Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces (Lecture Notes in Mathematics 1598)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen

1598

Jfirgen Berndt Franco Tricerri Lieven Vanhecke

Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces

~ Springer

Authors Jiirgen Berndt Mathematisches Institut Universit~it zu K61n Weyertal 86-90 D-50931 K61n, Germany E-mail: berndt @mi.uni-koeln.de Franco Tricerri t formerly: Dipartimento di Matematica "U. Dini" Universit~ di Firenze Lieven Vanhecke Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200 B B-3001 Leuven, Belgium E-mail: fgaga03 @cc 1 .kuleuven.ac.be

Mathematics Subject Classification (1991): 53C20, 53C25, 53C30, 53C40, 22E25

ISBN 3-540-59001-3 Springer-Verlag Berlin Heidelberg New York

CIP-Data applied for 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready TEX output provided by the authors SPIN: 10130255 46/3142-543210 - Printed on acid-free paper

Preface The fundamental conjecture about harmonic manifolds has been a source of intensive research during the past decades. Curvature theory plays a fundamental role in this field and is intimately related to the study of the Jaeobi operator and its role in the geometry of geodesic symmetries and reflections on a Riemannian manifold. Our research about harmonic manifolds led in a natural way to the study of spaces with volume-preserving geodesic symmetries and several related classes of manifolds, in particular commutative spaces and Riemannian manifolds all of whose geodesics are orbits of one-parameter groups of isometries. It was also a part of our motivation for developing the theory of homogeneous structures. In this work, the classical and the generalized Heisenberg groups provided a rich collection of examples and counterexamples. It is also well-known that the latter ones take a nice and important place in the florishing research about nilpotent Lie groups and nilmanifolds. Recently the picture has changed drastically on the one hand by the positive results of Z.I. Szabd and on the other hand by the discovery of the Damek-Ricci harmonic spaces which are the first counterexamples to the fundamental conjecture. These manifolds are Lie groups whose Lie algebras are solvable extensions of generalized Heisenberg algebras. The discovery of these spaces led to a renewed interest in the field, in particular because, just as in the case of the generalized Heisenberg groups, they were found during the work in harmonic analysis and not much attention was given to the detailed study of their geometry and the properties of their curvature as reflected in those of the Jacobi operator. These notes present a more detailed treatment of this aspect for both classes of manifolds. We do this by relating our study to the several classes of Riemannian manifolds which we have introduced or studied recently in the field of the geometry of the Jaeobi operator. It will be shown that they have a rich geometry and provide again answers, examples and counterexamples for several other conjectures and open problems. It is our hope that these notes will stimulate further fruitful research in this area. During our work in this field, many friends, collaborators and colleagues have contributed by means of their lectures, discussions, joint work, encouragement and interest. They all made this result possible. We are very grateful for their help and for sharing with us their interest and love for mathematics and in particular for

geometry. In particular, we thank O. Kowalski, F. Prfifer and F. Ricci. We also take the opportunity to thank our respective universities, the Consiglio Nazionale delle Richerche (Italy) and the National Fund for Scientific Research (Belgium) for their continued financial support. Finally we express our deep gratitude to our families for giving us the time needed to do what we enjoy so much.

Kgln, Firenze, Leuven May 1994 Jiirgen Berndt, Franco Tricerri, Lieven Vanhecke

To our deep sorrow Franco Tricerri, his wife and his two children died in an airplane crash two weeks after completion of this manuscript. Our loss is immeasurable.

Jiirgen Berndt and Lieven Vanhecke

vi

Contents Preface

v

Contents

vii

I Introduction

1

2 S y m m e t r i c - l i k e R i e m a n n i a n manifolds

4

2.1 N a t u r a l l y r e d u c t i v e R i e m a n n i a n h o m o g e n e o u s s p a c e s . . . . . . . . . . . . . . . .

4

2.2 R i e m a n n i a n g.o. s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.3 W e a k l y s y m m e t r i c s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4 C o m m u t a t i v e s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.5 P r o b a b i l i s t i c c o m m u t a t i v e s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.6 H a r m o n i c s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.7 D ' A t r i s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.8 r

14

a n d g3-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.0 C0-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.10 ~ g - s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.11 ~ : - s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.12 O s s e r m a n s p a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3 Generalized Heisenberg groups 3.1 G e n e r a l i z e d H e i s e n b e r g a l g e b r a s a n d g r o u p s . . . . . . . . . . . . . . . . . . . . . . . .

21 22

3.1.1 D e f i n i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.1.2 C l a s s i f i c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.1.3 A l g e b r a i c f e a t u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.1.4 Lie e x p o n e n t i a l m a p

26

...........................................

3.1.5 S o m e g l o b a l c o o r d i n a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.1.6 Levi C i v i t a c o n n e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.1.7 C u r v a t u r e

.....................................................

28

3.1.8 T h e J a c o b i o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.1.9 G e o d e s i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

vii

3.1.10 I n t e g r a b i l i t y of ~ a n d 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.1.11 I r r e d u c i b i l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.1.12 T h e o p e r a t o r K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.1.13 I s o m e t r y g r o u p

34

................................................

3.1.14 K~ihler s t r u c t u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.2 S o m e c l a s s i f i c a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.3 S p e c t r a l p r o p e r t i e s of t h e J a c o b i o p e r a t o r

..........................

36

3.4 C o n s t a n c y of t h e s p e c t r u m a l o n g geodesics . . . . . . . . . . . . . . . . . . . . . . . . .

42

3.5 R o t a t i o n of t h e e i g e n s p a c e s a l o n g g e o d e s i c s . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.6 S o m e c o r o l l a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.7 J a c o b i fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.8 C o n j u g a t e p o i n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.9 P r i n c i p a l c u r v a t u r e s of geodesic s p h e r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.10 M e t r i c t e n s o r w i t h r e s p e c t to n o r m a l c o o r d i n a t e s 4 Damek-Ricci

...................

spaces

74 78

4.1 B a s i c c o n c e p t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.1.1 D e f i n i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.1.2 C l a s s i f i c a t i o n a n d idea for c o n s t r u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.1.3 A l g e b r a i c f e a t u r e s a n d d i f f e o m o r p h i s m t y p e . . . . . . . . . . . . . . . . . . . .

79

4.1.4 Lie e x p o n e n t i a l m a p

...........................................

80

4.1.5 S o m e g l o b a l c o o r d i n a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.1.6 Levi C i v i t a c o n n e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.1.7 C u r v a t u r e

84

.....................................................

4.1.8 T h e J a c o b i o p e r a t o r 4.1.9 S y m m e t r y

...........................................

.....................................................

4.1.10 I n t e g r a b i l i t y of c e r t a i n s u b b u n d l e s

.............................

85 86 87

4.1.11 G e o d e s i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.1.12 I s o m e t r y g r o u p

................................................

94

structures .......................................

95

4.1.13 N e a r l y K s

4.2 S p e c t r a l p r o p e r t i e s of t h e J a c o b i o p e r a t o r

..........................

96

4.3 E i g e n v a l u e s a n d e i g e n v e c t o r s a l o n g geodesics . . . . . . . . . . . . . . . . . . . . . .

105

4.4 H a r m o n i c i t y

108

......................................................

4.5 G e o m e t r i c a l c o n s e q u e n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

Bibliography

115

Index

123

viii

Chapter 1 Introduction Nilpotent and solvable Lie groups with left-invariant Riemannian metrics play a remarkable role in Riemannian geometry. At many occasions they arise quite naturally. For example, they appear in the Iwasawa decomposition of the isometry group of a non-compact Riemannian symmetric space. Also, every connected homogeneous Riemannian manifold of non-positive sectional curvature can be represented as a connected solvable Lie group with a left-invariant metric. Among the nilpotent Lie groups the two-step ones are of particular significance. Some aspects of the geometry of the latter ones, equipped with a left-invariant Riemannian metric, have been treated recently in [Ebe2] and [Ebe3]. The generalized Heisenberg groups form a subclass of the simply connected two-step nilpotent Lie groups with a left-invariant Riemannian metric and, as mentioned by P. Eberlein [Ebe2], should be regarded as the model spaces among these two-step nilpotent Lie groups in a similar way as the Riemannian symmetric spaces among all Riemannian manifolds. Generalized Heisenberg groups were introduced by A. Kaplan [Kapl] around 1980 in the framework of his research about hypoelliptic partial differential equations. Starting from a composition of quadratic forms, Kaplan defined a class of simply connected two-step nilpotent Lie groups with a left-invariant Riemannian metric which includes the classical Heisenberg groups. The composition of quadratic forms is intimately related to the theory of Clifford modules. In fact, to each representation of the Clifford algebra of IR" with respect to a negative definite quadratic form one can associate a generalized Heisenberg group with m-dimensional center. This assignment is one-to-one if m ~ 3(mod4). When m =- 3(mod4), non-equivalent Clifford modules may yield isometric generalized Heisenberg groups. Nevertheless, using representation theory of Clifford algebras, a complete classification of generalized Heisenberg groups can be achieved. Both the harmonic analysis and the geometry of the three-dimensional Heisenberg group and its immediate higher-dimensional generalizations have been fruitful fields for research in the past. Since their introduction by A. Kaplan many mathematicians were attracted by the generalized Heisenberg groups in relation with these two fields. See the reference list in [DaRi2]. One of the first remarkable results con-

cerning the geometry of generalized Heisenberg groups was that they are D'Atri spaces, that is, have volume-preserving geodesic symmetries (up to sign). On the other hand, these groups are not naturally reductive as a Riemannian homogeneous space unless the dimension rn of the center is one or three. This answered negatively the question whether a D'Atri space is always locally isometric to a naturally reduetive Riemannian homogeneous space or not. Moreover, nilmanifolds arising as compact quotients from generalized Heisenberg groups have attracted considerable attention in spectral geometry. For more details see IGor2], where the author provides, by using suitable compact quotients of generalized Heisenberg groups, the first known examples of closed isospectral Riemannian manifolds which are not locally isometric to each other. As will be shown in Chapter 3 of these notes, generalized Heisenberg groups also provide examples and counterexamples for other questions and conjectures. But up to now, a systematic study of the geometry of these groups, in particular the aspects relating to the Jacobi operator, was not available. One of the purposes of these notes is to provide a thorough treatment of these aspects based on the explicit research about the spectral theory of this operator and the explicit computation of the Jacobi vector fields vanishing at a point. This method of attack does not only give new geometrical properties but also yields new and more geometrical proofs of known results. Moreover, by doing this, we will relate our research to the different classes of Riemannian manifolds which have been introduced recently in the framework of the study of the geometry of the Jacobi operator. Chapter 2 contains a short survey about these classes including their definitions, known classifications, various characterizations and relations between them. Our interest in the treatment of D'Atri spaces, as introduced in [Dat], [DaNil], and [DaNi2], came from the research about harmonic spaces. The fundamental conjecture about harmonic spaces (also referred to as the conjecture of Liehnerowicz) stated that every Riemannian harmonic manifold is locally isometric to a two-point homogeneous space. It was shown that the condition of harmonicity is equivalent to two infinite series of conditions on the curvature tensor and its eovariant derivatives, known as the even and odd Ledger conditions. The D'Atri property is equivalent to the set of odd Ledger conditions. Only during the past five years there was a breakthrough in this field on the one hand by the positive results by Z.I. Szabd (see 2.6) and on the other hand by the negative one by E. Damek and F. Ricci. More precisely, the last two authors showed in [DaRil] that this conjecture is false by proving that there exist suitable extensions of arbitrary generalized Heisenberg groups which are harmonic. Any such extension is a simply connected solvable Lie group with a left-invariant Riemannian metric. Among these Lie groups are the complex hyperbolic spaces, the quaternionic hyperbolic spaces and the Cayley hyperbolic plane. Their horospheres provide realizations of the Heisenberg groups in the complex case and of suitable generalized Heisenberg groups with three- and seven-dimensional center, respectively, in the two other cases. In each of these particular cases the corresponding classical or generalized Heisenberg group is precisely the nilpotent part in the Iwasawa decomposition of the isometry group of the hyperbolic space. The above mentioned extension is then the solvable part in the Iwasawa decomposition and, as a group, is a semi-direct product of the nilpotent group and

the real numbers. By imitating this construction of the hyperbolic spaces as solvable Lie groups one obtains from each generalized Heisenberg group a solvable Lie group with a left-invariant Riemannian metric. These particular extensions have been called Damek-Ricci spaces. Any of these spaces is a Hadamard manifold with the corresponding generalized Heisenberg group embedded as a horosphere, and is either one of the above hyperbolic spaces or is non-symmetric. In the latter case each one provides a counterexample to the fundamental conjecture about harmonic spaces. Moreover, as is mentioned in IGor2], the study of the closed geodesic balls in Damek-Rieci spaces by Z.I. Szab5 yielded the first examples of closed isospectral Riemannian manifolds with boundary which are not locally isometric to each other. As concerns the harmonic analysis on the Damek-Ricci spaces we again refer to

[DaRi2]. But also here, a detailed study of the geometry of the Damek-Ricci spaces is appealing. Some aspects of it have already been considered by several authors (for more details see Chapter 4). Using again the Jacobi operator, in Chapter 4 we will consider those aspects which are related to some of the classes of manifolds considered in Chapter 2. This leads to several new geometrical characterizations of the symmetric Damek-Ricci spaces. It will also be proved that the DalnekRicei spaces provide, as in the case for generalized Heisenberg groups, examples and counterexamples to open questions and conjectures. All this gives support for the belief that a further study of their geometry will lead to the discovery of other nice geometrical properties. A more detailed description of the contents of each chapter will be given at the beginning of each of the respective chapters.

Chapter 2 Symmetric-like Riemannian manifolds In this chapter we provide some basic material about various classes of R i e m a n n i a n manifolds which m a y be regarded as generalizations of R i e m a n n i a n (locally) symmetric spaces. O u r list of such generalizations is not exhaustive. For example, we do not talk a b o u t the class of k-symmetric spaces [Kowl] which are n a t u r a l generalizations of s y m m e t r i c spaces too. Our selection contains only those spaces which are related to our research on generalized Heisenberg groups and their Damek-Ricci h a r m o n i c extensions. Concerning the material about the spaces presented here we have tried to be r a t h e r complete as regards known classifications and characterizations. T h e basic references given here will guide the reader to further results and details on these spaces. See also [Van2] for a selection. All manifolds are supposed to be connected and of class C ~ O u r sign convention for the R i e m a n n i a n curvature tensor R is given by R(X, Y) = [Vx, Vv] - V[xy] for all t a n g e n t vector fields X, ]/', where V denotes the Levi Civita connection.

2.1

N a t u r a l l y reductive R i e m a n n i a n h o m o g e n e o u s spaces

Let M = G / H be a R i e m a n n i a n homogeneous space endowed with a G-invariant R i e m a n n i a n metric g. The Lie group G is supposed to be connected a n d to act effectively on M . A decomposition of the Lie algebra g of G into ~ = ~ (~ m, where is the Lie algebra of H , is said to be reductive if Ad(H)m C m. If H is connected, a decomposition ~ = b G ra is reductive if and only if [O,m] C m. Note t h a t in the present situation there always exists a reductive decomposition. For X, Y C m we denote by [X, Y]m the projection of IX, Y] onto m. Each X e ~ generates a o n e - p a r a m e t e r s u b g r o u p of the group I(M) of isometrics of M via p ~-* (exp tX) 9p and hence induces a Killing vector field X" on M. If g = ~ 9 rn is a reductive decomposition of g, the natural torsion-free connection V with respect to this decomposition is

defined by :

Ylm

for all X, Y E m, where 7r(H) = o for ~r : G ~ G / H . structure on M is a tensor field T of type (1,2) such that

Finally, a homogeneous

~Tg = ~ R = ~TT = 0 for ~7 := V - T, where ~7 is the Levi Civita connection of (M,g) and R the corresponding Riemannian curvature tensor. Then we have the following characterizations (or definitions) of naturally reductive Riemannian homogeneous spaces (for (i) and (ii) see [KoNo,Chapter X,3]; for (iii) see [AmSi, Theorem 5.4] and [TrVal, Theorem 6.2 and the subsequent remark]). P r o p o s i t i o n 1 [Kono], [AmSi], [TrVal] Let (M, g) be a homogeneous RiemannJan manifold. Then ( M , g ) is a naturally reductive Riemannian homogeneous space if and only if there exist a connected Lie subgroup G of I ( M ) acting transitively and effectively on M and a reductive decomposition ~ = b | m of g, where ~ is the Lie algebra of the isotropy group H at some point in M , such that one of the following equivalent statements holds: (i) g([X, Zlm, Y ) + g(X, [Z, rim) = 0 for all X, Y, Z 9 m; Oi) the Levi Civita connection of ( M , g ) and the natural torsion-free connection with respect to the decomposition are the same; (iii) every geodesic in M is the orbit of a one-parameter subgroup of I( M ) generated by some X G ra. An important observation is that a Riemannian homogeneous space M = G / H might be naturally reductive although for any reductive decomposition ~ = ~ | m of none of the statements in the proposition holds. The point is that there might exist another appropriate subgroup G of I ( M ) such that M = G / / t and with respect to which a reductive decomposition satisfies the required conditions. Because of this ambiguity the following result has been proved worthwhile for verifying that certain Riemannian homogeneous spaces are naturally reductive without knowing their isometry group and its transitive subgroups explicitly (see [BeVa4], [B1Va], [GoGoVal], [GoGoVa2], [Nag], [ToVa], [TrVal], [TrVa2] for applications). P r o p o s i t i o n 2 [TrVal] Let (M, g) be a complete and simply connected Riemannian manifold. Then ( M , g ) is a naturally reductive Riemannian homogeneous space if and only if there exists a homogeneous structure T on M with Tvv = 0 for all tangent vectors v of M . Every Riemannian symmetric space is naturally reductive. As the classification of Riemannian symmetric spaces is known since the work of E. Cartan, we concentrate now on non-symmetric naturally reductive spaces. For dimension two the situation is clear since any Riemannian homogeneous space obviously has constant curvature and hence is a locally symmetric space. For non-symmetric naturally reductive Riemannian homogeneous spaces in dimensions three, four and five there

are the following results (for dimension three see [TrVal, Theorem 6.5] and in a more explicit way [Kow3]; for the geometric realizations see [BeVa4]). T h e o r e m 1 [TrVal], [Kow3], [BeVa4] Let (M, g) be a three-dimensional simply connected Riemannian manifold. Then (M,g) is a non-symmetric naturally reductire Riemannian homogeneous space if and only if it is one of the following spaces: (i) the Lie group SU(2) with some special left-invariant Riemannian metric g. There is a two-parameter family of left-invariant Riemannian metrics on SU(2) making it into a naturally reductive Riemannian homogeneous space. These metrics are precisely those obtained by considering SU(2) ~ S 3 as a geodesic sphere in some two-dimensional complex projective or hyperbolic space equipped with some Fubini-Study metric of constant holomorphic sectional curvature; (ii) the Lie group SL(2,]R) with some special left-invariant Riemannian metric g. There is a two-parameter family of left-invariant Riemannian metrics on SL(2, JR) making it into a naturally reductive Riemannian homogeneous space. These metrics are precisely those obtained by taking the universal covering of any tube around a one-dimensional complex hyperbolic space embedded totally geodesically in a two-dimensional complex hyperbolic space equipped with some Fubini-Study metric of constant holomorphic sectional curvature. In explicit form, these spaces are given by M = IR3[t, x, y] with 1 ds~ - ]a + b---~dr2 ] + ]a + b]e-2tdx 2 + (dy + x / ~ e - t d x ) 2, where a,b E IR with b > 0 and a + b < O. Geometrically, a and b are the eigenvalues of the Ricci tensor of M , the first one with multiplicity two," (iii) the three-dimensional Heisenberg group H3 with any left-invariant Riemannian metric g. There is a one-parameter family of such metrics on H3 and they are obtained by realizing H3 as a horosphere in some two-dimensional complex hyperbolic space equipped with some Fubini-Study metric of constant holomorphic sectional curvature. Explicitly, M = ]R3[x, y, z] with =

1

2 (dx + dz + (dy - xdz)

),

where b E IR+. Here, - b and b arc the eigenvalues of the Ricci tensor of M , the first one with multiplicity two.

T h e o r e m 2 [KoVal] Let ( M , g ) be a four-dimensional simply connected Riemannian manifold. Then ( M , g) is a non-symmetric naturally reductive Riemannian homogeneous space if and only if it is isometric to some Riemannian product SU(2) x IR, SL(2, IR) x ]R, H3 x IR, where the first factor is equipped with a naturally reductive Riemannian metric according to the classification in dimension three.

T h e o r e m 3 [KoVah] Every five-dimensional simply connected non-symmetric naturally reductive Riemannian homogeneous space is either a Riemannian product M1 x Ms, where M, is SU(2), SL(2, JR) or H3 with some naturally reductive metric and Ms is some standard space of constant curvature, or locally isometric to one of the following spaces: (i) (SO(3) x S 0 ( 3 ) ) / S 0 ( 2 ) ~

or (SO(3) x SL(2,]R))/SO(2)~

or (SL(2,]R)x

SL(2, ~t))/SO(2). where SO(2)r denotes the subgroup consisting of pairs of matrices of the form

(cossin 0)(co sinrt0) sin t 0

cos t 0

0 1

x

sin rt 0

cos rt 0

0 1

(t E IR)

and r is a rational number. On each of these spaces there is a family of naturally reductive invariant Riemannian metrics depending on two real parameters. For each of the three types the whole family of locally non-isometric spaces depends on two real parameters and one rational parameter," (ii) (H3 x S 0 ( 3 ) ) / S 0 ( 2 ) (~) or (H3 x SL(2, ]R))/SO(2) (~), where SO(2) (~) denotes the subgroup consisting of all pairs of matrices of the form

(lO ) (cosr sin o) 0 1 0 0 0 1

x

sinrt 0

cosrt 0

0 1

(tE]R)

and r is a rational number. On each of these spaces there is a family of naturally reductive invariant Riemannian metrics depending on two real parsmeters. For each of the two types the whole family of locally non-isometric spaces depends on two real parameters and one rational parameter; (iii) the five-dimensional Heisenberg group Hh. The naturally reductive left-invariant Riemannian metrics on H~ form a two-parameter family. Explicitly, these spaces are M = lRS[x,y,z,u,v] with

ds ~ = ~-(du ~ + dx ~) + ~(dv ~ + dy ~) + (udx + vd~ - dz) ~ P and )~, p E ]R+; (iv) SU(3)/SU(2) or SU(1, 2)/SU(2), and on each space there is a family of naturally reductive invariant Riemannian metrics depending on two real parameters.

Geodesic spheres in two-point homogeneous spaces except CayP 2 and CayH 2 are naturally reductive Riemannian homogeneous spaces (see [Zil2] and [TrVa2]). Every simply connected ~7-umbilieal hypersurface of a complex space form is naturally reductive [BeVa4]. This has been extended by Nagai [Nag] to the so-called hypersurfaces of type (A) in complex projective spaces and their corresponding ones in complex hyperbolic spaces. Every simply connected ~2-symmetric space (that is,

Sasakian manifold with complete characteristic field such that the reflections with respect to the integral curves of that field are global isometries) is naturally reducrive [B1Va]. Every simply connected Killing-transversally symmetric space (that is, a space equipped with a complete unit Killing vector field such that the reflections with respect to the flow lines of that field are global isometries) is naturally reductive (see [GoGoVal] and [GoGoVa2]). Note that each ~2-symmetric space is a Killing-transversally symmetric space. a

For further results and references on naturally reductive Riemannian homogeneous spaces we refer to J.E. D'Atri and W. Ziller [DaZi], who also classified all naturally reductive compact simple Lie groups. For a treatment of the non-compact semisimple case, see C. Gordon [Gorl].

2.2

Riemannian

g.o. s p a c e s

A Riemannian manifold (M, g) is said to be a Riemannian g.o. space [KoVa7] if every geodesic in M is the orbit of a one-parameter subgroup of the group of isometries of M. Clearly, any such space is homogeneous. From Proposition l(iii) in 2.1 we derive immediately P r o p o s i t i o n Every naturally reductive Riemannian homogeneous space is a Riemannian g.o. space. O. Kowalski and the third author [KoVa7] have proved that the converse holds if the dimension is less than six. T h e o r e m 1 [KoVa7] Every simply connected Riemannian g.o. space of dimension ~ 5 is a naturally reductive Riemannian homogeneous space. Combining this with Theorems 1, 2 and 3 in 2.1 yields a classification of all simply connected Riemannian g.o. spaces of dimension less than six. For dimension six the converse does not hold. In fact, there is the following result: T h e o r e m 2 [KoVa7] The following six-dimensional simply connected RiemannJan g.o. spaces (and only those) are never naturally reductive: (i) ( M , g ) is a two-step nilpotent Lie group with two-dimensional center, provided with a left-invariant Riemannian metric such that the maximal connected isotropy group is isomorphic to SU(2) or U(2). All these Riemannian g.o. spaces depend on three real parameters; (ii) ( M , g ) is the universal covering space of a homogeneous Riemannian manifold of the form M = S 0 ( 5 ) / U ( 2 ) or M = SO(4, 1)/U(2), where SO(5) or S 0 ( 4 , 1) is the identity component of the full isometry group, respectively. In each case all admissible Riemannian metrics depend on two real parameters. Every geodesic sphere in a two-point homogeneous space except CayP 2 or CayH 2 is a Riemannian g.o. space since it is naturally reductive. For CayP 2 and CayH 2 it

is still an open problem whether the geodesic spheres are g.o. spaces or not.

2.3

Weakly symmetric spaces

A Riemannian manifold M is said to be a weakly symmetric space [Self if there exist a subgroup G of the isometry group I ( M ) of M acting transitively on M and an isometry f of M with f2 6 G and f G f -1 = G such that for all p,q E M there exists a g E G with g(p) = f ( q ) and g(q) = f(p). It can easily be seen that any Riemannian symmetric space is weakly symmetric. There are the following geometrical characterizations of weakly symmetric spaces: P r o p o s i t i o n [BePrVal], [BeVa5] Let ( M , g ) be a Riemannian manifold. the following statements are equivalent:

Then

(i) M is a weakly symmetric space; (ii) for any two points p, q E M there exists an isometry of M mapping p to q and q to p; (iii) for every maximal geodesic 7 in M and any point m of 7 there exists an isometry of M which is an involution on 7 with m as fixed point.

Note that Riemannian manifolds having property (iii) have been introduced by Szab6 [Sza2] as ray symmetric spaces. In dimensions three and four the simply connected weakly symmetric spaces are completely classified. T h e o r e m 1 [BeVa5] A three- or four-dimensional simply connected Riemannian manifold is a weakly symmetric space if and only if it is a naturally reductive Riemannian homogeneous space (see Theorems 1 and 2 in 2.1). We also have the following further examples of non-symmetric weakly symmetric spaces. T h e o r e m 2 [BeVa5] Each of the following hypersurfaces, endowed with the induced Riemannian metric of the ambient space, is a weakly symmetric space for n>2: ambient space hypersurface @ptube around tube around ]I-IP" CayP 2 tube around horosphere; CH" horosphere; CayH ~ horosphere;

{p}, C P ' , . . . , or C P " - ' {p}, ]PIP',..., or ]HP "-1 {p} or CayP 1 tube around {p}, @ H i , . . . , or @H"-' tube around { p } , ] H H ' , . . . ,or I H H " - ' tube around {p} or CayH 1 .

Other examples, which were discovered only very recently, will be treated in forthcoming papers.

2.4

Commutative

spaces

A commutative space is a Riemannian homogeneous space whose algebra of all invariant (with respect to the connected component of the full isometry group) differential operators is commutative. I.M. Gelfand [Gel] has proved that any Riemannian symmetric space is commutative (see also [Hell, p. 396]). This was generalized by A. Selberg to the class of weakly symmetric spaces.

Proposition

[Self Every weakly symmetric space is a commutative space.

Note that it is unknown whether the converse holds. For dimension less or equal than five the simply connected commutative spaces are well-known: T h e o r e m [Kow3], [KoVa3], [Bie] A simply connected Riemannian manifold of dimension =

,

Polarization of the first equation gives J x J v + JYJx = - 2 < X , Y > i d D .

Interchanging U and V in the second equation shows that ,Ix is a skew-symmetric endomorphism, that is, <JxU, V > + = 0 .

Replacing here V by Jx V implies <JxU, J x V > = 1xl2 . Another consequence of the skew-symmetry of ,Ix and the third equation is < J x U , J r V > + <JyU, J x V > = 2 < U , V > < X , Y >

Replacing here V by U gives <JxU, J v U > = [UI~<X,Y> 9

24

.

T h e equation before the last one is equivalent to [JxU, V] - [U, Jx V] = -2 X .

Replacing V by Jx V implies [JxU, YxV] = -IXl~[U, v] - 2X,

and replacing U by V gives

[V, JxV] = IY[2X 9 These equations will be used frequently in the subsequent sections without referring to t h e m explicitly. Let V E ~ be a non-zero vector. We denote by ker ad(V) the kernel of the linear map a d ( V ) : v --~ ~, U H [U,V] and by ker a d ( V ) • the orthogonal complement of ker ad(V) in ~. Since U E ker a d ( V ) if a n d only if 0 = = <JzV, U> for all Z E 3, we see t h a t ker ad(V) • = J~V . If, in addition, V is a unit vector, the m a p 3 --+ ker ad(V) • , Z ~ J z V is a linear isometry with inverse m a p ker ad(V) • --* 3, U ~ [V, U] . Of p a r t i c u l a r i m p o r t a n c e are the generalized Heisenberg algebras which satisfy the so-called J2-condition: For all X, Y E 3 with < X , Y > = 0 and all non-zero U r there exists a Z E 3 so that d x J y U = JzU, that is, so t h a t J x J y U E ker a d ( U ) • T h e n we have Theorem

[CoDoKoRi] A generalized Heisenberg algebra a satisfies the J2-con-

dition if and only if (i) r e = l ,

or

(ii) m = 3 and n = .(k,O) ~ .(O,k) for some k E IN, or (iii)

m = 7 and n = n ( 1 , 0 )

-

n(O, 1).

These particular generalized Heisenberg algebras are isomorphic to the nilpotent part in the Iwasawa decomposition of ~he Lie algebra of the isometry group of C H "+1, I[-IH "+1 and C a y H 2, respectively.

25

3.1.4

Lie exponential

map

From Proposition 3.1.3 we immediately get Proposition group.

1 Every generalized Heisenberg group is a two-step nilpotent Lie

Let N be a generalized Heisenberg group. We consider elements in n also as left-invariant vector fields on N. For U + X E n denote by 7u+x the one-parameter group generated by U + X. Then the Lie exponential map exPn is defined by exPn:n~N,

U+X~Tu+x(1).

As the Lie exponential map of every connected, simply connected, nilpotent Lie group is a diffeomorphism (see for example [Hell, p. 269] or [Rag, p. 6]), we conclude

P r o p o s i t i o n 2 The Lie exponential map expn : n ---* N i3 a diffeomorphism. This implies

Corollary N is diffeomorphic to IRm+". The group structure on N can be described via expn by using the CampbellHausdorff formula. The two-step nilpotency of N gives the simple formula

expn(U+X).expn(V+Y)=expn(U+V+X+Y+~[U,V])

3.1.5

Some

.

global coordinates

We now introduce some global coordinates on N. Let Vl . . . . , V,, 111. . . . , Ym be an orthonormM basis of the Lie algebra n and V l , - . . , ~3,, Ya. . . . , ym the corresponding coordinate functions on n. The diffeomorphism expn : n --* N then yields global coordinates vl,. 99 v,, Yl, 99-, Y,, on N via the relation

(~,,..., ~,, v,,..., v~) o exp, = (~,,..., ~,, ~ , . . . , % ) . L e m r n a We have 0

1 9

0 Y,

_

Oyi ' "where

A ijk

:~



26

Oyk

Proof. Let p := expn(U + X) E N be arbitrary and Lp the left translation on N by p. Using the formula for the multiplication in N according to 3.1.4 we obtain V,(p) =

L.oV,(e)

=

L.~

r ~ exp.(~))(O

=

0 N ( t ~ L,(expdt~)/)(O)

=

a,(0)

with

ai(t)=exp.(U + t V i + X

+~t[U, Vi]) .

We have

(~ o~,,)(t)

=

~,(U)+&kt,

(v~ o,~,)(t)

=

~(x)+

~t 1

= O~(x)- ~t = , ) ~ ( x ) - ~ t E < j r . v . E > < v , vj> .I

1

=

~)i(X)- ~tXA~9~(U ) . 2

Thus, (vk o a,)'(O)

=

*,~ ,

(w o . , ) ' ( 0 )

=

-~ ~%v,(y),

1

k~

3

and therefore, v,(p)

=

< U , V > - 43-1[u,Ell ~ - -~< JxU, Jv V > . In particular, it can easily be seen that the sectional curvature attains both positive and negative values. The equation for V Q shows that (Vv+rp)(V + II, V + Y) = 0, that is, the Ricci tensor is invariant under the geodesic flow of N. Polarization of the latter equation shows that the cyclic sum over all entries in Vp vanishes, that is, the Ricci tensor of N is a Killing tensor. L e m m a The Ricci tensor of any generalized Heisenberg group is a Killing tensor, or equivalently, is invariant under the geodesic flow. As V Q never vanishes, we also have

Proposition None of the generalized Heisenberg groups is a locally symmetric space.

3.1.8

T h e Jacobi operator

One of the central objects in our studies is the Jacobi operator defined by Rv+v := R(., V + Y)(V + Y) for all V + Y G n. The above expression for the curvature tensor implies as a special case

Rv+y(U + X)

:

~Jtu,v~V +

- tv, J vl +

lvt2x + 1 U,

.

By a straightforward computation we get for the covariant derivative

R'v+v := (Vv+vR)(., V + Y)(V + Y) of the Jacobi operator Rv+r the expression

R'v+v(U + X)

= 3[jusyv]V2 '

+ ~Jtu'v]JvV- (~,V, 2 + ,Y,~) J x V

- ( < U , V > - < X , Y > ) J v V - V

+ (~[VI2 + IYI2) [U,V] + Y . The spectrum of the Jacobi operator will be computed explicitly in 3.3.

29

'

3.1.9

Geodesics

Let V + Y E T , N -~ a be a unit vector and 3, : IR ---* N the geodesic in N with 3'(0) = e a n d ~(0) = V + Y. We consider subspaces of n also as subbundles of T N via left translation. Since V v V = 0 and V y Y = 0, the integral curves of V and Y are geodesics in N . This implies t h a t 3,(t) = expa(tY)

, if Y = 0 ,

3,(t) = expn(tY )

, if V = 0 .

Now suppose t h a t V # 0 # Y. The vectors V, J v V and Y span a three-dimensional Heisenberg algebra ha. It is easy to check that n3 is an autoparallel s u b b u n d l e of T N . Therefore 3' lies in the leaf of n3 through e, which is a t o t a l l y geodesically e m b e d d e d three-dimensional Heisenberg group. This shows P r o p o s i t i o n Every geodesic 3, in N lies in a totally geodesically embedded threedimensional Heisenberg group. The latter one is uniquely determined if and only if ;y is not tangent to v or 3. So, the d e t e r m i n a t i o n of geodesics in generalized Heisenberg groups can be reduced to the one in a three-dimensional Heisenberg group. We continue with the case V # 0 # Y and put V ,12

Y

T h e n 3, is of the form 3,(t) = e x p n ( a ( t ) ? + b(t)J~.? + c(t)?) with some functions a, b, c satisfying a(0) = 0 , a'(0) = I V l , b(0) = 0 , b'(0) = 0 , c(0) = 0 , c'(0) = I Y I . Let u , v , x be the global coordinates on the three-dimensional Heisenberg group d e t e r m i n e d by ? , J ~ ? , 1) and in accordance with 3.1.5. By means of L e m m a 3.1.5 we have

o

? + ~v?

o

Jr? ~u?

Ou Ov 0 Ox'

-

-

y,

and therefore,

=

(uoT)'OoT+(voT)'~--~oT+(xo3")'~xO3,

=

a'?+b'J~.?+(~(a'b-ab')+c')] 30

z .

Differentiating this again gives the system of equations 0 =

a" + la'bb' - ~ab '2 + b'c' 2 1 ,% 1 , , 0 = b"-~a +-~aab - a ' c ' , 0 =

~I ( a ,b - a b ' ) ' +

c"

.

The last equation implies 1 Cl = 2(ab' - a'b) +

IYI 9

Inserting this into the first two ones then gives a"= -IYIb' , b"= IYla'

and thus a" = -IYl2a .

Successively we may now compute a(t), b(t), c(t), and we obtain a(t)

-

[Vl sin(IYIt)

b(t)

=

IWl,, cos(IYIt)) -y-ill -

c(t) =

[rl

IYlt + ~'V'2 ( t - ~1l sin(,Y]t)) .

Summing up, we obtain (see also [Kap2] and the correction in [Kap3]) T h e o r e m Let V + Y E TeN ~- n be a unit vector and 7 : IR --~ N the geodesic inN

with 7 ( 0 ) = e

and ~/(O) = V + Y . Then

7(t) 7(t)

= expn(tV), i f Y = O , = exPn(tY ) , i f V = O ,

7(t)

= expn

sin(IYlt)V + ~y-~(1 - cos(JYIt))JvV

+ ( t + 2 - ~lv[~ ( t - ~ y l s i n ( l Y , t ) ) ) Y ) , i f V r In addition, when we identify different tangent spaces of N along ~[ via left translation, we have +(t)

=

V,

ifY=O,

+(t)

=

Y,

if y=o

;/(t)

=

cos(IYIt)V + ~ 1 s i n ( i Y i t ) j y V + y

,

31

, if vy~ocY

.

3.1.10

I n t e g r a b i l i t y of v and 3

Consider ~ and ~ as left-invariant distributions of T N .

Proposition The distribution ~ is not integrable. The distribution 3 is autoparallel and hence integrable. Each leaf of 3 i~ a totally geodesically embedded ]R ~ (m = dims) endowed with the standard Euclidean metric. Proof. The non-integrability of v follows from Iv,v] - 3. The explicit expression of the Levi Civita connection shows that V X, Y E3: V v X = O , whence ~ is autoparallel. Therefore ~ is integrable and each leaf M of 3 is a totally geodesic submanifold of N. As

V X , Y, Z E 3 : R ( X , Y ) Z = 0 , the Gauss equation of second order implies that M is flat. Recall that the Lie exponential map expn : n --~ N is a diffeomorphism. For each Y E ~ the integral curve through e is a geodesic in N. Thus, the leaf of 3 through e is expn(~), which is diffeomorphic to ]R"~. Eventually, by left-invariance, we see that each leaf of ~ is diffeomorphic to ]Rm. []

3.1.11

Irreducibility

Proposition Every generalized Heisenberg group i8 irreducible as a Riemannian manifold. Proof. Suppose that a generalized Heisenberg group (N, g) is a Riemannian product M1 • M2 with dim Mi >_ 1. Let VI + Y1 and V2 T Y~ be tangent vectors to M1 and M~ at some point p E N, respectively, where the decomposition is with respect to the decomposition of TpN into v | 3 obtained by left translation from e. Then O = (Vvl+ylp)(Vl + Y1, V2 + y2) -

2 m 8+ n g(Jy, V~, V2) - 2m ~ g ( J+y n,

V:, V~).

This shows that Jy, V1 is tangent to M1 and Jy, V2 is tangent to M2 at p. Analogously, Jy~V2 is tangent to M~ and Jy~V1 is tangent to M1 at p. As 0 -- g(V~, V2) + g(Y~, Y2), this implies

0 = g(Jy, V~, JY, Y2) + g(Jy, V~, Jy~V2) =

2g(Y~, Y2)g(V~, V2) = -2g(Y~, Y2) ~ = -2g(V~, V2) 2 .

Therefore V1, Y1 are tangent to M1 and V2, Y2 are tangent to M~ at p. Now it is clear that there exist i , j E {1,2}, i ~ j, such that V~ and ~ may be chosen as unit vectors. The plane cr spanned by Vi and Yj has sectional curvature equal to zero, but, on the other hand, the formula for K ( a ) in 3.1.7 gives K ( a ) = 1//4, which is a contradiction. [] 32

3.1.12

The operator K

Let V + Y be a vector in n with V # 0 # Y and let Y " denote the o r t h o g o n a l complement of the span of Y in 3. We put y

1>

Y

and define an e n d o m o r p h i s m

Kv,r : y z ~ Y •

~ [~',JxJ~,tz] .

For X, Z E Y• we have

"



=

= < J z g , JxJ~,9> = - < J x ~ ' , J z J ~ ? >

=

- < [ ~ ' , JzJ~,?], X > = - < K v . r Z , X > ,

which shows t h a t Kv, y is skew-symmetric. Thus K V,Y 2 is a s y m m e t r i c e n d o m o r p h i s m of Y• Let X be an eigenvector of K ~ y of unit length with corresponding eigenvalue p. T h e n

I.t = < g ~ y X , X > = - < g v y X ,

Kv, y X > = --I[~z, JxJ~,Y]t ~ 9

Putting JxJ~f/ = u + Jz?

with U E ker ad('V) and Z E ~ then gives = -1[17, Jzl?]] 2 = - I Z l ~ . As 1 = I J x J ~ 9 1 ~ = IUI ~ + IZl ~ ,

we deduce t h a t # ~ [ - 1 , 0 ] and v/=-# is the length of the p r o j e c t i o n of JxJe'~" onto ker a d ( l f ) • = ker a d ( V ) I . In particular,

JxJvV E kerad(V) ,

#=0 #=-1

JxJrV E kerad(V) • 9

In the l a t t e r case we then have JxJrV

= [YlJ~cv,~xV 9

F u r t h e r , we see t h a t n satisfies the J2-condition if and only if Kr.2 r = -idy~. for all V + Y E .with V #O# Y. Suppose now t h a t V + Y is a unit vector and let 3' : IR ~ N be the geodesic in N with 7(0) = e a n d ~(0) = V + Y. Then

+(t) = v ( t ) + Y

33

with

V(t) -- cos(lYlt)V + ~1 sin(iYlt)Jv y where different tangent spaces along 7 are identified via left translation. We define a skew-symmetric Yl-valued tensor field Kx along 7 by

K,(t) := Kv(,)y Then

I~;(t)x

= [P(t), JxJ~P(t)] = cos2(lYlt)[ff, JxJ~,(/] + sin2([Y[t)[J~, JxJ~,(/] + sin(lY[t) cos(lY[t)([V, JxJ~/] + [J~V, JxJ~,~z]) = =

[~, JxJ~P] K,(O)X,

since

[ g ~ , Jx J ~ ] = - [ J ~ , Jx~] = - [ J ~ J ~ , ~] and

[V, Jx J~V] + [J,2V, Jx J~ V] = -[~/, JxV] -4- X = O . This implies L e m m a The eigenvalues of K~ are constant along 7.

3.1.13

Isometry

group

We denote by A ( N ) the group of automorphisms of N whose differential at e is an orthogonal map and by L(N) the group consisting of the left translations on N. Both A ( N ) and L(N) act on N via isometries. Then we have

Proposition [Kap2] The isometry group of a generalized Heisenberg group ( N, g) is the semidirect product of A(N) and L(N), where A(N) acts on L ( N ) via conjugation. The algebraic structure of A(N) has been determined by C. Riehm in [Riel].

3.1.14

K~ihler structures

The question whether there exists a K/~hler structure on some generalized Heisenberg group has to be answered in the negative. In fact, according to Lemma 3.1.7 the Ricci tensor of a generalized Heisenberg group N is a Killing tensor. If N carries the structure of a Ks manifold compatible with the left-invariant Riemannian metric g, then its Ricci tensor must necessarily be parallel (see [SeVa3]). Since none of the generalized Heisenberg groups has a parallel Ricci tensor, we conclude:

Proposition None of the generalized Heisenberg groups carries a KShler structure which is compatible with its left-invariant Riemannian metric. 34

3.2

Some classifications

In this section we present the known classifications of generalized Heisenberg groups which belong to one of the various classes of symmetric-like Pdemannian manifolds discussed in C h a p t e r 2. T h e o r e m 1 [Kap3] A generalized Heisenberg group is a naturally reductive Riemannian homogeneous space if and only if dim 3 E {1,3}. See also [TrVal] for an alternative proof. Theorem

2 [Rie2] A generalized Heisenberg group N is a tliemannian g.o. space

if and only if (i) dim $ 9 {1,2,3}, or (ii) dim $ = 5 and d i m N = 13, or (iii) dim $ = 6 and dim N = 14, or (/v) d i m 3 = 7 and

(I) d i m N = 15, or (2) dim g = 23 and n = n(2, 0) Z n(0, 2), or

(3) dim N = 31 and a = n(3, 0) Z ,(0, 3). For d i m 3 = 2 and dim g = 6 see also [TrVal] and [Kap3]. F. Ricci [Ric] has classified all generalized Heisenberg groups N for which the convolution algebra L~(N) is commutative; here, A ( N ) is the group of a u t o m o r p h i s m s 1 of N t h a t act as orthogonal transformations on n and La(u) is the algebra of all L 1functions t h a t are invariant under A(N). According to A. K a p l a n a n d F. Ricci [KaRi], this convolution algebra is commutative if and only if the algebra of all invariant differential operators is commutative. Hence we get Theorem

3 [Ric] A generalized Heisenberg group N is a commutative space if

and only if (i) dim ~ 9 {1,2,3}, or (ii) dim 3 = 5 and d i m N = 13, or (iii) d i m 3 = 6 and d i m N = 14, or (iv) d i m 3 = 7 and (1) dim N = 15, or (2) dim Y = 23 and n = n(2, 0) = n(0, 2).

35

From the preceding two theorems we get an example of a Riemannian g.o. space which is not a commutative space, namely the 31-dimensional generalized Heisenberg group N with seven-dimensional center and n ~ n(3,0). In 4.1.10 it will be shown that N is isometric to a horosphere in a complex hyperbolic space if dim3 = 1, in a quaternionic hyperbolic space if dims = 3 and n = n(0, k) = n(k,0) for some k E IN, and in Cayley hyperbolic plane if dim3 = 7 and d i m N = 15. So from Theorem 2 in 2.3 we obtMn

Proposition 1 A generalized Heisenberg group N is a weakly symmetric space

i] (i) dim3 = 1, or Oi) dim~ = 3 and n = n(0, k) - n(k,0) for some k E IN, or (iii) dim3 = 7 and d i m N = 15. As a weakly symmetric space is always commutative (see 2.4), one could therefore check the remaining cases in Theorem 3 in order to obtain a complete classification of the weakly symmetric generalized Heisenberg groups. Up to now we were not able to settle this question. Finally we state

Theorem 4 [Kap3] Every generalized Heisenberg group is a D'Atri space. In Sections 6, 9 and 10 we will give alternative proofs of this result. As every harmonic space is an Einstein manifold, but none of the generalized Heisenberg groups is an Einstein space, we obtain

Proposition 2 None of the generalized Heisenberg groups is a harmonic space. Up to now we do not know which of the generalized Heisenberg groups are probabilistic commutative. We also mention that in [TrVal] and [Kap3] the geometry of the six-dimensional generalized Heisenberg group with two-dimensional center is studied and further properties are obtained. In particular, it is proved that the eigenvalues of (g#) in normal coordinates have antipodal symmetry. In Section 10 we will show that this property holds in fact on every generalized Heisenberg group.

3.3

Spectral properties of the Jacobi operator

In this section we compute the eigenvalues and the corresponding eigenspaces of the Jacobi operators of an arbitrary generalized Heisenberg group N at the identity e.

Theorem Let V + Y be a unit vector in n. (i) Y = O. Then Rv has three distinct eigenvalues O, - 3 / 4 and 1/4; the corresponding eigenspaces are kerad(V), kerad(V) • and ~, respectively. 36

(ii) V = O. Then Rv has two distinct eigenvalues 0 and 1/4; the corresponding eigenspaces are 3 and v, respectively. (iii) V ?t 0 ?t y . We decompose n orthogonally into n=n3~)pGq, where

n3 :=

span{V, JvV, Y} ,

p :=

kerad(V) N k e r a d ( J v V ) ,

q :=

span{Y • ,Jv~V, J v ~ J v V } 9

The spaces as, p and q are invariant under the action of Rv+v and we have: (1) Rv+vl,3 has two (if IV1 ~ = 1/4) or three (if Iyl ~ # 1/4) distinct eigenvalues, namely 1 Oa,,d ~

1 , i f Iyl ~ = ~ ,

_

1

O, 1 and 41 _ iYl~ , i f Iyl ~ # ~ ; the corresponding eigenspaces are span{V + Y, g r v } and I R ( - I Y [ : V +

IVIW)

IR(V + Y ) , I R ( - I Y I ~ V + [VI2Y) and I R J y V

1 , i f [Y[ 2 = ~ , 1 , i f [Y[ 2 r -~ ,

respectively. (2) (if p r {0}) nv+yIp has only one eigenvalue, namely (1 - 1 V 1 2 ) / 4 = 1Y12/4. (3) (if q y~ {0}) We put I f := I i v y and decompose y . t orthogonally into Y• = Lo @ . . . @ Lk , where Lj := ker(K 2 - ~ / d y ~ ) ,

(i = 0 , . . . , k)

and 0>_#o > p l > . . .

>#k_>--I

are the distinct eigenvalues of K 2. It can easily be seen that X E Li ~

K X E Lj

(j=O,...,k)

,

whence d i m L j is even provided that #j # O. We now define qj :=

span{L~,Jr, V, J r , J v V } , j = O , . . . , k , #k # - 1 ,

qk :=

span{Lk, J L ~ V } , i f P k = - - l . 37

Then

q=q0~...@qk

, dimqj =

0(mod3) 0(mod4) 0(mod6)

, if th=O , if #i=-I , otherwise

and each space qj is invariant under the action of Rv+y. Finally, we put p~ :=

1 ~ - I V I 2,

p~ :=

~(1+ ffl + 321VPlYl~),

p3 :=

8(1 - ffl + 321VI~IYI2) ,

(A) (if j = k and #k = - 1 ) Rv+elqk has two distinct eigenvalues I~kl and xk~, which are the solutions of

the corresponding eigenspaces are {(xk,-Px)X+~IYIJNxVIXEL,}

(i=1,2).

We always have

'~kl + 'r = ~lV[ ~ + pl = ~ (1 - 3lVI =) . (B) (otherwise) Rv+ylqs has three distinct eigenvalues ~1, ~2 and ~ja, which are the solutions of 27 4 2 (p - p,)(p -- p~)(P -- P3) = ~ l V l IYI m 9 We always have 1 ,r

+ , . ~ + ,.~3 = p~ + p~ + p3 = ~ - IVl ~

9

The correspondin 9 eigenspaces are 9wen by (a) (if j = 0 and tto = O) JLoV

for ~ol := pl ,

{ ( 4 p , - I Y I ~ ) X + 3 J x J v V I X E Lo}

for ~o, := P, (i = 2,3);

(b) (otherwise) ((Pl-

aj,)((4ai,-

Igl~)X + 3 J x J r V ) +

38

~lVl~lglJ~xV l X e Lj}.

Remark. As d i m Y • = m - 1, the dimension of q can be estimated by 2(m - 1) _< dimq _< 3 ( m - 1 ) , and the first inequality is an equality for all V + Y E n with V ~ 0 ~ Y precisely if n satisfies the J~-condition.

Proof. We recall from 3.1.8 that Rv+r(U + X )

= 43JtuvlV + ~ J x J v V + ] [Y[~U

+

lWx +

1

v>Y ,

a n d consider three cases. (i) Y = 0. T h e n

IVl'

= 1 ~d

Rv(U + X ) =

JT,v]V +-~X .

This implies that V U E ker ad(V) : RvU = 0 and

V X Ea:RvX = 1X 9 4 If U E ker a d ( V ) ' , then there exists a Z E 3 so that U = J z V and we get

3 RvU=-~J[jzv, vl V = - ~ J z V = - ~ U3. (ii) V = O. T h e n IYl 2 = 1 and

1

R~(U + X) = ~U Thus

1

V U E ~ : RyU = : U 4 and

VXEs:RvX=O. (iii) V ~ 0 r Y. Since

Jv,V

=

kerad(V) • M(JvV) z ,

Jy,JvV

=

k e r a d ( J v V ) ~ M(V) • ,

it follows easily that q = span{Y z, Jr*V, Jv*JvV} is orthogonal to na and p. 39

The cases (1) and (2) may be checked without difficulties by a straightforward computation. From now on we suppose that q ~ {0}. For X E Lr we have { ~[V[2X+3[Y[JKxV

.Rv+vX =

~IVI2X + '*~JxJvV

, ifj=kand#k=-I ,otherwise,

and i f j < k or (j = k and #k r - 1 ) ,

Rv +v Jx Jv Y = -~31V121Y12X - -~3[VI21YIJKx V + ~ ]Y[2Jx Jv V This shows that qj is invariant under the action of

Rv+v for each j E { 0 , . . . , k}.

Next, we compute the eigenvalues and the eigenspaces of Rv+v [qj. We start with the case j = k and #~ = - 1 . Let X E Lk be a non-zero vector. Then we see from the above formulae that span{X, JgxV} is invariant under Rv+v. Thus, there exist a,/~ E IR such that aX + j3JKxV is an eigenvector of Rv+y, say with corresponding eigenvalue ~. Then we get the equations

~a = 4[V[~a+3[VI2[Y[/~,

The second equation yields

(~ -

p,)•

= ~lYl~

9

As t3 = 0 is impossible (since it implies also a = 0), we may normalize the eigenvector such that fl = 3[Yl/4. Then a = ~ - Pl, and from the first equation we get

The assertions stated in (A) now follow easily. From now on we assume that j < k or #k ~ - 1 . We first consider the case when j = 0 and #0 = O. Let X be a non-zero vector in L0. Then K X = O, as [KX[ 2 = - = O, and hence

Rv+Y JxV = ( ~ - I V ' 2 ) JxV = plJxV , whence p~ is an eigenvalue of Rv+r with JLoV consisting of corresponding eigenvectors. Next, we see that span{X, JxJvV} is invariant under Rv+v. Hence, there

40

exist a,/3 E IR such that aX +/3JxJvV is an eigenvector of equations we get in this case are

Rv+v.

T h e resulting

nil = ~a + l[Y,~/3 . The second equation gives

As/3 = 0 is impossible (since it implies a = 0), we may normalize the eigenvector so t h a t / 3 = 3/4. Then a = ~ - I Y I 2 / 4 and hence (~ _ 1 , y , ~ ) (~ _ ~,y,~) = ~ l V l ~ , y l ~ or equivalently,

(~ - m ) ( ~ - p3) = o . From this the assertion in (B)(a) follows immediately. It remains to study the case j > 0 or #0 # 0. Let X be a non-zero vector in Lr From the above computations we see that there exist a,/3, "y E IR such that

aX +/3JrxV + 7JxJvV is an eigenvector of

Rv+v.

/~Ot

r7

Here, the eigenvector equations are

= I[V,2a-~[V,~[Y[#~/3+~[V[2[Y[~'7, =

p l / 3 - 41vI~IYIT,

=

~+~IYI~7.

1

3

The second equation gives 3

(~ - pl)/3 = -41VI~IYI7. As fl = 0 is impossible (since this implies 7 = 0 and hence a = 0), we may normalize the eigenvector by putting/3 = 9WI~IYI/4. Then 7 = 3(p~ - ~) and hence, using the third equation, a = (4~ - J Y l 2 ) ( m - x ) . The first equation then gives (x-l[V,~)(4x-[Yl2)(p,-~)=~lVl2[Y[~(pl-x) 41

2 7 V 4l[y 2 _~.~[

Pi

,

that is,

= (~ - p,)(~ - p~)(~ - p~). From this the assertion in (B)(b) then follows. [] An immediate consequence of the Theorem is C o r o l l a r y None of the generalized Heisenberg groups is a pointwise Osserman

space.

3.4

Constancy

of the

spectrum

along

geodesics

We shall now prove that in any generalized Heisenberg group the eigenvalues of the Jacobi operator are constant along geodesics. Theorem

Every generalized Heisenberg group is a e-space.

We will provide two alternative proofs, one using eigenvalues and the other one using eigenvectors of the Jacobi operator along geodesics.

First proof. Let V + Y E n be a unit vector and 7 : IR ~ N the geodesic in N with 7(0) = e and ~(0) = V + Y. According to Theorem 3.1.9 we have ~(t)=

V(tl+Y

, ifY#O

V

, if Y = 0 ,

where

]

v(t) = cos(IYlt)V + I~l sin(tilt)JrV Thus IV(t)l and IY(t)l, the lengths of the projections of ~(t) onto v and 3, respectively, are constant. Further, from Lemma 3.1.12, we know that the eigenvalues of K~ are constant Mong 7. As according to Theorem 3.3 the eigenvulues of the Jacobi operator P~ along 7 depend only on IY(t)l, IY(t)l and the eigenvalues of K~, we conclude that the eigenvalues of R~ are constant. []

Second proof. We shall apply characterization (iv) in Proposition 2 of 2.8. Let V + Y E n be a unit vector. We have to prove that for each eigenvalue ~ of Rv+y there exists a corresponding eigenvector U + X so that < R~+y (U + X), U + X > = 0. In order to accomplish this we use the expressions for the eigenspaces of Rv+r which have been computed in Theorem 3.3. From 3.1.8 we recall that R~+y(U+X)

= ~ J t u J ~ v l V + 32j [ u v' l J y V - ( ~ I V ] ~ + I Y ] 2 ) - ( < U , V > - < X , Y > ) J y V - V

+ (~IVI2 + 'YI2) [U, V] + Y 42

and consider again three cases. (i) Y = 0. Then -i

Rv(U + X) = - ! J x V + 2[U, V] 2 and hence, V U E k e r a d ( V ) : = 0, 1 V U E k e r a d ( V ) l : = - ~ < J x V , X > = 0. (ii) V = 0. In this case the assertion follows from R~, = 0. (iii) V # 0 # Y. On as we have

R;+r(V + Y) = 0 ,

R;+vJrV

= -~IVI~(-IYIZV+ IVI~Y),

RLy(-IYI:V + IVI:Y) = - ~ l V l 2 J v V . From this we conclude that Y and U 9 (ker ad(V) Mker ad(JyV)) G IRV @ IRJvV. This implies that

JyU, JxV(t)

[u, v(t)] c q•

and hence

T,(t)(U + X ) = - 1 j y U +

~JxV(t) + ~[U,V(t)] E q• .

45

Thus, T~ maps q.L into itself. It can easily be checked that T~[q • is skew-symmetric. Further, we have (J}V(t) - J~,V(t)) = 0

IYI and, as U e (ker ad(V) A kerad(JyV)) G IRV @IRJyV,

([JrV, V(t)]

+ IV, JrV(t)])

= [JrU, V(t)] + Y 1 - [yliY + Y -

1

2

,~,{<J~.U, Y(t)>Y

+ Y

: 0 .

Next, if U + X 6 q, then also J r V , JxV(t) , [U, V(t)] 6 q,

and hence

So, T~ maps q into q. Again, it follows easily that T~Iq is skew-symmetric. Finally, we obtMn = 0 and = O, and this finishes the proof. D

Lemma

2 R'~ = [R~, T~].

Proof. If Y = 0, then tL(t)(U + X ) = Rv(U + X ) =

l-x,

J[u,vlV + 4 1

R'~(t)(U + X) = n'v(U + X) = --~JxV + [U, V] , T~(t)(U + X) = -~JxY + [U, V] . Thus

1 3 V ,V] 3 V + ~[U,V] - ~JxV - -~[J[v.v] [P~, T,](t)(U + X) = -~J[~xv.v] = - ~ : x1 =

V + ~[U, v]

R~( t )(V + X)

If V = O, we have

I_U,

n , ( t ) ( u + x ) = Ry(U + X ) = 4

t R,(t)(U + X ) = R'~(U + X ) = O,

T~(U + X) = - ~ J v V . Therefore,

[R~,T,](t)(U + X) = - JyU + -~JvU = 0 = R'Jt)(U + X) . Next, let V # 0 -~ Y. First suppose that U + X E q. Then

3 v(,)~V(t ) + ~JxJvV(t) + ~[Y[2U - ~[U, JvV(t)] + ~IV]2X , R~(t)(V + X ) = ~Jtu, 47

and each term on the right-hand side is in q. Further,

R'~(t)(U + X)

= ~J[v,j.v(olU(t) + -~Jiv,vmlJyU(t) -

IVI ~ + Iyl ~

JxU(t)

+ (~lVl~ + lYl2) [u, v(t)] and

T~(t)(U + X) = 3 j~.U + ~JxV(t) + ~[U, V(t)] . We now obtain

lilt, T~](t)(V + X)

=

+

+w3j [jxv(,).v(,)]V(t) + ~lYl~JxV(t)

~[ffxV(t),J~,V(t)]

+~ JE~,v(,,lJ,,v(o + ~lVI2[U,v(t)] - w9 j YJE~.~(,)]v(t)- ~[JEu,v(olV(t), V(t)]

-~J,,JxJyV(t)- ~[JxJ~V(t), v(~)] -~lYl~:Yu - ~lyl~[U, v(t)] 3 j i~,~,.~(,)lV(t)- ~lVl~JxV(t) +-~ = Rjt)(u + x ) , !

since

+ ~Jt~.,yv(,)jv(t) = Jtv, v(t)lJvY(t)- -~ 9 Jv J[vy(o)V(t) = -~ 3 J[u,v(o]JyY(t) , ~ J[Jxv(o,v(olV ( t ) - ~ [Vl~Jx V ( t ) = - ~ lV]2 Jx V ( t ) ,

81Yl'Jx V(t) - ~ Jv Jx JyV(t) = -IYl'Jx v(t) , ~lVl2[V, v(t)] - ~[4~.~(,)~v(t), 3 v(t)] -w

J.V(t)] -

-~[+xV(t), +~v(~)] -

=

~lVl~[U,

v(t)]

IYI2[U,v(t)] -- IYl~[U, v(t)],

~[+x :Y v(t), v(t)] = o.

Finally, let U + X E q.L, that is, 1

X = ~--~<x, Y>Y and U e (ker ad(V) n ker ad(JvV)) @ IRV @ 1RJyV. 48

Then 1 1 [U, V(t)] = ~ - ~ < [ U , V(t)], Y > Y = ~-~p<JyU, V ( t l > Y -

1 y iyl~

and 1 [U, JyV(t)] = ]-V~ Y = 7~-~<JxU, JyV(t)>V = Y . 141-

Therefore, 3

1

i

tL(t)(U + X) = ~Jcu,v(ojV(t) + JxJrV(t)+ ~[YI~U + ~<X,Y>V(t)

3 4l < x , Y > V ( t ) _ 4--~JyV(t) + 4[Y]~U

-

-~Y

+

~ Ivr~x ,

and each term on the right-hand side is in q• Further,

3

R'~(t)(V + X) = ~J[u,j~v(t)]Y(t) + -~J[u,v(t)]JvY(t) -

(~

)

[y[2 + [y]2 JxY(t)

- ( < V , V(t)> - < Z , Y>)JrY(t) - V(t)

+ (I[v[2 + IY]~) [U,V(t)] + Y = ~Y(t) + -~JvY(t)-

]Y[2JxY(t)

Iv?

- 21Y[5Y and

T4t)(U + x) = -~JyU + JxV(t)+ ~[u,v(t)] 1 1 JyU + =JYV(t) --:---- = WI2IYI

= IVI2IYIt~j # O.

But we get a ~0-parallel orthonormal frame field E l , . . . ,E6 of q~(X) along 7 by defining

El(t) := X , E2(t) :=

1 ~/-~ K X ,

1

E3(t) := ~ J x V ( t ) , E4(t) :=

Ivl l _ ~ J K •

1 Es(t) := IVI]Yl lx/T__~_~(JxJyV(t) -IYIgKxY(t)) , Es(t)

:=

1 I.Vllyl _x/_~x/y~(JKxJrY(t) - #jlYJJxY(t)) .

We continue the computations to get

T~(t)( Jx JyV(t) - IYIJKxV(t)) = ~[YI(JKxJyV(t) -- pilrlJxV(t)) + Ig[2(1 +#~)JxV(t), T~(t)(JKxJvV(t) - #~ [YIJxV(t)) 3

31Y[2(I+ pj)J,(xV(t), = fftt~lY[(JxJvV(t) -IY[JgxV(t)) + -~ 58

and

&(t)x [~(t)KX it,(t)JxV(t) &(t)J,,xV(t) ~ ( t ) ( J x J v V ( t ) - IYIJ, cxV(t)) [~(t)( Jgx JrV(t) -- #j [YIJxV(t)) So, with respect to E , , . . . , E 6 , becomes

=

0,

=

0~

=

-(1+

IYl=)JxV(t),

= -(1 + IYl~)J,cxV(t), = -21Yl=(JxJyV(t)- IYlJ,r =

-2lYl2(JKxJyV(t)

- pjlYlJxV(t)).

the Jacobi equation along 7, restricted to qj(X), z"-Pz'+Qz=O

with z := (zl, z2, z3, z4, zb, z6) T,

o o IVI

P:=

0 0 0

o o

-IVI o 0

o -IVl 31YI~-~

o IVI -31YI-v~--~ 0 -3]Yh/l + #j 0

o o 31YIx/1-~r

0 o

0 3IYIv/r-+--~ -31Yl -vl=-~

31Y[-x/-z-~

0

0 o

0

-3[Y[~

o o

and O 0 0 0 o 0

Q:=

0 0 0 0 0 0 0 0 0 0 0 - ( I + I Y I ~) 0 0 0 0 0 - ( I + I Y I ~) o o o 0 o -21YI ~ 0 0 0 0 0 -2IY[ 2

This gives the system of differential equations o

=

z;'+lVIz'~,

0

=

z~ + I V l z : , ,

0

Z3"

- -

IYlz', - 31YI -x/Z-~z~ - 31YIx/1 + mz~, - (1 + IYl~)z3

0 = z'; - IVlz'~ + 31YI -f2-~z; - 31Y1~/1 + ~z~ - (1 + IYl~)z~, 0 =

z;' + 31YI~/1 + ~,r

+ 31YI -f2-~,4 - 21Yl~z~,

0 =

z;' + 31Y1~/1 + g j ~ - 31YI - ~ j z ~

- 21Yl~zo 9

Integration of the first two equations yields z',

=

- I V l z 3 + z',(O) ,

z'~

=

-IVlz4

+ z'~(o) .

Inserting this into the equations for za and z4 provides 0 =

%' - 2[Y['z3 - 3 [ Y l - ~ z ' 4

0 =

z~' - 2fYl~z4 + 3fY[ - x / ~ z ; - 3[YIx/1 + pCz'6 - IVlz'~(O) 9 59

- 3[Y[v~-+ #iz; - [V[z',(0) ,

'~

J

We differentiate the first one and insert the last equation for z4 and the one for z~ to obtain z;' + 7]Y[2z'3 - 6[YI3

- ~ j z 4 - 6[YI3~/1 + # j z 5 - 3 [ Y [ [ Y [ ~ - p j z ' 2 ( O

) = O .

Differentiating again and inserting the last second order equation for z3 gives llll

2

II

4

z~ + 51vl z~ + 41Yl z~ + 2 1 y l l v ? z ' , ( o )

= o.

As z 4 + 5lYl~z ~ + 4lYI 4 _- (z ~ + lvl~)(z ~ + 4[YI 2) , we have the general solution z3(t) = a c o s ( I Y [ t )

+ fl sin(lYlt) + ~ cos(2JYlt) + ~ sin(2[Ylt )

Wig(o) 2lY[ 2

We now compute analogously for z4 to obtain

z~' + 71Yl2z'~ + 6lYI3-~L-~z~ - 6[Y[~v~-+ ,jz~ + 3[vllYl

-~z'~(o) = O.

Differentiating again and inserting the last second order equation for z4 leads to z~'" § 5[Y]2z~' § 4[YI4z4 § 2[V[[Y[2z'2(O) = 0 . So, we have the general solution z , ( t ) = ~ cos(IYIt) + 3 sin(IYIt) + ~ cos(21YIt) + ~ sin(21Ylt)

We now have to take account of the initial values. For B~(0): The initial values for z~ are 0=z~(0)

=

~+5-

0=z'3(0)

=

Ivl 21VP ' 1Y1(#+27),

IVl = z;'(0)

=

- I v l 2 ( a + 45),

0 = z;"(0) =

-[Y13(3 + 8 ~ ) .

A simple computation gives

IVl IYP' ~ = ~

IVl 2lYi~ , ~ = o .

Inserting this in the general form of the solution yields

From z 1' = - I V l z ~ + 1 and zl(O) = o we then deduce 1 + IY[2t . z l ( t ) = ~IVl~ ( ~ sin(21YIt ) - sin([Ylt)) + - -

2[Y[ 2

60

IVlz~(0)

21YI~

For z4 we have the initial values

0 = z~(0) = ~ + ~ , o = 4(0) = 1Y1(3+2~), o = d'(o) = -IV}S(& + 4 ~ ) , - 3 1 v I I Y I - ~ , = d"(0) = -1Y1"(3+8~). A straightforward computation shows that

=o

~-

IVl

IVl

7 -

Inserting this in the general solution yields now

IV] r - - / 1 ~ -~sin(2lYit ) - sin(lYI0) z4(t) = ~--~V-#~ From

z'2 = -]Vlz4 and z2(0) = 0 we then get

_-

Iv] ~ r - - §

4"-)

Next, for z5 and z6 we find

z~(t) =

1 (z;"(t) + 7lYl~z~(t) 6iYi3v/y_4_~

61rI"-~z4(t))

IV[ ~/1 .c------[yl2 + ttj (lsin(2lYit) - sin([Y]t))

1 (z2'(t) + 71Yl=z'4(t)+ 61Yl"-C-~z~(t) + 3IVIIYIr 6lYpx/I--4-~ = O.

z~(t)-

Summing up, we get

B.(t)

=

(~ (lsin~21Yl,~-sio~,YI,))+1 2-~-~t) § IYI~ El(t) iyi3

+ivI 2 r---- f l lYl (cos(IYl0- 1 IV[ F - - - (~ sin(2lYit) - sin(iYlt)) E4(t) ~--~V-#i

IVl

sin(lYlO)

IV* 1 +IY[".~ X [y P (~sin(2iYIt) -- sin(lYit)) + 2"--"~-~'t] , IVI"

61

- sin(Wit)) JxJvV(t)

1 v 2 .

1 + IYP.~

+~(cos(iY[t

) - 1)2KX

21YI 1 +~-~ cos(lylt)(1 -

cos(lYIt))JxV(t)

1

+ ,~,~ sin(IYIt)(cos(IYI t) - 1)Jx J y V ( t ) .

For BE,(0): For z3 we have the initial values z3(0) = 0, z~(0)= 1, z;'(0)= 0, z;"(0)=-7lVl 2 . Here we obtain

1 1 ~5=0 IY[' ' ~ -IYI Inserting this in the general solution gives

a=O, ~=---

z3(t) = ~yj(sin(2lYIt ) - sin(Wit))

o

From z~ = -IVlz~ ~ d z,(0) = 0 we deduce zl(t) = ~

(~cos(21YIt)-cos(lYIt)+ ~).

The initial values for z4 are z4(0) -- 0 , z~(0) -- 0, z'4'(0) = -3[YI -~/-L--~, z~"(0) = 0. A straightforward computation shows that 0.

Inserting this in the general solution gives

z~(t)

=

i~--]-X/~i(cos(21YIt) - cos(lYIt)).

From z~ = - I V l z 4 and z2(0) = 0 we get z2(t) = t~[[2-~--~ ( s i n ( l r l t ) - ~ sin(2lYlt)) .

62

Eventually, for z5 and z6 we obtain 1

zs(t) = 6lYl%/i--c~(zg'(t ) + 7[Yl2z'3(t)- 6[Yl~-~z4(tl) 1 ~/1 + ~r - cos(lYlt)) IYI 1

z6(t) = 6lglav/i_T_~(z4"(t ) + 7[YI2z'4(t) + 6lYI3~/rL-~z3(t)) =

O.

Summing up, we get

B~,(o)(t)

-

IVl (~ cos(2lr[t) - cos(lYlt) +

IYP

1) E,(t)

1 +~y](sin(2lYlt) - sin(lYlt))E3(t)

+ ~---~-~(~os(21YIt) - r + i~---~1 + gj(cos(2lYIt) - cos(IYIt))Es(t) , and hence

B~v(t)

-

IYt 2

irP (~ cos(2lYlt) - cos(tYIt) + ~) X IV[ (sin(lYi Q _ 1 sin(2lY]t) ) h'X

+ [--~1(sin(21YIt) - sin(lYIt))YxV(t) 1

+]-~(cos(21Ytt) -

cos(lYIt))yxy~v(t)

tVlZ c o s ( I Y l t ) ( c o s ( I Y l t ) fVl ~

-

1)x

+ v__.~sin(IYIt)(1 - cos(IYlt))KX 1

+~-~ sin(lYlt)(2 cos(IYI t) - 1)JxV(t) 1 +~T(cos([YLt) - 1)(2 cos([Ylt) + 1)JxJvV(t) . For B~,(0): For za we have the initial values z3(0) = 0 , z;(0) = 0 , z;'(0) = slYly/1 + ~s , z~"(0) = 0 . 63

This implies a=

+~j,

/~=0,

~=-

l+~j,

r/=0.

Inserting this in the general form of the solution provides 1

z3(t) = ~y-~X/1 + p~(cos(Iglt) - cos(2lYlt)). From z'1 =

-IVIz3 and zl(0) = 0 we deduce IVI

zl(t) = ~ y - ~ / 1 + #, (~ sin(2lYlt) - sin(lYlt)) 9 T h e initial values for z4 are z4(0) = 0 , z;(0) = 0 , z;'(0) = 0 , z;"(0) = 0 . Thus 24=0

,

and from z~ = -IVlz4 and z~(0) = 0 we get Z 2 =0

.

Eventually, for z5 and z6 we get

zs(t)

= =

1

61El ~

, +/i_4__v(z~ (t + 71Y12z'3(t))

1 ~y-~(sin(2lYl* ) - sin(lY[*)), 1

z6(t) = 6]ypx/i_~j6lYI3vfl2-~z3(t _

)

1 _~-2-~(cos(lYlt) _ cos(2tYlt) ) .

IVl

S u m m i n g up, we get BE.0~(t)

=

IV[ r - - -+ttr - - - / 1[,5 sin(2[Ylt) - sin([Y[t)) El(t) iy[2~/1 / + 7 1 - ~ 1 -I- Hi(cos(It It) 1

+~-~(sin(2iYlt) + i~-~ - ~ j ( c o s ( I Y I t )

64

cos(21Ylt))E3(t )

sin( lY lt ) ) Es( t ) -

cos(2lYit))E6(t),

or equivalently,

IVl~(,

Bsx ~,vV-lrlSKxv(t )

\

~y-T, ~ + sr (2 sin(2lYlt)

-

sin(IYlt)) X

+(1 + ~r - cos(21Ylt))JxV(t) 1 -4-~-~(sin(2[Y[t) - sin( IYIt ) )( Jx Jr V ( t ) - IYl J~x V ( t ) )

1

+~--~(cos(IYlt) - cos(21YIt))( JKx JrW(t) -- #r

IYl=r

[~.,1 + tlr (~ sin(21Ylt) - sin(lYlt)] X

-

+(cos(rrl~)

-

cos(21YIt))yxV(t)

+(sin(Irlt) - sin(2lY[t) )JKxV(t ) 1

+~-~(sin(2lYlt) - sin( lYlt ) )Jx Jr W( t ) 1 +~-~(cos([Ylt) - cos(2[Ylt))JKxJrW(t). As (replace X by IYIKX in the case Be~(0))

Blvl~xv(t)

[VI~ (~ cos(2[YIt) - cos(IY[t) + 2 ) K X

IYI

vl2

(sin([Y[t) - 1 sin(2[Y[t)) X

+(sin(2[Y[t) - sin(lYlt) )JxxV(t ) 1

+T-?-[(cos(21YIt) - cos(IYrt))J,~xJyV(t), we finally get

-

Iyl ~

(~ sin(2lYlt)

-

sin(Ir]t)) X

Iv[ ~

+(cos(IYlt) - cos(2[Y[t))JxV(t) 1 +~y--~(sin(2lYIt) - sin([Y[t))JxJrV(t) -

]Y[~ sin([Y]t)(cos([y[t)- 1)X fYJ

IYl ~ +~-~ cos(lY[t)(cos([Ylt)- I) K X

+(I - cos(lYlt))(2cos(lYlt)+ l)JxV(t) 1 + ~ - [ sin(lY[t)(2 cos(IYit) - l)JxJvV(t). 65

The cases #0 = 0 and pk = - 1 can be treated as follows. First, if #0 = 0, then K X = 0 for all X E L0. For a unit vector X E L0 define q0(X) := span{X, J i V ( t ) , Jx J y V ( t ) } . The Jacobi equation along 7, restricted to %(X), is then given by the subsystem of the system of equations of the preceding general case obtained by considering the ~70-parallel orthonormal frame field El, Ea, Es of q0(X) along 7. Secondly, if pk = - 1 , then J x J v V ( t ) = IYIJKxV(t) for all X E L~. For a unit vector X ELk define qk(X) := span{X, Ix'X, J x V ( t ) , J K x V ( t ) } 9 Here, the Jaeobi equation along 7, restricted to qk(X), is given by the subsystem of the system of equations of the preceding general case obtained by considering the ~'0parallel orthonormal frame field El, E2, E3, E4 of q~(X) along 7. In both special cases the solutions are as in the general case with K X = 0 or J x J y V ( t ) = ]Y[JKxY(t), respectively. By this the theorem is now proved. []

3.8

Conjugate points

As a first application of Theorem 3.7 we will now compute some conjugate points in generalized Heisenberg groups. We denote the exponential map at the identity e by exp~ and its differential by exp,.. T h e o r e m Let V + Y C n be a unit vector and 7 : ]R ~ N the geodesic in N with 7(0) = e and ~(O) = V + Y. We have (i) Y = O. There are no conjugate points along 7. (it) V = O. The conjugate points along 7 are at t E 2~r7/*. The multiplicity of these conjugate points is n and the kernel of exp,. at t Y is ~. (iii) V ~ O ~ Y . (1) Every t with IY]t e 2~r77" determines a conjugate point along 7 with multiplicity n - 1, and the kernel of exp,. at t ( V + Y ) is the orthogonal complement V • of ]RV in ~.

(2) Every

t satisfying the equation IVp

2 cot

determines a conjugate point along 7. The first positive t with this property is greater than 27r/[Y[.

66

Proof. (i) Y = 0. Let U + X E n be a n o n - z e r o vector. A c c o r d i n g to T h e o r e m 3.7 we have

12 1 ~JxV + (t - ~t3) X + -~t[U,V] Bv+x(t) = tU + ~t 9 If U a n d JxV are linearly i n d e p e n d e n t , t h e n Bv+x(t) vanishes if a n d o n l y if t = 0. Now s u p p o s e t h a t U a n d JxV are linearly d e p e n d e n t . If JxV = 0, t h e n Bv+x(t) = 0 holds o n l y if t = 0. If JxV r O, t h e n U = aJxV for some c~ E IR, a n d we o b t a i n

Bv+x(t)=(a+~t)tJxV+(1-1t2-~at) If Bv+x(t) = 0, t h e n the JxV-component gives t = 0 or t = - 2 a . If t = - 2 a , the X - c o m p o n e n t is - 2 ( 1 + a 2 / 3 ) a a n d vanishes precisely w h e n a = 0. So we c o n c l u d e t h a t Bv+x(t) = 0 if a n d only if t = 0, whence there are n o c o n j u g a t e p o i n t s along 3'(ii) V = 0. Let U + X E , be a non-zero vector. T h e n

Bu+x(t) Here one sees readily t h a t t E 2~rZ'.

= s i n ( t ) U + (1 -

Bv+x(t)

cos(t))JyU + tX .

vanishes at some t # 0 if a n d o n l y if X = 0 a n d

(iii) V # 0 # Y . Let U + X E n b e n o n - z e r o . We m a y a s s u m e t h a t U + X is o r t h o g o n a l to V + Y, because otherwise Bv+x(t) vanishes only if t = 0. If U + X is in ha, p or q, then, according to T h e o r e m 3.7, Bv+x(t) is in n3, p a n d q, respectively, for all t. T h u s , in order to find the conjugate points along 3' it suffices to investigate the J a c o b i fields Bv+x with initial values Bv+x(O) = 0 a n d B'v+x(O) in n3, p or q. T h e s i m p l e s t case to deal with is w h e n U + X C p (so X = 0). By m e a n s of T h e o r e m 3.7 we have

By(t) Clearly,

By(t)

= ~1

sin(lglt)g+

= 0 precisely if

1 ~-~(1 -

cos(IVlt))JyU.

IYIt E 2Tr7l.

Next, if U + X C ha, we p u t

E,(t)

:=

1 IYI, -T~]vt 't')+ M IY[y ' E,(t).- IVIIYI JYV(t)

with

V(t)

1 sin(IYlt)JrV : = cos(lY[t)V + ~y~

and

u + x = ~EI(0) +/~E~(0) w i t h some a,/3 E IR, n o t b o t h equal to zero (recall t h a t U + X is a n o n - z e r o vector o r t h o g o n a l to V + Y). F r o m T h e o r e m 3.7 we t h e n o b t a i n

Bv+x(t)

=

1 [yl3(a(sin(lYIt) -Ivl~lYI

1

t) + ~[Yl(cos(lYlt) -

+ , - m , (~(1 - cos(lYlt)) +/~[Y[ IIV 67

sin(lYlt))E2(t).

1))E,(t)

The determinant of the matrix

IYl(cos(IYIt)

sin(IYlt -IYl2JYlt

IYl(1-eos(IYIt))

- 1)

IYI ~sin(IYIt)

/

is equal to IYI2(2(1 - cos(IYIt)) - IVl=lYI t sin(IYlt)) =

IYl2(4sin2(~-)-2lVl2lYltsin(I--~-)cos(I-~)),

and vanishes precisely if

Iglt e 2~7z or

IVP -

Note that the first condition implies a = 0 and hence U + X = flJyV. Further, since IVI = < 1, the first positive t satisfying the second equation is subjected to the i n e q u a l i t y IYIt > 2r. So we have obtained all conjugate points along 7 arising from initial values in a3 N (V + Y)• Finally, if IYIt E 2r7] and X E Y• then B x ( t ) _ _ _+1 [Y]2tX , B j x v ( t ) = O, nsxs,.v(t ) = 0 . 21YI ~

This shows that the multiplicity of the conjugate points with IYIt E 2~rTZ" is n - 1, and the kernel of exp,. at t ( V + Y ) is

9 tdvV • p G (q n o) = V • . By this also statement (iii) is proved. D Theoretically the same method may be used to decide whether there arise further conjugate points from initial values in q, but the explicit computations become much more complicated. We did not pursue this work in order to have a complete answer. On the other hand, q is trivial if dim3 = 1 and then we obtain as a corollary a complete classification of the conjugate points in Heisenberg groups.

Corollary Let N be a (2k + 1)-dimensional Heisenberg group, V + Y E n a unit vector and "/: ]R --+ N the geodesic in N with 3'(0) = e and "~(0) = V + Y . We have (i) Y = O. There are no conjugate points along % (ii) V = O. The conjugate points along "7 are at t E 27r7]*, and their multiplicity is 2k. (iii) Y ~ 0 ?~ Y . The conjugate points along 7 are at all t satisfying Iglt e 2~rZ" or

iVl~ -

In the first case the multiplicity of the conjugate point is 2k - 1, in the second case it is one. The first conjugate point is at 27r/IY I. Partial information about conjugate points in generalized Heisenberg groups has also been obtained by J. Boggino [Bog]. 68

3.9

Principal

curvatures

of geodesic

spheres

The main result of this section says that the principal curvatures of small geodesic spheres in generalized Heisenberg groups are the same at antipodal points. To prove this, we start with the following L e m m a Let M be a Riemannian manifold with Levi Civita connection V , p 6 M , 7 : [0, b] --~ M a geodesic parametrized by arc length and with "/(0) = p, and r 610, b] so that the geodesic sphere Gv(r ) centered at p and with radius r is a hypersurface of M . Denote by ~ the "outward" unit normal field of Gp(r) and by A the shape operator orGy(r) with respect to ~, that is, Av = - V ~ for all v 6 TGp(r). Let T~ be a parallel skew-symmetric tensor field along 7. As in 3.7 we define ~'o:=Vo+T~

and R~ := R~ + T~ .

Let D be the solution of the End(~/l )-valued second order equation Vo(Yob - 2T~ZoD + R , D = 0 with initial values 9 ( 0 ) = 0 a~d (V0D)(0) = id,(0)~ Then

A~(r) = -(%D)(r) o 9-'(r) + T~(r).

Proof. Let E be a ~'o-parallel vector field along 7 which is perpendicular to "~. For Y := D E we then have V a V o Y + R.rY

=

Vo~Ta(/)/~) - 2 T ~ ' o ( / ) / ~ ) + / ~ / ) / ~

= (n + rs and

1 1 [U+X+rA, V+Y+sA] := [U, V]n + ~rV - ~sU + r Y - ~X .

In this way ~ becomes a Lie algebra with an inner product. The attached simply connected Lie group, equipped with the induced left-invariant metric, is denoted by S and is called a Damek-Ricci space.

4.1.2

Classification

and

idea

for construction

Of course, the classification of all Damek-Ricci spaces follows from the one of generalized Heisenberg groups established in 3.1.2. The dimensions of the Damek-Ricci spaces can be determined from the tables in 3.1.2 by just adding one dimension. Also here the case when m, the dimension of 3, is congruent to 3(mod 4) is of particular interest. In such a situation we define

~(k~, k~) := .(k,, k~) 9 where kl, k: are non-negative integers, and denote the corresponding Damek-Ricci space by S(kl, k~). It follows from 3.1.2 that ~(kl, k2) is isomorphic to ~(kl,_k:) if and only if for both Lie algebras the dimension of 3 is the same and (kl, k2) E {(k~, k2), (k2, k~)}; this is also the precise condition in order that two Damek-Ricci spaces S(kl, ks) and S(~h, Ir are isometric to each other. The idea for the construction of the Lie algebras s stems from the following considerations. Let m = 1, that is, n is a (2k + 1)-dimensional Heisenberg algebra in the classical sense. Then n is isomorphic to the nilpotent part in the Iwasawa decompositi6n of the Lie algebra of the isometry group of the complex hyperbolic space C H TM. The solvable part in this Iwasawa decomposition is isomorphic to ~ = n 9 a with the Lie algebra structure as defined above. This shows that for m = 1 the Damek-Ricci spaces are isometric to the complex hyperbolic spaces. Similarily, if m = 3, the Damek-Ricci space S(k, O) ~- S(O, k) is isometric to the quaternionic hyperbolic space ]HHk+I; and if m = 7, the Damek-Ricci space S(1, 0) ~ S(0, 1) is isometric to the Cayley hyperbolic plane C a y H 2. So the construction of the Damek-Ricci spaces arises from imitating the construction of the non-compact rankone symmetric spaces of non-constant curvature from their Iwasawa decomposition. Note that the real hyperbolic space can be obtained via this procedure by starting with a zero-dimensional vector space 3 and, therefore, an Abelian Lie algebra n = v.

4.1.3

Algebraic

features

and

diffeomorphism

type

The derived subalgebra [5, ~] of , is equal to the generalized Heisenberg algebra n and hence nilpotent. This shows that s is a solvable Lie algebra and, therefore, 79

Proposition

1 Each Damek-Ricci apace is a solvable Lie group.

T h e definition of the Lie algebra structure on n implies that, as a Lie algebra, a is the semi-direct sum ~ = n + l a of n and a with regard to the ]R-algebra h o m o m o r p h i s m f:a

~ der(n) , sA ~ (n -o n , V + Y H l s v + s Y )

2

.

C a r r y i n g this over to the group level means that S is a semi-direct p r o d u c t S = N XF ]R of the generalized Heisenberg group N attached to S and ]R, where F : 11:{~ A u t ( N ) , s ~ ( N ~ N , e x p n ( V + Y ) ~ expn(e'/2V+e'Y))

,

expn is the Lie exponential m a p of N and 11% is considered in the canonical way as the simply connected Lie group attached to o. Writing this down explicitly, the group s t r u c t u r e on S = N x F ]R is given by ( e x p n ( U + X ) , r ) . (expn(V+Y), s)

:

(oxp~

Since expn : n --~ N is a diffeomorphism, the m a p exPn x e x p a : n • 0 --* N x 11:[, V + Y + s A

~ ( e x p n ( V + Y ) , s)

is a diffeomorphism. Therefore the Lie group S is often identified with ~ e q u i p p e d with the group s t r u c t u r e obtained from S via this diffeomorphism. We will not use this identification here. A consequence of the preceding consideration is Proposition

2 S is diffeomorphic to ]Rrn+n+l.

We finally mention that the endomorphisms Jz play also an i m p o r t a n t role for Damek-Ricci spaces, and their algebraic features stated in 3.1.3 will also be used frequently in this chapter without referring to t h e m explicitly.

4.1.4

Lie exponential

map

Let V + Y + s A 6 ~ be arbitrary. We want to compute now the corresponding onep a r a m e t e r subgroup of S, that is, the integral curve a : ]R ---* S of V + Y + s A with a ( 0 ) = e. According to 4.1.3 we may write c~(t) = (expn(V(t) + Y(t)), s(t)) with some functions V : ]R -+ 0, Y : ]R --+ 3 and s : ]R -+ ]R. T h e n the condition

. ( t + {) = ~ ( t ) .

~(~)

leads to (expn(V(t + {) + Y(t + t)), s(t + t))

=

(expn ( V ( t ) + e'(0/~V(t) + Y ( t ) + e ' ( ' ) Y ( { ) + ~e'(O/2[V(t), V ( { ) ] ) , s ( t ) + 80

s({)) .

First, the second component gives s(t + t) = s(t) + s ( t ) , and from 4 0 ) = O, s'(O) = s we then get s(t) = st.

As expn is a diffeomorphism, the first component then provides the conditions V(t + t)

=

V(t) + e"/~V(t) ,

Y ( t + t)

=

Y ( t ) + e"Y(t) + ~e"l~[V(t), V(t-)] .

Further, we have the initial conditions V(O) -- O, Y(O) = O, and, as the differential of expn at 0 E n is just the identity transformation of Ton ~ n, V'(O) = V , Y'(O) = Y . It can easily be checked that if s = 0

tV V(t) :=

2 ( e " / ~ - 1)V s

, ifsr

and if s = 0

tY 1 - ( e " - 1)Y

Y ( t ) :=

, ifsr

S

are the solutions of these functional equations. Thus we summarize P r o p o s i t i o n The one-parameter subgroup of S = N X F IR induced by the vector V + Y + s A E ~ is t~(expn(t(Y+Y))

, if s = O ,

,0)

, ifsr

t~-* @ x P n ( ~ ( e ' q ~ - l ) v + l ( e ' t - 1 ) Y )

In particular~ the Lie exponential map exps : s -* S satisfies exp~(V+Y+sA) =

{ (expa(~/Y)'0) expn

1

(e ' / ~ - I ) V T - ' ( e ' - I ) Y

)

) ,s

, i f s : 0 i f0 ,

sr

.

S

Using the fact that the Lie exponential map expn of N is a diffeomorphism, a simple calculation implies C o r o l l a r y The Lie exponential map exp~ : ~ -~ S of S is a diffeomorphism. The Lie exponential map on Damek-Ricci spaces has also been derived by J. Boggino [Bog]. Notice that he used another model of S, which is why his formula is not the same as ours. 81

4.1.5

Some

global coordinates

We now introduce some global coordinates on S. Let G . . . . , V, LY1,-.., Ym, A be an o r t h o n o r m a l basis of the Lie a l g e b r a , and ~3~,..., ~?,, ~)~,..., ~)m,A the corresponding c o o r d i n a t e functions on ~. Using the diffeomorphism expn x expa we o b t a i n global coordinates v ~ , . . . , v,, y ~ , . . . , Ym, ~ on S via the relation ( v ~ , . . . , v , ~ , y ~ , . . . , y ~ , , X ) o (expn x expa ) = ( ' b ~ , . . . , # , , # , , . . . , y m , . ~ ) 9 Lemma

We have

Vii Yi

1 ~12~ vAu ~e

e~12a~~ =

~--

-

a

~ Oyk

,

eA 0

Oy, 0 A = -Oh'

'

where A~j := .

Proof. Let p :-- (expn(U + X ) , r ) e S -- N x r ]R be a r b i t r a r y a n d Lp the left t r a n s l a t i o n on S by p. Then, using Proposition 4.1.4, we have V~(p) =

np,,V~(e)

H (exp.(tv,), 0))(0) 0

=

N(~ ~ L.(exp.(tY,),0))(0)

=

,s,(o)

with

c~i(t) = (expn (U + e'/2tV~ + X + ~e~/2t[U, V.]) ,r) 9 We have

(y,~ o o,)(t)

=

1 ,-12 ~,~(X) + ~e t

=

I ,-12t < J y k V , , U > ~k(X) - ~r

=

#~(X)-~e

1 ,/2

~<Jy~Y,,Yj> J

1 r j2

= #k(X)-- ~e" ty~A~iS~(U), J (~ o cq)(t)

=

r.

82

Thus,

(v~ o . , ) ' ( o ) (y~

o.,)'(0)

=

,~,//~,

=

-~.~'~AU/U),

=

o,

J

(~o~,)'(o)

and therefore, Y,(p)

=

~,(o)

: Z(v~ o o,),(o@~)+ z(~ o o,),(o)~(~)+ (~ o o,),(o)~(~) k

k

ayk

= e~(p)/20~i(p) - lex(,)/2y~ A~jvi(p)_O----(p). 2

j,k

uY k

Similarily, Y,(p)

=

L.oY,(e)

=

Lp.,

=

0 ~ ( t H Lp(expn(tY~),0))(0)

=

3,(0)

(t H (expn(tY~) ,0))(0)

with /3,(t) = (expn(U + X + e~tYi), r) . Here we have

(,,~

o ~,)(t) = ~(u)

, (yk o 9 , ) ( 0 = ~ ( x )

+ 6,/t

, (~ o 9 , ) ( t ) = , . ,

and hence

(,k o Z,)'(o) = o , (w o/~,)'(o) = 6 , S ,

(A o/~,)'(o) = o .

This implies Y,(p) = 3,(o) = ~'r

Yi

Finally, A(p)

=

Lp.~A(e)

=

Lp.~

_

0 (t ~ Lp(exPn(0),t))(0) Ot

=

a(o)

(t H (exp.(O), t))(O)

83

with

c(t) : (exp.(U + x ) , ~ + t ) . Here,

(~

o ~)(t) = ~ ( u ) ,

(y~ o c)(t) = ~ ( x ) ,

(A o c)(t) = ~ + t ,

and hence (vk o c)'(0) = 0 , (yk o c)'(0) = 0 , (,~ oc)'(0) = 1 . This implies 0

A(p) = ~(0) = -~(p) Thus the L e m m a is proved. [] We will use the L e m m a also in the following converse form. Corollary

We have 0

Ovi 0 cgy~ 0 - OA

- -

4.1.6

Levi Civita

1

)~

k

e-~/2Vi + -~e- ~y,kAijvyYk '

z

~-

~

A

"~ Y i

.

connection

Let V be the Levi Civita connection of a Damek-Ricci space (S, g). Using the same m e t h o d as for the generalized Heisenberg groups we obtain

Vv+Y+,A(U+X+rA) = - ~ J x V - 1 j v U - I-~rV- ~[U, V ] - rY + ~A 1 + <X,Y>A. 2 4.1.7

Curvature

Let R be the R i e m a n n i a n curvature tensor, Q the (1, 1)-Ricci tensor and r the scalar curvature of a Damek-Ricci space (S, g). By a straightforward c o m p u t a t i o n we get

R(U+X+rA, V+Y+sA)(W+Z+tA) = -21jxJgW + l r Jv

+

~JzJvU - ~JzJxV + 1-j[uvlW-~ '

4

_ 1 1 1 1 1 ~sJxW - ~sJzV + -~tJvU + ~rJzV - -~tJxV

1 X , Y>W +~
2- ~(3JXl2lYJ 2 +6<JxU,

JyV> + 1 ) .

A more detailed discussion of the sectional curvature can be found at the end of 4.2. In p a r t i c u l a r it will be proved there that

Proposition

2

Every Damek-Ricci space is a Hadamard manifold.

Recall t h a t a Hadamard manifold is a complete and simply connected R i e m a n n Jan manifold with non-positive sectional curvature. It is worthwhile to mention t h a t a homogeneous manifold of non-positive curvature can always be represented as a solvable Lie group with a left-invariant metric (see for instance [Hei]). For a nice survey a b o u t the geometry of Riemannian manifolds of non-positive curvature see [Ebel]. A consequence of Proposition 2 is Corollary

The exponential map exp~ : T~S ---* S of S at the identity e is a

diffeomorphism.

4.1.8

T h e Jacobi operator

Let V + Y + s A C ~ be a unit vector. For the Jacobi o p e r a t o r and its covariant derivative with respect to V + Y + s A we then have

85

Rv+v+,A(U+X+rA) 3 = ~ J x J r V + ~Jtu,v]V + ~rJrV -~U + 1 3 < X , Y > V

~sJxV

~ ) } A and R v ,+ Y + , A ( U + X + r A

4.1.9

3 (J[u ,v]Jv V ) = -~

+ J[u,j~v]V - J v V - V )

Symmetry

T h e J2-condition, which has been introduced in 3.1.3, can now be given a beautiful geometrical interpretation. T h e o r e m A Damek-Ricci space S is a Riemannian symmetric space if and only if the attached generalized Heisenberg algebra a satisfies the J~-condition. More precisely, S is a Riemannian symmetric space if and only if

(i) m = 1; then S is isometric to the complex hyperbolic space CH k+l, 2k = n, with constant holomorphic sectional curvature -1; or (ii) m = 3 and n = n(k,0) ---- a(O,k); then S is isometric to the quaternionic hyperbolic space ]I-IHT M with constant quaternionic sectional curvature -1; or (ill) m = 7 and n = n(1, 0) ~ n(0, 1); then S is isometric the the Cayley hyperbolic plane C a y H 2 with minimal sectional curvature -1. Note t h a t the simply connectedness of the Damek-Ricci spaces implies t h a t if such a space were locally symmetric it would also be globally symmetric.

Proof. F i r s t suppose t h a t S is a Riemannian symmetric space. Let V + Y E n be a u n i t vector with V # 0 # Y and K := Kv,v as in 3.1.12. Further, let X be a non-zero vector in Y• Using the above expression for the covariant d e r i w t i v e of the Jacobi o p e r a t o r we o b t a i n 0 = R'~+vJxV = - ~ l Y l ~ ( J x J y V - Z.v~KxY) . This shows t h a t n satisfies the J2-condition. Conversely, suppose that n satisfies the J2-condition. Let V + Y + s A E z be a unit vector. T h e n we have

R'v+y+,A(X+rA)

= O, 86

R'v+r+,aV

=

O,

R'~+r+,~JrV

=

O,

Rv+r+,AU

=

0 for all U E kerad(V) O V" ,

R v' + v + ' a J x V

-

23 1 V [ 2 ( Y x J y V - J,v,~xV) = 0 for all X E y z .

Since u satisfies the J~-condition, ~ can be decomposed orthogonally into

= (3 G a) | IRV @ ]RJvV 9 (ker ad(Y) O V • | J v ~ V , and we conclude that RV+Y+sA = 0

for all unit vectors V + Y + s A E ~. This implies that V R = 0 (see for instance L e m m a 5.1 in [VaWi2]), that is, S is locally symmetric. As S is simply connected, it is also globally symmetric. The assertion then follows from Theorem 3.1.3, the discussion at the end of 4.1.2 and the remarks about the values of sectional curvature at the end of 4.2. []

4.1.10

Integrability of certain s u b b u n d l e s

In this section we discuss the integrability of the subbundles v,

3 , a,

vOa,

3Ga,

n

of T S obtained by left translation of the corresponding subspaces of T~S ~- ~ and the geometric structure of the induced foliations in case of integrability. Therefore we first recall the notions of Riemannian foliations, spherical submanifolds and isoparametric hypersurfaces. Let ( M , g ) and B be Riemannian manifolds. A submersion ~r : M ~ B is called a Riemannian submersion if the differential of ~ preserves the length of all tangent vectors of M which are perpendicular to the fibers of 7r. Now, let L be a foliation of M and B the set of all leaves of L equipped with the quotient topology with respect to the canonical projection 7r : M --+ B. Then L is called a Riemannian foliation if 7r can be made locally a Riemannian submersion. Geometrically this means that the leaves of L are locally equidistant (or parallel) to each other. If V is the Levi Civita connection of M , then L is Riemannian if and only if for each vector field X tangent to L and all vector fields Y, Z orthogonal to L the equation

9 ( v , x , z) + 9(Y, v ~ x ) = 0 holds (see Theorem 5.19 in [Ton]). This analytic characterization of Riemannian foliations will be used in the proof of the subsequent Proposition. A submanifold B of a Riemannian manifold M is called spherical (or an extrinsic sphere) if B is totally umbilical in M and the mean curvature vector field of B is parallel in the normal bundle of B. 87

A hypersurface B of a Riemannian manifold M is called isoparametric if its principal curvatures are constant. P r o p o s i t i o n Concerning the integrability of the following left-invariant subbundles of T S we have

5) (ii)

D is not integrable; 3 is integrable and the induced foliation of S is Riemannian; each leaf is a spherical submanifold of S with mean curvature vector A and isometric to fit m with its standard Euclidean metric;

(iii) a is integrable and the induced foliation of S is not Riemannian; each leaf is a totally geodesic submanifold of S and isometric to IR;

(iv)

G a is not integrable;

(v)

3 G a is integrable and the induced foliation of S is not Riemannian; each leaf is a totally geodesic submanifold of S and isometric to the real hyperbolic space ]RH "~+1 of constant curvature - 1 ;

(vi)

D (~ 3 = n is integrable and the induced foliation of S is Riemannian; the foliation is the horosphere foliation of S by the horospheres centered at the point at infinity determined by the integral curves of A; each leaf is isometric to the corresponding generalized Heisenberg group N and an isoparametric hypersurface of S with two constant principal curvatures, 1/2 and 1, and corresponding eigenspaces ~ and 3, respectively.

Proof. The statements (i) and (iv) follow from the fact t h a t a d ( V ) : ~ ~ 3 is surjective for all non-zero V E ~. For all U + X , V + Y 6 n we have Vv+x(U+X) = -

J x V - -~JrU -

[U, V] + ~ A + < X , Y > A ,

a n d hence

[U + X , V + Y]

e ..

Therefore, , is integrable. Moreover, as + = 0,



n induces a R i e m a n n i a n foliation L. Now, according to 4.1.3, the multiplication on S r e s t r i c t e d to N x F {0} is just the multiplication on N x~- {0} induced from the one on N. Thus, the leaf of L through e is the hypersurface N x F {0} and hence isometric to the generalized Heisenberg group N a t t a c h e d to S. By left-invariance of ,, each leaf of n is isometric to N. At each point A is a unit n o r m a l vector of the leaf t h r o u g h t h a t point, and from

VvA=-=V

1 z

and V v A = - Y 88

the statement about the principal curvatures follows. Finally, the statement about the horosphere foliation follows by applying Theorem 4.2 in [Wo12] to S. So statement (vi) is proved. For all X, Y E 3 we have VxY = <X,Y>A.

This implies that 3 is integrable, each leaf B of the induced foliation L is totally umbilical and A restricted to B is the mean curvature vector field of B. Since VvA = -Y one sees that A is parallel in the normal bundle of B. Therefore each leaf of L is a spherical submanifold of S. Denote by R B the Riemannian curvature tensor of B and by h the second fundamental form of B. Then h(X,Y) = <X,Y>A,

and the Gauss equation of second order implies, for Z E 3, R ~ ( X , Y ) Z = R ( X , Y ) Z + X - < X , Z > Y = 0 ,

that is, B is a flat manifold. Now, let B be the leaf through the identity of S. As C n, the leaf B is a submanifold of the generalized Heisenberg group N embedded in S as N xF {0}, the leaf of n through e. According to 4.1.3 the multiplication on N coincides with the one on S restricted to N xF {0}. Thus we may apply Proposition 3.1.10 to 3IN and obtain that B is isometric to IRm with the standard Euclidean metric. By left-invariance of 3 it follows that each leaf of L is isometric to IR'~. Further, for X E 3 and V, W E ~ we have 1 1 + < V + s A , V w + , A X > = - ~ < d x V , W > - ~

= 0,

which shows that L is a Riemannian foliation. Thus (ii) is proved. As VAA = 0 , each integral curve of A is a geodesic in S. Since S is complete, simply connected and of non-positive curvature it follows that each leaf of a is a totally geodesic submanifold of S diffeomorphic (and hence isometric) to IR. Further, since < V v A , V > + = - I V I 2 r 0

for V # 0, the foliation cannot be Riemannian. This proves statement (iii). Next, we have Vv+,A(X+rA) = -rY + <X,Y>A

89

E ~G a .

This shows that ~ @ a is integrable and the induced foliation L is totally geodesic. Let B be the leaf of 3 9 a through e. Then the Gauss equation of second order implies

RB(X+rA, Y+ s A ) ( Z+ t A ) = - < Y + s A , Z+ t A > ( X + r A ) + <X+rA, Z + t A > ( Y + s A ) , which is precisely the curvature tensor of a space form with curvature - 1 . According to Corollary 4.1.7 the exponential map exp~ of S at e is a diffeomorphism. This implies that B = exp,(3 G a) is diffeomorphic to ]Rm+l and hence isometric to the real hyperbolic space ]RH m+l with constant sectional curvature - 1 . As

+ = -IVI 2 r 0 for V r 0, L is not Riemannian. This proves statement (v). []

4.1.11

Geodesics

Let V + Y + s A be a unit vector in T~S ~- ~ and 7 : ]R ~ S the geodesic in S with 7(0) = e and "~(0) = V + Y+ s A . We define a subalgebra ~4 of ~ by ~4 := span{V, Jy V, :I/,A}

.

This algebra is one-dimensional if V = 0 = Y, two-dimensional if either V = 0 or Y = 0, and four-dimensional otherwise. A straightforward computation shows that the left-invariant subbundle 24 of TS is autoparallel, that is, it is integrable and its leaves are totally geodesic. Clearly, ~ lies in the leaf By of the induced foliation of S through the identity e. Furthermore, By is simply connected, for By = expe(,4 ) and expe is a diffeomorphism. If V # 0 # Y, the Lie algebra 24 is the extension of the three-dimensional Heisenberg algebra as described in 4.1.1. Thus By is isometric to the two-dimensional complex hyperbolic space C H 2 with constant holomorphic sectional curvature - 1 . If V = 0 and Y # 0, then the Gauss equation of second order implies R~

A,

~-~=-A

and

,A

A-

]y[ .

Thus By is isometric to the one-dimensional complex hyperbolic space C H 1 with constant sectional curvature - 1 . Furthermore, B~ is the intersection of all totally geodesic C H ~ in S containing e and at which Y and A are tangent. If Y = 0 and V r 0, then the Gauss equation of second order implies R B" A,

iV[-

4

~V-~,A A -

4IVI

Thus B~ is isometric to the two-dimensional real hyperbolic space IRH 2 with constant sectional curvature - 1 / 4 . Furthermore, B~ is the intersection of all totally geodesic C H 2 in S containing e and at which V and A are tangent. Finally, if 90

V = 0 = Y, 7 is the integral curve of A and B~ is the intersection of all totally geodesic r 2 in S containing e and at which A is tangent. The situation described above is very similar to the one in symmetric spaces, where the role of CH 2 is taken over by the flats. Summing up we have shown P r o p o s i t i o n Every geodesic in a Damek-Ricci space S lies in a suitable totally geodesically embedded complez hyperbolic space @H2 with constant holomorphic sectional curvature - 1 . As the geodesics in CH 2 are well-known, the preceding Proposition enables one to write down explicitly the geodesics in Damek-Ricci spaces. The idea how to do this is due to [CoDoKoRi] and is as follows. As a model for the two-dimensional complex hyperbolic space we take the open unit ball D :-- {z E c~ I Izl < 1} in C ~ equipped with the Bergman metric d s 2 = 4(1

-

Izl~)dz 9 d z - 5 d z . z d ~

(1-1zl~) ~ of constant holomorphic sectional curvature - 1 . Any geodesic ~ : ~ ~ D parametrized by arc length and with a(0) = 0 is of the form ~(t)=o(t)z

with some z E OD and 0(t) := tanh(t/2) . Next, consider the Cayley transform C : D ~ L) from D onto the Siegel domain

given by

c ( z l , z~) =

( 1 2zl l_z ) + z2' 1 T

'

The map C is biholomorphic with inverse map C-~(z,,z~) =

( l + z2' l-+ l_z )

'

Now equip D with the Riemannian metric for which U becomes an isometry. This gives the Siegel domain model of the two-dimensional complex hyperbolic space equipped with the Bergman metric of constant holomorphic sectional curvature - 1 . We then deduce that

Z:=Co~= k 1 u is the general form of a geodesic/3 : ]R ~ / ) 8(0) = (0, 1). 91

l+ez2] parametrized by arc length and with

Next, we fix unit vectors 17 6 v and l~ 6 3 such that

IYl?

V = IY[l~ and Y =

and denote by C H ~ the two-dimensional totally geodesic complex hyperbolic subspace in S determined by span{I?, JglT, I~,A}. Note that G H 2 = S~ if V f i 0 # Y, and B~ is strictly contained in r 2 otherwise. We define a bijection @ : D ~ r 2

by r

~(z,)Jf~V + ~(z2)]z) ,In

z 2 ) : : (exp n (~(zl)17 +

(~(z~)1

Its inverse m a p is given by 9 -l(expn(alT+bJ~.V+cY),t) The Lie group r

1 2 +b2)+ ic) . : (a+ib, e'+-~(a

2 acts simply transitively o n / ) by r

CH 2 x b ~ D,

(g,p) H @ - ' ( g . @(p)) ,

or explicitly, by r

+

bJ~.9 + c?), t), (z,, z~))

1 1 = (aTibTet/2zl,ic+ ~(a2+b 2)+ ~e '/~(a - ib)zl + etz2) .

Thus C H 2 acts on /) by biholomorphic transformations. As every biholomorphic transformation o f / ) is an isometry we conclude that C H ; acts on /9 simply transitively by the isometrics Cg, g 6 C H 2. Next, since /31(0) = 0 and /32(0) = 1, we have (I).(0,1)/~(0) = ~(B'I(0))~" + ~(/3~(0))J~l? + ~(/3'2(0))1? + ~ ( ~ ; ( 0 ) ) A . From this we deduce that ~.(0,1) : T(0,1)/) ~

T~eH 2

is a linear isometry. Since

(I) oCg =Lgo(I) for all g 6 C H 2, where L 9 denotes left translation by g, and both La and Cg are isometrics, we conclude that r is a linear isometry for each z 6 D. Thus we have proved that 9 is an isometry. So the geodesic 7 fixed at the beginning of this p a r a g r a p h is given by 7 = ~ o/3, where/3 is determined by the initial values IWl = y , ( o )

= z~

and

~ + ilYI

= y=(O) = -z=.

As

1 1+

8z2

1 1 - sO - i l Y l 8

-

1 X

(1 - sO+ilYlO

with x : = (~ - ~e) ~ + I Y l ~ e ~ ,

92

)

we eventually obtain

21vie 7

-- r

l+se+ilYlQ

1-~--ilYl6'

1~/I--~/

= (expa(20(l~SO)V+~JvV+2~xY),ln(~x~2))

.

So we have proved T h e o r e m 1 [CoDoKoRi] Let V+Y+sA E s be a unit vector and 7 : K:t ---+S = N xv IR the geodesic in S with 7(0) = e and ~(0)= V+Y+sA. Then

7 = (exp" (20(I ~ sO)V + 202jYV + 2~xY) 'ln (~----X~2) with O(t) := tanh(t/2)

and X := (1 - sS) 2 +

IYl=~ ~ .

Note that in [CoDoKoRi] the second sign in the formula for the geodesics is not correct. Note also that in [CoDoKoRi] the model for S is a semi-direct product of N and IR+ which can be obtained from ours by identifying ]R with IR+ via the real exponential map. We will now compute the tangent vector field ~ of 7, where we still identify different tangent spaces along 7 via left translation. Therefore, let IK, ~" and C H 2 as above. Further, let v, u, y, A be the coordinate functions on C H 2 induced by ~z, J~.V,Y,A (see 4.1.5). We define h.-

1 - 82

x and obtain, by using Corollary 4.1.5 and the preceding Theorem 1, ,0

,0

,0

,0

= (v oT)No~+(uoTlNo~+(yoT)NoT+(~oT) ~ o 7 =

1

~(v

1

o ~)'? + ~ ( u

o ~)'y~

1 1 ((~ o 7)' + 5(v

+(~ o 7)'A. By a straightforward computation we get

(v o ~)' =

IVlh-((1 - ~O)~ - IYI~O~), x

(u o 7)'

=

2lVllYl~e(1 - s ~ ) ,

(y o 7)' =

IyI-h(1 - (IYI ~ + s~)e ~) ,

(AoT)'

(lnh)'.

=

:Y

93

Inserting these expressions into the general formula for -) we o b t a i n

+ = ---~--~((1 - sO) 2 -JYl2O2)lvl~ + 2 v/h0(1 - so)lvllY[Je9 + hlY[~" + (ln h)'A. X

X

Therefore we have proved T h e o r e m 2 Let V + Y + s A E s be a unit vector and 7 : IR ---* S = N • geodesic in S with 7(0) = e and ~(0) = V + Y + s A . Then =

the

. v/-h((1 - sS) 2 - ]YI20~)V + 2--~-~0(1 - sO)JyV + h Y + (ln h)'A , X X

where

1 - 02 8(t) : = t a n h ( t / 2 ) , X := (1 - sS) 2 + IY]202 and h . - - X In particular, when we decompose ~f(t) into ~(t) = V(t) + Y ( t ) + A(t) with respect to ~ = v @ 3 @ a, then

IV(t)l 2 = ]Vl2h(t) and

IY(t)l

2

= ]Y]2h2(t) .

T h e last s t a t e m e n t can be checked easily by using the explicit formula for ~. In p a r t i c u l a r we see now that in the generic case V # 0 # Y the length of the projections of ~ onto v and ~ is not constant along 7.

4.1.12

Isometry

group

T h e i s o m e t r y groups of the symmetric Damek-Ricci spaces, t h a t is, of C H k, ]I-IHk (k >_ 2) a n d C a y H 2, are well-known from the classical theory of s y m m e t r i c spaces. If S is non-symmetric, the isometry group of S is as small as possible. Denote by A ( S ) the group of a u t o m o r p h i s m s of S whose differential at e preserves the inner p r o d u c t on T , S ~ 5, and by L ( S ) the group of left translations on S. Clearly, any m a p belonging to A ( S ) or L ( S ) is an isometry of S. T h e o r e m [Dam2] If S is a non-symmetric Damek-Ricci ~pace, the isometry group of S is the semi-direct product L ( S ) x F A ( S ) with F : A ( S ) ~ A u t ( L ( S ) ) , r ~ ( L ( S ) ~ L ( S ) , L 9 ~ CLgr -~ = L~(g)) .

94

4.1.13

N e a r l y K~ihler s t r u c t u r e s

Recall that a nearly K~hler structure on a R i e m a n n i a n manifold (M, g) is a skewsymmetric tensor field J of type (1,1) on M satisfying j2 = - - i d T M and (V,J)v = 0, or equivalently, by polarization,

(V~J)w + (V~J)v = 0 for all t a n g e n t vectors v, w E TpM and all p C M. Suppose now that J is a nearly Ks structure on a Damek-Ricci space S which is invariant u n d e r the group L(S) of left translations on S. So each left t r a n s l a t i o n is a holomorphic m a p of S with respect to J, and J maps left-invariant vector fields on S to left-invariant ones. We denote by ( )~ and ( )3 the projections onto 0 a n d 3, respectively. From

0 = -VA(JY)+ JVAY = -(VAJ)Y =

(VrJ)A

=

Vy(JA)-

JVyA

1

= - ~ J y ( J A ) ~ + <JA, Y > A + J Y we get the relations 2(JY)o

( JY)3

Jr(Jg)o,

=

= O.

Similarily, we o b t a i n from

O = - ( V AJ)V = (Vv J)A = - ~ J(jA)3V - ~[( JA)o, V] + I < j A , V> A + I j v the relation

(JV)3 = [(JA)0, Y] . Using these relations we obtain

- < J Y , V> = <JV, Y > = = <Jy(JA)o, V > = 2 < J Y , V > , a n d therefore also (JY)D = 0 . We conclude that J3 C a and hence dim3 = 1, that is, S is a complex hyperbolic space. We summarize T h e o r e m A Damek-Ricci space S admits a nearly KShler structure which is invariant under the group L( S) of left translations on S if and only if S is isometric to a complex hyperbolic space. 95

C o m p l e x hyperbolic spaces are known to carry even an invariant Kghler structure, t h a t is, an invariant nearly Kghler structure for which also V J = 0 holds. T h a t none of the Damek-Ricci spaces with dima >_ 2 carries a K/ihler s t r u c t u r e (which is invariant under L ( S ) ) follows also from a more general theory. Indeed, since the scalar curvature of a Damek-Ricci space S is non-zero, any K/ihler structure on S has to be invariant. Moreover, since S is a solvable Lie group it must be a homogeneous d o m a i n and finally, because it has non-positive curvature, S has to be symmetric. See [DaDo], [Dor], [DoNa], [Lid], and [Lie2] for more details. Recently an analogue of the above mentioned result by D ' A t r i and Dotti Miatello has been discovered in the framework of quaternionic geometry. In [Cor] it is proved t h a t a (non-flat) real solvable Lie group endowed with an invariant quaternionic K/ihler structure (called an Alekseev~kii apace) and of non-positive curvature is symmetric. This implies that a n o n - s y m m e t r i c D a m e k - R i c c i space cannot be equipped with an invariant quaternionic Kiihler structure. A b o u t the classification of the Alekseevskii spaces, see [Ale], [Cor] and [dWVP].

4.2

Spectral properties of the Jacobi operator

In this section we compute the eigenvalues and the corresponding eigenspaces of the Jacobi operators of an a r b i t r a r y Damek-Rieci space S at the identity e. For a p a r t i a l result see also [Sza2]. Theorem

Let V + Y + s A

be a unit vector in 2.

(i) V = 0, Y = 0. The eigenvaluea and eigenspaces of R,a are 0 -1/4 -I

(ii)

V

=

O, ~ =

O.

, a; ,

D ;

,

a.

The eigenvalues and eigenspaces of Ry are 0

, IRY;

-1/4

, ~;

-1

, Y•

(iii) Y = O, s = O. The eigenvalues and eigenspaces of R v are O,

IRV;

-1/4

, (kerad(V)flV •174174

-1

, kerad(V) • .

(iv) V = O, Y 7~ O, s 7~ O. The eigenvalues and eigenapacea of n y + , a are 0

, IR(Y+sA)

-1/4

, D;

-1

, (a| 96

;

N(Y+sA)

a

.

(v) V ~ 0, Y = 0, s ~ 0. The eigenvalues and eigenspaces of Rv+,A are 0

~(V+sA)

,

;

, ( k e r a d ( V ) | a) A ( V + s A ) • ~ { I v l ~ x

-1/4

-1,

~JxV I x 9 3) ;

-

{sX+JxVIX 9

(vi) v # o # Y. we decompose ~ orthogonaUy into ~=S4GP~)q

,

where

span{V, JyV, Y , A } ,

~4 : = p :=

kerad(V) Akerad(JvV) ,

q :=

span{Y • ,Yy~V, J r ~ J y V } 9

The spaces ~4, P and q are invariant under Rv+Y+,A and we have (1) the eigenvalues and eigenspaces of Rv+Y+,a[~4 are 0 , lR(V+Y+sA); -1/4 -I

, {(as+fl[Y[2)V+(fls-a)JrV-fl[V]2Y-a[V[2A

, ]R(JrV + s Y -

Ja, fl E lR} ',

IYI~A) ;

(2) (if p # {0}) Rv+y+,A[p has only one eigenvalue, namely - 1 / 4 ; (3) (if q ~ {0}) We put I{ := Kv.r and decompose Y• orthogonally into Y•

Lo|174

,

where Lj : = k e r ( K 2 - # j i d r ~ ) ,

(j=O,...,k)

and O>_#o> pl > . . . >

#k >_-1

are the distinct eigenvalues of K 2. It can easily be seen that X E Lj ~

K X E L,

(j=0,...,k),

whence d i m L j is even provided that #j ~ O. We now define qj

:=

qk : =

span{L i,JLJV,JLjJYV} , j = 0 , . . . , k ,

#k ~ - 1

span{Lk,JLkV} , i f # k = - l .

Then q = q0 | -. - G qk , dim qj

~

0(mod3) { 0(mod4) 0(mod6)

, i f #j = 0 , i f #~ = - 1 , otherwise

and each space qj is invariant under Rv+r+,A. Further, we have 97

,

( A ) (if j = k and #k = - 1 ) the eigenvalues and eigenspaces of Rv+r+,A]qk a r e

-1/4 -1

, {IVI2X+Jx(JvV-sV)

I XCLk}

, {(IVI 2 - 1)X + J x ( J v V -

;

sV) I X ELk} 9

( B ) (otherwise) Rv+v+,a]qi has two or three distinct eigenvalues ~,, i = 1,2,3, and 3 1 - 1 < ~ < - ~