jacobi operator is Lie D-parallel has been proved, i.e. LX l = 0, where

X ∈ D = ker(η) ( see [6]). Additional, a classification of real hypersurfaces in CP 2 and CH2

equipped with pseudo-parallel structure Jacobi operator, i.e. R(X,Y ) · l = L(X ∧Y ) · l, where

L is a function and L 6= 0, is given (see [7]).

References

[1] J. BERNDT: Real hypersurfaces with constant principal curvatures in complex hyperbolic

space, J. Reine Angew. Math., 395 (1989), 132-141.

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[2] T. CECIL AND P. J. RYAN: Focal sets and real hypersurfaces in complex projective spaces,

Trans. Amer. Math. Soc., 269 (1982), 481-499.

[3] M. KIMURA: Real Hypersurfaces and Complex Submanifolds in complex projective spaces,

Trans. Amer. Math. Soc., 296 (1986), 137-149.

[4] R. NIEBERGALL AND P. J. RYAN: Semi-parallel and Semi-symmetric Real Hypersurfaces in

Complex Space Forms, Kyungpook Math. J., 38 (1998), 227-234.

[5] M. ORTEGA, J. D. PEREZ AND F. G. SANTOS: Non-existence of real hypersurfaces with

parallel structure Jacobi operator in nonflat complex space forms, Rocky Mountain J. Math.,

36 (2006), no. 5, 1603-1613.

[6] K. PANAGIOTIDOU AND PH. J. XENOS: Real hypersurfaces in CP 2 and CH2 whose struc-

ture Jacobi operator is Lie D-parallel, to appear in Note di Matematica.

[7] K. PANAGIOTIDOU AND PH. J. XENOS: Real hypersurfaces equipped with pseudo-parallel

structure Jacobi operator in CP 2 and CH2, submitted for publication.

[8] J. D. PEREZ AND F. G. SANTOS: On the Lie derivative of structure Jacobi operator of real

hypersurfaces in complex projective space, Publ. Math. Debrecen, 66 (2005), 269-282.

[9] J. D. PEREZ, F. G. SANTOS AND Y. J. SUH: Real hypersurfaces in complex projective space

whose structure Jacobi operator is Lie ξ-parallel, Differential Geom. Appl., 22 (2005), no. 2,

181-188.

[10] J. D. PEREZ, F. G. SANTOS AND Y. J. SUH: Real hypersurfaces in complex projective

space whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc. Simon Stevin, 13(2006), no. 3, 459-469.

[11] J. D. PEREZ AND F. G. SANTOS: Real hypersurfaces in Complex Projective Space Whose

Structure Jacobi Operator is Cyclic-Ryan Parallel, Kyungpook Math. J., 49 (2009), 211-219.

[12] R. TAKAGI: On homogeneous real hypersurfaces in a complex projective space, Osaka J.

Math., 10 (1973), 495-506.

[13] R. TAKAGI: Real hypersurfacesin a complex prjective space with constant principal curva-

tures, J. Math. Soc, Japan, 27 (1975), 43-53.

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ON ηα-EINSTEIN PARA-S MANIFOLDS

BENIAMINO CAPPELLETTI MONTANO, LUIS M. FENANDEZ, AND ALICIA PRIETO-MARTIN

Abstract. In paracontact geometry, an n-dimensional smooth manifold Mn has an al-most paracontact f -structure (f, ξα, ηα) if it admits a tensor field f of type (1,1), r vectorfields ξ1, · · · , ξr, (r < n), and 1-forms η1, · · · , ηr satisfying the following compatibilityconditions:

ηα(ξβ) = δαβ ,

ηα f = 0,

f2 = Id−r∑

α=1

ηα ⊗ ξα; α, β = 1, · · · , r.

It is known that here exist a Riemannian metric g on Mn such that

g(fX, fY ) = g(X,Y ) −r∑

α=1

ηα(X)ηα(Y ).

Then, Mn is called an almost r-paracontact metric manifold.

In this communication, we first introduced the notion of a para-S manifold, by fol-lowing the above definition but considering a pseudo Riemannian metric g and we studysome of its properties.

More precisely, since we obtain that there are no Einstein para-S manifolds, we definethe notion of ηα-Einstein para-S manifold as para-S manifold (M2n+s, f, ξα, ηα, g) suchthat the Ricci tensor satisfies:

Ric = ag + b

s∑α=1

ηα ⊗ ηα + (a+ b)∑α6=β

ηα ⊗ ηβ ,

being a, b two differentiable functions.We prove that M2n+s can be projected onto an Einstein para-Kaehler manifold. Finally,we study the conditions for an ηα-Einstein para-S manifold to be ξα-conformally flat andwe obtain a characterization theorem.

Universita degli Studi di Cagliari, Dipartimento di Matematica e Informatica, CagliaryE-mail address: b.cappellettimontano@unica.it

Departmento de Geometrıa y Topologıa, Facultad de Matematicas, Universidad de Sevilla,Apartado de Correos 1160, 41080 Sevilla, Spain

E-mail address: lmfer@us.es, aliciaprieto@us.es

2000 Mathematics Subject Classification. 53C25, 53C40.

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On nearly pseudo-Kahler manifoldsLars Schafer

Leibniz Universitat Hannover

A pseudo-hermitian manifold (M,J, g) is called nearly pseudo-Kahler manifold,if (∇g

XJ)Y is skew-symmetric. The semi-Riemannian setting is more flexibleas the well-studied Riemannian case. This is for instance underlined by theexistence of flat non-Kahlerian examples of nearly pseudo-Kahler manifolds. Inthis talk we report on the ongoing analysis of this class of pseudo-hermitianmanifolds.

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