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Hodge—dirac operators for hyperbolic spacesP. Cerejeiras a & J. Cnops ba Departamento de Matemática , Universidade de Aveiro , Aveiro, 3810, Portugalb VWA/RUG, Rijksuniversiteit Gent , Galglaan 2, Gent, B-9000, BelgiumPublished online: 29 May 2007.

To cite this article: P. Cerejeiras & J. Cnops (2000) Hodge—dirac operators for hyperbolic spaces, Complex Variables, Theory andApplication: An International Journal: An International Journal, 41:3, 267-278

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Hodge-Dirac Operators for Hyperbolic Spaces

F? CEREJEIRAS a'* and J. CNOPS

a Departamento de Matematica, Universidade de Aveiro, 3810 Aveiro, Portugal; VWAIRUG, Rijksuniversiteit Gent, Galglaan 2,B-9000 Gent, Belgium

Communicated by R. Delanghe

(Received February 1999)

In this article we extend the concept of hyperbolically harmonic functions for the upper half plane, first introduced by Leutwiler, to an arbitrary conformally flat manifold. We shall consider the special cases of the two standard models for hyperbolic spaces, namely the Poincare model for the upper half space and the spherical model. For these two mani- folds, an analogue of the Cauchy-Riemann equations, obtained via the Euclidean Dirac operator, is studied, along with particular decompositions of its solutions.

Keywords: Laplace-Beltrarni operator; Hodge system; Poincare model; Hodge-Dirac operators; Cauchy-Riemann operators

1991 Mathematics Subjects Classification: 30635

1 OVERVIEW

In the last decade, several attempts have been made to generalise the framework of Clifford analysis from flat space to more general mani- folds. In [Ll,L2], Leutwiler proposed an original approach for this problem, combining the Hodge-deRham system (d,d*) with the standard Dirac operator to describe the Laplace-Beltrami operator associated to the hyperbolic space IW: obtaining there a system of equa- tions equivalent to Cauchy-Riemann equations. Together with the

*Corresponding author.

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characterisation of Mobius transformations in terms of Vahlen matri- ces and the subsequent establishment of a class of invariant operators under the action of given fractional linear groups (see Cnops, [Cl]), one can perform the extension of Leutwiler's approach to arbitrary n-dimensional conformally flat manifolds.

2 PRELIMINARIES

We introduce now some notations and background material concern- ing Clifford algebras.

Let Rn stand for the vector space with orthonormal base e l , . . . ,en. The real Clifford algebra generated by the vector space Rn, together with the multiplication rules eiej + ejei = -2Sv, i, j = 1 , . . . , n, shall be denoted Ce(n). We define the inner product of two vectors 2 and y' as 2. y' = - (1/2)(2y'+ 92) and set -121' = 2.2. But there is also an outer (exterior) product 2 A y' = ( 1 12) (Ty' - jZ), which can be extended in an associative way. This is the product of exterior algebra, and it is not too hard to show that the Clifford algebra has a basis of elements of the form:

where jl < . . . < jk (if k = 0, we get the element 1). Such an element is called a k-vector; so is a linear combination of several of these elements with equal length k. The space of k-vectors is written clk(n) , and is isomorphic to A ~ R ~ . For an element a of the Clifford algebra, talk will be the k-vector part of a.

A function f : Q -, Ce(n) is said to be monogenic in R c Rn if it is c'(R) and satisfies df=O in 0, where 8 = eidi is the Dirac operator.

For an arbitrary bijective c2 function T from R onto T(0 ) and an operator C acting on C2(R,Ce(n)), we shall denote by 2 the conjugate operator which acts on C 2 ( ~ ( R ) , C e ( n ) ) as C( f o T-' ) = ( L f ) o T - l . The operator C is said to be invariant under the action of T if and only if c = L.

A conformally flat manifold is a domain M in Rn where the metric is changed by a conformal factor X which is a strictly positive C" function.

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HODGE-DIRAC OPERATORS

The metric dsM is then given by

We shall use the notation (M, A) for such a manifold. It is possible to give a slightly more general definition, where the metric is given locally by such a formula; the results below apply to this case as is easily shown glueing together coordinate patches. Notice that conformally flat mani- folds in our sense are automatically orientable.

Let GM be the isometry group of such a manifold. Obviously, GM is also an element of the conformal group of (M, l), and so, if the dimen- sion is at least three, it is a Mobius transformation. It is an elementary consequence of the transformation formula of the Euclidean Laplacian A under conformal transformations (see [C2]) that

THEOREM 1 The operator

is invariant under for all T E GM.

3 HODGE EQUATIONS FOR CONFORMALLY FLAT MANIFOLDS

The factorisation of the Laplacian in terms of the Dirac operator for the quaternionic case by Fueter [Fu], opened the way for a generali- sation of the concept of holomorphic functions to higher dimensions. In fact, the Dirac equation plays, on Euclidean spaces, a role similar to that of the Cauchy-Riemann equations in the complex plane (see [BDS,GM]), its solutions being monogenic functions.

On an n-dimensional Riemannian manifold, the natural counter- parts for harmonic functions are the solutions of the Laplace- Beltrami equation corresponding to such a manifold. It is known that in such case the Laplace-Beltrami operator for scalar functions is given by

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where gq stands for the inverse of the matrix gq associated to the metric g, and )gl = det(go). In our case gq(x) = Sq/A(x) and we obtain for the Laplace-Beltrami operator the expression

since gq = X21n, g'i= A21n and Igl= X2". Leutwiler's approach is based on the decomposition of the Laplace-

Beltrami operator in terms of the Hodge-deRham system for differen- tial forms. In fact, the exterior differential d and its adjoint d* act in the following directions:

and the Laplace-Beltrami operator is defined as (see [L3])

ALB = (d + d*)2

= d*d+ dd*.

We identify real valued functions with 0-forms, having values in A ~ ( R ~ ) , and have the following definition:

DEFINITION 1 (A-harmonic function) Let (M, A) be a conformallyflat manifold. A real valued function f is said to be a A-harmonic function on R c M if it satisfies ALB f = 0 on a. The set of all A-harmonic functions on R will be denoted by 'Flx(f2).

But we can also identify vector valued functions with 1-forms having values in A@?) as follows: using orthogonal coordinates in Rn we identify in a point x the orthogonal unit vectors ei with the elements dxi/A(x) which are orthogonal units in that point since

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HODGE-DIRAC OPERATORS 271

This way, for a scalar function f we can identify (d+d*) f(x) with A(x)df (x) (notice that d* f = 0 anyway), while for a vector valued func- tion ii it can be shown (see [C3]) that (d + d")G(x) can be identified with

This prompts the definition:

DEFINITION 2 (A-monogenic function) On the same assumptions, a vector valued function ii is said to be a A-monogenic function on R c M if it satisfies

on fl. The set of all A-monogenics on R will be denoted by Mx(R).

Remark T h s procedure can, of course, be extended to k-vectors with k > 1 (see again [C3]). However, if we limit ourselves to vector func- tions, we have the important property that locally a A-monogenic func- tion has the form df, where f is A-harmonic. This is not true for higher order k-vectors where curvature terms play an important role. There- fore, we limit ourselves here to scalar and vector functions. This way, the Hodge equation provides a system of equations similar to the Cauchy-Riemann system in the complex plane, and allows to study several properties of the A-harmonic and A-monogenic solutions. It has been proved, for the case of the upper half space, (that is, A(x) = x,), there exists a basis for the regular polynomials in Leutwiler's sense.

Also.

(i) The notations here adopted generalise the concept of hyperboli- cally harmonic and hyperbolically monogenic functions, intro- duced by Leutwiler and Cnops, respectively, for the planar model.

(ii) Equation (4) stands for the equivalent on (M, A) of the Hodge system studied in [Ll]. We shall refer to it as Hodge equation for ( M , A).

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Finally, the relation (see [C2])

links the Laplace-Beltrami operator for the manifold (M, A) and the invariant Laplacian from Theorem 1. Since 1 is obviously a constant function we get

LEMMA 2 The Laplace-Beltrami operator is invariant under the action

of GM.

There is a very special case where X is the 'inner product' with a sphere, X(x) = (x2 + m2 + r2 - 2x. m( for a finite sphere with centre m and radius r, or X(x) = Ix. m - bJ for a hyperplane with unit normal m and real height parameter b. In this case, the sphere separates Rn into two parts. Choosing M to be one of these, we have that the isometry group GM acts transitively on M (the fixgroup of a point is then iso- morphic to O(n)). We have here

n(n - 2)r2 (finite sphere of radius r),

(hyperplane).

4 HYPERBOLIC MODELS

These considerations provide the two main models for hyperbolic spaces, i.e. spaces with constant negative curvature K = -1, for which we now give a short description. The following models, both due to PoincarC, are obtained by introducing in an n-dimensional vector space a suitable metric which will induce the hyperbolic structure.

Planar model - Denote by R: the upper half space

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TABLE I Hyperbolic models

Metric Laplace-Beltrami operator

Upper half space W: AP(x) = Xn x:A + (2 - n)x,& Unit disk D A&) = (1 - 1x1~) (1 - Ix12)'A + 2(n - 2)(1 - (xIZ)a,

We obtain the manifold (R:, x,), on which, according to the previous section, the isometry group, which in this case simply is M (Rn-') , the group of Mobius transformations on It-', acts transitively. Notice that (R:, x,) has indeed constant curvature K = - 1.

Spherical model - let D be the unit disk

Here we have the manifold (D, (1 - 1x1~)) which is isometric to the first. An example of such an isometry is the Cayley transformation (up to a constant factor 4, which can be disregarded). Using the subscripts S and 7' for spherical and planar models we obtain Table I.

5 Xp-MONOGENIC FUNCTIONS

The Laplace-Beltrami equation for the planar model

is a particular case of the elliptical partial equations studied by Huber in [HI. The Xp-harmonic functions are then a particular case of gen- eralised axially symmetric potentials, and therefore are analytic in every region which does not intersect the hyperplane x, = 0. The Hodge equa- tion associated to (6) takes the form:

and so the last component of an Xp-monogenic is zero in Itn-'. It is relevant here to use decompositions of functions and oper-

ators relative to the direction perpendicular to IW"-'. We therefore split

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the variable x into 2 + xnen. Likewise, we write any vector valued func- tion u' as

and write

Let R = fix]-7, T [ be a cylindrical domain, with r > 0 and fi c Itn-'. Also, denote by VF(R) the space of all Xp-monogenic homo- geneous polynomials of degree m in R. Then each element &, of this space admits a decomposition of the form:

where each ak(2) = a:(2)el + . . . + al(2)en is a vector valued poly- nomial homogeneous of degree m - k .

THEOREM 3 A homogeneous polynomial &(x) = C;==, ak(jl)x,k of degree m 1 1 is Xp-monogenic i f and only if the associated polynomials ak = aLe, + . . . + a[en satisfy

(i) (n - 2)a; - 0 (ii) (n - 2 - k)a[ = c&' da:-l Idxi, for k = 1 , . . . , m

(iii) ka; =d~k"_~ /c?x~ , f o r k = l , . . . , m , i = l , ..., n - 1 (iv) a a ~ - l / a x j = a a ~ - l / d x i , f o r k = l , ..., m,i , j=l , . . . , n - 1 , i # j

in fi.

Proof Using, for each ak the decomposition ak(%) = &(%) + ai(j?.)en the application of the Hodge Eq. (7) to the polynomial (8) gives Eqs. (i)-(iv).

An immediate consequence of this theorem is that each polynomial p', in V;(R) is generated by a real valued homogeneous polynomial a,"_, (of degree m - n + 2) and by a vector polynomial iio = aAel + . . . + a,"-'e,_l, homogeneous of degree m. Hence, we can define its

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characteristic function and - at least for polynomials - Leutwiler's polynomial decomposition established in [L2] holds for n > 3.

DEFINITION 3 For J?,,, = p;el + . . . + p i e n E V,"(R), we define its characteristic function QFm : fi -+ R as

Note that in terms of (8) the characteristic function equals a,"_,(%). With this definition, we construct the two spaces

and

having the property that if &, E V,"(R) then !Pflm E 0 if and only if p', = 0 in 0. Also, remark that V,"(R) is empty (apart from the zero element) whenever m < n - 2. Therefore,

and we are now in condition to determine the projection operators of V,"(R) onto U,"(R) and V r ( R ) , which shall be denoted Pun. and Py7, respectively. We have

m > n - 2 and n odd,

. m 2 n - 2 and n even,

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and

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pv;(flm) = { m 2 n - 2 and n odd, (14)

m 2 n - 2 and n even.

Moreover, due to Theorem 3 we obtain for the homogeneous poly- nomials of degree m - k the expressions

leading to the recursion formulas for the homogeneous polynomials in (14)

(en8)k+1a,"_2 (%)en an-3+2s = S = 1,2, . . . , 2$-'(s - l)! (n - 3 + 2j)

(en 8) 2"a,"_2 (2) en (16)

s = 1,2, . . . , 2~(s)! (n - 3 + 2j)

which, together with the fact that qFm(2) = ak2(2) , proves the exist- ence of an isomorphism between the space VT(R) and the space of all homogeneous real valued polynomial of degree m - n + 2, that is dim(VT(0)) = m!/((n - 2)!(m -n+2) ! ) fo rm>n-2 .

6 As-HARMONIC FUNCTIONS

For the spherical model, with the metric given by As = 1 - (x12, we obtain for the Laplace-Beltrami operator (see Table I)

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which is easily seen to be invariant under the action of the fixgroup Gs. This is the group of Mobius transformations leaving the unit ball invariant and, as already indicated, is isomorphic to M(IW"-'). An important subgroup is the group of orthogonal transformations, which suggests considering annular domains centred at the origin. For L2(S), as is well known, an orthonormal basis of spherical harmonics can be taken. which we can write

{s;, i = I , . . . , Nk, k E No}, (18)

where k is the degree of the spherical harmonic. We can decompose As-harmonic functions in terms of these, and study the coefficients.

DEFINITION 4 ((i, k)-coefficient) Let f be a As-harmonic function in an annular domain centred at the origin, with interior radius R1 and exterior radius R2. Then f can be written as

where each f;, denoted as the (i, k)-coefficient of the functionf, is given by

where (., .)s is the usual inner product of L2(S) and r = 1x1.

Then (see [C2,Ce]) we have that each f; satisfy the differential equation

(1 - r2)ry1' + [(n - 1 + 2k) + (n - 2k - 3)r2] y1 + 2(n - 2)kry = 0

(21)

f o r a l l k ~ N ~ a n d i = 1 , . . . ,Nk, wherer=JxJ €]R1,R2[. This equation has, for each n and k fixed, two linearly independent

solutions in ]R1, R2[, namely, for k = 0, f;,,(r) = 1 and

c;-~ r2s-n+2

( 2 ( - 1 ) ~ 2 ~ - n + 2 , n odd,

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Here c/-~ is the binomial symbol, and stands for ( n - 2 ) ! / (s!(n - 2 - s)!). For k > 0 w e have

n fi,k ( r ) = r 2-n-2k F 1 - - , 2 - i 2

and

n n 1 - - - + k ; r 2 ) n e v e n o r ( n o d d a n d r s l ) ,

f ; k M = 2 2

n oddand r > 1 .

THEOREM 4 Let f be a As-harmonic function in an annular domain cen- tred at the origin, with interior radius R1 and exterior radius R2. Then each coeficient fk is a linear combination of&\k and&$.

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[BDS] F. Brackx, R. Delanghe and F. Sommen. Clifford Analysis, Pitman Advanced Publishing Program, 1982.

[Cl] J. Cnops. Spherical geometry and Mobius transformations, pp. 75-84, in: Clif- ford Algebras and their Applications in Mathematical Physics, Kluwer Academic Publishers, Deinze, 1993.

[C2] J. Cnops. Hurwitz pairs and applications of Mobius transformations, Habili- tation Thesis - Ghent, 1994.

[C3] J. Cnops. Monogenic vector fields on the Poincari manifold, Complex Variables, 1999,39,255-277.

[Ce] P. Cerejeiras. 0 Operador de Dirac em Espapos Hiperbolicos, Ph.D. thesis, Aveiro, 1997.

[Fu] R. Fueter. Die Funktionentheorie der Differentialgleichungen Au = 0 und AAu = 0 mit vier reellen Variablen, Comment. Math. Helv., 1934/5,7, 307-330.

[GM] J. Gilbert and M. Murray. Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, 1991.

[HI A. Huber. On the uniqueness of generalized axially symmetric potentials, Ann. Math., 1954,60(2), September, 351 -358.

[Ll] H. Leutwiler. Modified Clifford analysis, Complex Variables, 1992,17, 153-171. [L2] H. Leutwiler. Modified quaternionic analysis in IR3, Complex Variables, 1991,20,

19-51. [L3] H. Leutwiler. Remarks on modified Clifford analysis, pp. 389-397, in: Potential

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