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 <_ 5 is a commutative space if and only if it is a naturally reductive Riemannian homogeneous space (see the theorems in 2.1). This result cannot be extended to higher dimensions. For example, the Riemannian homogeneous space SU(3)/T, where T is a maximal torus in SU(3), provides an example of a six-dimensional naturally reductive Riemannian homogeneous space which is not commutative [Jim1]. In [Jim2] J.A. Jim6nez also provides examples of non-commutative naturally reductive Riemannian homogeneous spaces for any odd dimension greater or equal than five. Other examples of this kind are provided by the Stiefel manifolds of orthonormal two-frames in lit ", n > 30 [Jim3]. On the other hand, the six-dimensional generalized Heisenberg group with two-dimensional center is a commutative space which is in no way naturally reductive (see Theorems 1 and 3 in 3.2). We do not know whether there are commutative spaces which are not Riemannian g.o. spaces. See also [Hel2] for further results and references about commutative spaces.
2.5
Probabilistic
commutative
spaces
A probabilistic commutative space is a Riemannian manifold M such that L, Lt = L~L, holds locally for all sufficiently small s, t C JR+. Here, the operator L is the second mean value operator on M defined by L , f ( p ) := - -
0.)n_ 1
n-l(1)
f(expp(t())d~ ,
where f E C ~ p E M, n = d i m M , S"-1(1) is the unit sphere in TpM, w,-1 is the volume of S"-1(1), expp is the exponential map of M at p, and d~ is the volume 10
element of Sn-l(1). This concept originated from a work by P.H. Roberts and H.D. Ursell [RoUr] on compact Riemannian manifolds on which any two random steps commute. The general case has then been treated by 0. Kowalski and F. Priifer [Kow2], [KoPr], [Pru]. P r o p o s i t i o n 1 [KoPr] [Kow2] An analytic Riemannian manifold M is probabilistic commutative if and only if all Euclidean Laplacians of order 2k (k E IN) commute. The Euclidean Laplacians are defined as follows (see for example [GrWi]). Let p be a point in an analytic Riemannian manifold M and x l , . . . , x, normal coordinates on M centered at p. Define a local differential operator ~ by n
02f
and put
~x(')f(P) := ( s
f(P) 9
Then/~(k) is a global analytic differential operator on Mr, called the Euclidean LapIacian of order 2k. Each /k (k) is a formally self-adjoint operator on the space of all C~176 on M with compact support. Proposition 1 immediately implies P r o p o s i t i o n 2 Every commutative space is also probabilistic commutative. The converse is not true, as we shall see in 4.5. Classifications of probabilistic commutative spaces are known only for dimension three. T h e o r e m [Kow3] A three-dimensional simply connected complete Riemannian manifold is a probabilistic commutative space if and only if it is a naturally reductive Riemannian homogeneous space (see Theorem 1 in 2.1).
2.6
Harmonic spaces
For a survey on these spaces containing results up to 1982 see [Van1], where the references for the following equivalent definitions or characterizations of harmonic spaces can be found. In particular, we refer to [Besl] and [RuWaWi]. P r o p o s i t i o n 1 A Riemannian manifold (M, g) is a harmonic space if and only if one the following equivalent statements holds:
(i) at each point m E M there exists a normal neighborhood of M on which the Laplace equation Au = 0 admits a real non-constant solution depending only upon the distance r to m and being analytic for r ~ O; (ii) for each point m E M the volume density function/9 = (detgq) 1/~ On normal coordinates centered at m ) is a radial function; 11
(iii) for each m E M the function Aa.~ is a function off'm, where a.~(p) := r~(p)/2 and r(p) is the distance from m to p; (iv) every sufficiently small geodesic sphere in M has constant mean curvature; (v) every sufficiently small geodesic sphere in M (where dim M > 2) has constant scalar curvature; (vi) M r f ( m ) = f ( m ) for all m E M, all sufficiently small r E JR+, and all harmonic functions f defined on a neighborhood containing the geodesic ball B,~(r); (vii) M , f ( m ) = L J ( m ) for all m E M, all sufficiently small r E JR+, and all harmonic functions f defined on a neighborhood containing the geodesic ball
Bin(r). Here, M r f ( m ) denotes the first mean value operator on M defined by 1 M r f ( m ) - volGm(r) fa .(r) f d o , where Gin(r) = expm(S"-l(r)) is the geodesic sphere of radius r and with center m, and do denotes the Riemannian volume element of G.,(r). Using results in [Kow2], [KoPr] or [Sza2] one can prove that P r o p o s i t i o n 2 Every harmonic space is a probabilistic commutative space. We shall see later that the word probabilistic in this statement cannot be dropped. In fact, every non-symmetric Damek-Ricci space is a harmonic space which is not commutative (see Corollary 2 in 4.5). Using characterization (iv) it is easy to see that each two-point homogeneous space is harmonic. For a long time a conjecture on harmonic spaces stated that every harmonic space is locally isometric to a two-point homogeneous space. This conjecture was known to be true for d i m M < 4, or if the universal covering space of M is compact, or if M is compact and has non-negative scalar curvature (see [Szal], [Sza2] for details). Moreover, in [Cao] it is proved that a compact harmonic Kiihler manifold of negative sectional curvature is isometric to a compact quotient of complex hyperbolic space, CH"/F, up to a constant scalar factor. A recent and more general result states that the Riemannian universal covering space 2~/of a compact Riemannian manifold M, such that M has strictly negative sectional curvature and is asymptotically harmonic (that is, all its horospheres have constant mean curvature), is a rank one symmetric space [BeCoGa]. Only recently E. Damek and F. Ricci [DaRil] found the examples of non-symmetric harmonic spaces which are the topic of Chapter 4. We also want to point out that the Riemannian product of two probabilistic commutative spaces is still commutative, whereas the analogue for harmonic spaces does not hold. So the class of probabilistic commutative spaces is strictly bigger than that formed by the harmonic spaces. Finally we mention that cvery harmonic space is an Einstein space. 12
2.7
D'Atri spaces
A Riemannian manifold is said to be a D'Atri space [VaWi2] if its local geodesic symmetries are volume-preserving (up to sign). Such spaces have been studied first by J.E. D'Atri and H.K. Nickerson [DaNil]. Nice characterizations of such spaces in terms of the mean curvature h,~(p) of geodesic spheres G~(r) with center m and radius r, p E G,~(r), are given by P r o p o s i t i o n 1 [Dat], [KoVa6], [VaWil] For any Riemannian manifold M the following statements are equivalent: (i) M is a D'Atri space; 5i) kin(p) = hp(m) for all m , p E M sufficiently close; 5ii) h,,(expm(r~) ) = hm(exp,~(-r~)) for all m E M , all unit vectors ~ E T,~M and r E JR+ sufficiently small; (iv) h~xp=(r~)(m) = hexp~(_~r r E JR+ sufficiently small.
for all m E M , all unit vectors ~ E TraM and
Classifications have been obtained only for dimensions less or equal than three. For dimension two the classification follows immediately from Proposition 5 below, and for dimension three we have T h e o r e m [Kow3] A three-dimensional simply connected complete Riemannian manifold is a D'Atri space if and only if it is a naturally reductive Riemannian homogeneous space (see Theorem 1 in 2.1). An open problem is whether any D'Atri space is locally homogeneous. From Proposition 1 in 2.6 and the above proposition it follows that P r o p o s i t i o n 2 Every harmonic Riemannian manifold is a D'Atri space. Further, we have P r o p o s i t i o n 3 [KoVa4] Every Riemannian g.o. space, and in particular every naturally reductive Riemannian homogeneous space, is a D'Atri space. In the special case of naturally reductive Riemannian homogeneous spaces this result was proved by J.E. D'Atri and H.K. Nickerson in [Dat] and [DaNi2]. Next, we have P r o p o s i t i o n 4 [KoPr] Every probabilistic commutative'space, and in particular every commutative space, is a D'Atri space. The special case of commutative spaces has been treated by O. Kowalski and the third author in [KoVa2]. Finally, we mention P r o p o s i t i o n 5 Every D'Atri space is analytic in normal coordinates and has constant scalar curvature. 13
In fact, the statement of Proposition 5 holds for the more general class of Riemannian manifolds whose Ricci tensor is a Killing tensor (that is, the cyclic sum over all entries in the covariant derivate of the Ricci tensor vanishes); see [BeVal] for more details. Of particular interest is the situation in dimension four. Partial classifications have been obtained by J.T. Cho, K. Sekigawa and the third author [SeVal], [SeVa2], [ChSeVa]; but up to now there is no complete classification. See also [Bes2], [Van2], [Will, and [SoPrVa] for further references.
2.8
Z- a n d N - s p a c e s
The concept of r and g~-spaces is due to the first and third author. A Riemannian manifold M is called a t-space if for every geodesic 7 in M the eigenvalues of the associated Jacobi operator field R~ := R(.,~)~ are constant; and M is called a gl-space if for any geodesic 7 in M the associated Jacobi operator field R~ is diagonalizable by a parallel orthonormal frame field along 7. Both classes generalize Riemannian symmetric spaces, for a Riemannian manifold is locally symmetric if and only if it is both a t-space and a ~3-space [BeVal]. A two-dimensional Riemannian manifold is a t-space if and only if it is of constant curvature. Every two-dimensional Riemannian manifold is a gS-space. For dimension three we have T h e o r e m 1 [BeVal] A three-dimensional complete and sirnply connected Riemannian manifold is a t-space if and only if it is a naturally reductive Riemannian homogeneous space (see Theorem 1 in 2.t). If M is a non-complete or a non-simply connected E-space, it is locally isometric to a complete and simply connected one. T h e o r e m 2 [BeVal] Let M be a three-dimensional ~3-space of class C ~. Then M is almost everywhere (namely on the set where the number of distinct eigenvalues of the Ricci tensor of M is locally constant; this is an open and dense subset of M ) locally isometric to one of the following spaces: (i) a space of constant Riemannian sectional curvature; (it) a warped product of the form M1 x! M2, where M1 is a one-dimensional Riemannian manifold, M2 is a two-dimensional Riemannian manifold, and f is a positive function on M1; (iii) a warped product of the form M2 x ! M1, where M1 is a one-dimensional Riemannish manifold, M2 is a Liouville surface, and f is given by f2(x, y) = lip(x). r
,
where the functions ~ and r come from a (local) Liouville form
(~(x) + r
~ + dy ~)
of the Riemannian metric of M2; 14
(iv) a three-dimensional Riemannian manifold with Riemannian metric of the form ds
= G,
s Fl(xl)lxl
- x ll l -
x ldx ,
where ~ denotes the cyclic sum and F1, F2, F~ are positive functions. Every real analytic Riemannian manifold of one of these types is a ~-space. For higher dimensions we do not know of any classification. Concerning examples of E-spaces we have P r o p o s i t i o n 1 [BeVal] Every Riemannian g.o. space and every commutative space is a E-space. All our known examples of E-spaces up to now belong to these two classes. We shall see in 3.4 that the generalized Heisenberg groups provide a third class of E-spaces. All these examples are homogeneous manifolds and it is still unknown whether any E-space is in fact a locally homogeneous space - in contrast to the ~spaces where we have a lot of examples which are not locally homogeneous (see also [ S o d ] , [Boe2], [Cho]). There are the following characterizations of E- and ~3-spaces. P r o p o s i t i o n 2 [BeVal], [BeVa2], [BePrVal] Let ( M , g ) be an n-dimensional Riemannian manifold. Then M is a E-space if and only if one of the following equivalent statements holds:
(i) for each p E M and each v E TpM there exists a (skew-symmetric) endomorphism Tv of TpM so that R'~ := (~7~R)(.,v)v = [Ro,T,] with R~ := R(.,v)v; (it) for each geodesic 7 in M there exists a skew-symmetric tensor field T~ along 7 such that R', = [R,, T~]; (iii) for each v E T M and eigenvalue g of R, the eigenspace of R, with respect to g is mapped by R I, into its orthogonal complement; (iv) for each v E T M and eigenvalue g of Ro there exists a corresponding eigenvector w r 0 such that g(R;zo, w) = O; (v) for each m E M every local geodesic symmetry Sr~ of M at m preserves the eigenvalues of R~(r), where 7 is an arbitrary geodesic in M with 7(0) = m and r E JR+ is sufficiently small; (vi) for each p E M and v E TpM there exists a skew-symmetric endomorphism T, of TpM such that T_, = -To and the eigenvalues of R'~ - [R~, T~] are independent of v; (vii) for every geodesic 7 in M defined at 0 and every eigenvalue function c of the associated Ledger tensor C we have c'"(O) = 0 (note that always c(O) = 1, c'(O) = 0 and c"(0) = - 2 ~ / 3 with some suitable eigenvalue ~ of ILr(O), and the Ledger tensor C is defined by C(r) := rA(r), where A(r) denotes the shape operator at v(r) of the geodesic sphere centered at v(O)); 15
(viii) for every p E M , v E TpM, eigenvalue ~ of R~ and corresponding eigenvector W w e h a v e gs(Rs(~)w TM H , w v) = O, where gs is the Sasaki metric on T M , S the geodesic spray of M and H and v denote the horizontal and vertical lift with respect to the canonical bundle map T M ~ M , respectively; for every k E { 1 , . . . , n - 1} the polynomial Pk : T M ~ lR, v ~ trace R~ is a first integral of the geodesic flow of M ; 1~ + ... + P,-I is a first integral of the geodesic flow of M ; for every k C { 1 , . . . , n - 1} the symmetric tensor field Sk on M obtained by polarization of Pk is a Killing tensor. P r o p o s i t i o n 3 [BeVal] Let (M, g) be an n-dimensional real analytic Riemanntan manifold. Then the following statements are equivalent:
(i)
M is a ~3-space;
(ii)
R~ o R'~ = R', o R~ for all v E T M ;
(iii)
the basic Jacobi fields on M are of the form as in locally symmetric spaces, that is, they arise from multiplying appropriate parallel vector fields with particular solutions of scalar Jacobi equations (here, a basic Jacobi field Y along a geodesic is a Jacobi field with Y(t) = 0 or Y'(t) = 0 at some point);
(iv)
the principal curvature spaces of any family of (sufficiently small) geodesic spheres in M are invariant with respect to parallel translation along the radial geodesics emanating from the center of this family; all (sufficiently small) geodesic spheres in M are curvature-adapted, that is, the shape operator and normal Jacobi operator commute.
Note that the symmetric tensor field $1 in statement (xi) of Proposition 2 is just the Ricci tensor. So the Ricci tensor on any t-space is a Killing tensor and hence we have (see also the remark after Proposition 5 in 2.7) P r o p o s i t i o n 4 Every E-space is analytic in normal coordinates and has constant scalar curvature.
2.9
r
The ~-spaces form a subclass of the t-spaces and were introduced in [BePrVal]. There are the following defining or characterizing properties of these spaces. P r o p o s i t i o n 1 [BePrVal] A Riemannian manifold (M,g) is a to-space if and only if one of the following equivalent statements holds: (i) for every geodesic 7 in M there exists a parallel skew-symmetric tensor field T~ along ~ such that R; = [ ~ , Td; 16
(ii) for every geodesic "y in M defined at 0 there exists a skew-symmetric endomorphism T of T~(o)M such that P
(t) =
e -'T o P
(O) o e 'T ,
where different tangent spaces along "7 are identified via parallel translation; (iii) for every geodesic ~/ in M defined at 0 there exists a skew-symmetric endomorphism T of T~(o)M such that all the higher order Jacobi operators R(~~), k 0, 1, 2 , . . . , satisfy --
= e-'* o
o e 'T ,
where different tangent spaces along "y are identified via parallel translation. Characterization (ii) says that co-spaces are precisely those Riemannian manifolds for which the Jacobi operator R~ along any geodesic -'f is obtained by conjugation of the Jacobi o p e r a t o r / ~ ( 0 ) at a single point with a suitable one-parameter subgroup of the orthogonal group. The space is locally symmetric if and only if this subgroup can always be chosen as the trivial one. As concerns examples of co-spaces we have P r o p o s i t i o n 2 [BePrVal] Every Riemannian g.o. space is a co-space. This immediately implies T h e o r e m A three-dimensional simply connected complete Riemannian manifold is a co-space if and only if it is a naturally reductive Riemannian homogeneous space (see Theorem 1 in 2.1). We know that commutative spaces are E-spaces, but we do not know whether they are also C,0-spaces. We also do not have a single example of a E-space which is not a C0-space. Another open problem is whether any E-space is a D'Atri space. For the subclass of all co-spaces this is known to be true. P r o p o s i t i o n 3 [BePrVal] Every Co-space is a D'Atri space. According to J.E. D'Atri and H.K. Nickerson [DaNi2] we say that a Riemannian manifold M has a special curvature tensor if it admits a tensor field T of type (1,2) satisfying
( V x R ) ( Y , X ) X = T x R ( Y , X ) X - R(TxY, X ) X
and ( V x T ) x = 0 .
It follows from Proposition l(i) that every space with a special curvature tensor is a Co-space.
2.10
:
-spaces
Consider the following geometric configuration. Let 7 be a geodesic in a Riemannian manifold M parametrized by arc length and defined on an open interval containing 17
0. We put m := 7(0) and choose r e IR+ such that p := 7(r) and q := 7 ( - r ) are defined. If r is sufficiently small, the geodesic spheres Gp(r) and Gq(r) of radius r around p and q, respectively, are smoothly embedded hypersurfaces in M. Further, rn lies on both geodesic spheres and Gp(r) and Gq(r) are tangent to each other at m. M is called a '~2E-space if for any such configuration the principal curvatures (counted with multiplicities) of Gp(r) and G,(r) are the same. Classifications of TE-spaces in low dimensions follow from the following result. T h e o r e m [BeVa3] Let M be a simply connected complete Riemannian manifold with dimension <_ 3. Then M is a qiE-space if and only if it is a naturally reductive Riemannian homogeneous space (see Theorem 1 in 2.1). (Note that the assumption of analyticity in [BeVa3] is unnecessary since every TE-spaee is a D'Atri space (see below) and hence real analytic in normal coordinates.) Examples of ~2E-spaces arise from the following two results. P r o p o s i t i o n 1 [BePrVal] Every weakly symmetric space is a ~E-space. P r o p o s i t i o n 2 [BePrVal] Every Riemannian manifold which is locally homothetic to a Sasakian space form is a ~E-space. It is an open problem whether naturally reductive Riemannian homogeneous spaces, Riemannian g.o. spaces, E0-spaces, E-spaces or commutative spaces are ~Espaces. From the classifications we just have the partial result [BePrVal] that every simply connected naturally reductive Riemannian homogeneous space, Riemannian g.o. space or commutative space of dimension < 4 is a ~:E-space. Consequences of the ~E-property are the E- and the D'Atri-property. P r o p o s i t i o n 3 [BePrVal] Every ~E-space is a E-space. P r o p o s i t i o n 4 [BeVa3] Every ~r
is a D'Atri space.
The motivation for studying ~r emerged from a result by the third author and T.J. Willmore [VaWi2] stating that a Riemannian manifold is locally symmetric if and only if for any configuration of geodesic spheres (as described in the beginning) the shape operators of Gp(r) and Gq(r) coincide at m. The last condition means that the two shape operators have the same eigenvalues and are simultaneously diagonalizable. It is natural to split up here the two conditions on the shape operators, the first one leading to the ~r and the second one to the so-called ~g3-spaces. In [BeVa3] it was proved that in fact the classes gl and ~:gl coincide in the real analytic case.
2.11
~-spaces
A Riemannian manifold (M,g) is said to be an ~r if small geodesic spheres in M have the same principal curvatures (counted with multiplicities) at antipodal points. Concerning Sr we have similar results as for ~r
18
T h e o r e m [BePrVal] Let M be a simply connected complete Riemannian manifold of dimension <_ 3. Then M is an ~r if and only if it is a naturally reductive Riemannian homogeneous space (see Theorem 1 in 2.1). P r o p o s i t i o n 1 [BePrVal] [BeVa5] Every weakly symmetric space is an | P r o p o s i t i o n 2 [BePrVal] Every Riemannian manifold which is locally homothetic to a $asakian space form is an 6E-space. P r o p o s i t i o n 3 [BePrVal] Every ~E-space is a E-space. P r o p o s i t i o n 4 [BePrVal] Every ~E-space is a D'Atri space. It is not yet clear whether the classes of ~E- and | partial answer is given by
coincide or not. A
P r o p o s i t i o n 5 [BePrVal] Let M be a ~-space or a commutative space. Then M is a ~E-space if and only if it is an ~r
2.12
Osserman
spaces
The notion of an Osserman space is due to the O s s e r m a n c o n j e c t u r e lOss] Every Riemannian manifold with globally constant eigenvalues for the Yacobi operators is locally isometric to a two-point homogeneous space. Note that in the conjecture only Jacobi operators with respect to unit tangent vectors are considered. It is easy to see that each two-point homogeneous space has the property stated in the conjecture. A Riemannian manifold with globally constant eigenvalues for the Jacobi operators is called a globally Osserman space [GiSwVa]. Up to now the conjecture is known to be true in the following cases: T h e o r e m [Chil], [Chi2], [Chi3] Let M be an n-dimensional globally Osserman space. Then M is locally isometric to a two-point homogeneous space in the following eases:
(i) n is odd; (ii) n =_ 2(rood 4); ( i i i ) n = 4;
(iv) n = 4k, k > 1, and M is a simply connected compact quaternionic Kdhler manifold with vanishing second Betti number; (v) M is a Kdhler manifold of non-negative or non-positive sectional curvature; (vi) M satisfies the •llowing axioms:
19
(1) R. has precisely two different constant eigenvalues independent of v E S M (the unit sphere bundle in TM); (2) let )~ and # be the two eigenvalues and for v E S M denote by E,(v) the span ofv and the eigenspace of R~ with eigenvalue #; then E , ( w ) : E~(v) whenever w E E,(v). One might also consider a local version, namely that for each p C M the eigenvalues of the Jacobi operator are independent of unit vectors v E TpM (but may vary with p E M). Such spaces are called pointwise Osserman spaces. Surprisingly there are pointwise Osserman spaces which are not globally Osserman spaces. For example, four-dimensional self-dual Einstein manifolds are always pointwise Osserman spaces, but not locally isometric to a two-point homogeneous space in general. See [GiSwVa] for more details in the local situation.
20
Chapter 3 Generalized Heisenberg groups In Section 1 we provide basic material about generalized Heisenberg groups, most of which has been presented previously in [Kapl], [Nap2], [Kap3] and [Riel]. In Section 2 we summarize the known results, which are due to [Kap3], [Ric] and [aie2], about the question which of the generalized Heisenberg groups belong to some of the classes of Riemannian manifolds discussed in Chapter 2. We also add a new result concerning weakly symmetric spaces. In Section 3 we determine the spectrum of the Jacobi operator and the corresponding eigenspaces. From this we conclude that none of the generalized Heisenberg groups is a pointwise Osserman space. The explicit knowledge of these data enables us to provide in Section 4 two alternative proofs for the fact that every generalized Heisenberg group is a C-space. Consequently, none of the generalized Heisenberg groups is a ~-space. In Section 5 we show that every generalized Heisenberg group is even a C0-space. The explicit expression of the tensor field T there is perhaps the crucial point in these notes, as without it the results in the subsequent sections would hardly be obtained. In Section 6 we draw some further conclusions from the results which have been obtained so far. In Section 7 we determine all Jacobi fields in generalized Heisenberg groups which vanish at the identity. The method introduced here for doing this generalizes the one used by I. Chavel [Cha] for normal homogeneous Riemannian manifolds and by W. Ziller [Zill] for naturally reductive Riemannian homogeneous spaces. The subsequent sections then contain applications of the explicit knowledge of these Jacobi fields. In Section 8 we compute some conjugate points in generalized Heisenberg groups. For the Heisenberg groups we obtain a complete classification of the conjugate points. The shape operator of geodesic spheres in generalized Heisenberg groups is the topic of Section 9. From the structure of the Jacobi fields we conclude that the principal curvatures of small geodesic spheres in generalized Heisenberg groups are the same at antipodal points. This means that every generalized Heisenberg group is an 6C-space. Combining this with a result of Section 5 then implies that every generalized Heisenberg group is also a fie-space. In the final Section 10 we prove that in generalized Heisenberg groups the metric tensor with respect to normal coordinates has the same eigenvalues at antipodal points (with respect to the center of the normal coordinates). This answers the question whether Riemannian g.o. spaces are characterized by this property of the metric tensor in the negative.
21
3.1 3.1.1
Generalized Heisenberg algebras and groups Definition
Let , and 3 be real vector spaces of dimensions n, m E IN, respectively, a n d fl : 0 x 0 ~ 3 a skew-symmetric bilinear map. We endow the direct sum u = 0 9 3 with an inner product <., .> such that 0 and 3 are perpendicular a n d define a n ]R-algebra homomorphism J : 3 - - ' E n d ( 0 ) , Z ~ Jz
by V U, V E o, Z E 3 : <JzU, V > = ~(U, V), Z > . We define a Lie algebra structure on n by
V V , V E o , X , Y E 3: [ V + X , V + V ] := ~ ( U , V ) . The Lie algebra n is said to be a generalized Heisenberg algebra if
V Z E 3 : J~ = - < Z , Z > i d o
9
The attached simply connected Lie group N, endowed with the induced left-invariant R i e m a n n i a n metric g, is called a generalized Heisenberg group.
3.1.2
Classification
End(D) , Z ~-* Jz be an ]R-algebra homomorphism satisfying J~ = - < Z , Z>idu for all Z E 3- T h e n J can be extended to an ]R-algebra h o m o m o r p h i s m jc~ from the Clifford algebra Cl(3, q) of 3 with respect to the quadratic form q(Z) := - < Z , Z > into End(0). Thus J induces a representation jcr of the Clifford algebra Cl(3, q) in 0, that is, 0 is a Clifford module over Cl(3 , q). Conversely, suppose that j c s is a representation of a Clifford algebra Cl(3, q) in 0, where 3 is a n m - d i m e n s i o n a l vector space a n d q is a negative definite quadratic form on 3. T h e restriction of jcz to 3 gives an ]R-algebra homomorphism J : 3 ---* End(0) satisfying J~ = q(Z)ido for Let J : 3 ~
all Z E 3- Equip 3 with the inner product obtained by polarization of - q a n d extend this to an inner product <., .> on n := 0 9 3 so that 0 and ~ are perpendicular and Jz acts as an orthogonal map for unit vectors Z E ~. T h e n define a skew-symmetric bilinear map/3 : 0 x 0 ~ 3 by <;3(U, V), Z > := <JzU, V > for all U, V E 0 a n d Z E 3. In this way we o b t a i n a generalized Heisenberg algebra n. Thus the classification of generalized Heisenberg algebras and the classification of representations of Clifford algebras of real vector spaces equipped with negative definite quadratic forms are in a one-to-one correspondence up to some equivalence which will be explained below. The latter classification is known (see for example [AtBoSh]) a n d as follows: a) If m ~ 3(mod 4), then there exists (up to equivalence) precisely one irreducible Clifford module 0 over Cl(3, q). Every Clifford module 0 over Cl(3, q) is isomorphic to the k-fold direct sum of 0, that is,
0 ~ Ok0. 22
b) If m _= 3(mod 4), then there exist (up to equivalence) precisely two nonequivalent irreducible Clifford modules ~1, ~2 over Cl(~, q). The modules ~1 and ~ have the same dimension and every Clifford module ~ over Cl(3 , q) is isomorphic to
for some non-negative integers kl, k~. The dimension no of ~ or ~1, ~ can be taken from the following table:
mlsp 8p+,lsp+218p+318p+418p+518p+0 8p+7 no
2 4p
2 4p+1
2 4p+~
2 4p+2
2 4p+a
2 4p+3
2 4p+a
2 4p+3
Thus we may conclude that for each m 6 IN there exist an infinite number of non-isomorphic generalized Heisenberg algebras with dim 3 = m. For the first numbers rn 6 { 1 , . . . , 11} we get the following dimensions for N: m
1 2 3 4 5 6 7 8 9 10 11
dim N 3 5 7 9 11 13 6 10 14 18 22 26 7 11 15 19 23 27 12 20 28 36 44 52 13 21 29 37 45 53 14 22 30 38 46 54 15 23 31 39 47 55 24 40 56 72 88 104 41 73 105 137 169 201 74 138 202 266 330 394 75 139 203 267 331 395
15 30 31 60 61 62 63 120 233 458 459
17 19 21 34 38 42 35 39 43 68 76 84 69 77 85 70 78 86 71 79 87 136 152 168 265 297 329 522 586 650 523 587 651
~ 1 7 6
~
o
9 1 7 6
If m = 3(mod 4), any two pairs (k,, k~) and (kl, k2) of non-negativeintegers with kl + k2 = ~:1 + ]% yield generalized Heisenberg algebras ,(kl, k~) and n(kl, k3) of the same dimension (kl + k2)no + m. These are isomorphic if and only if (k~, k2) 6 {(k~, k~), (ks, k~)}. In terms of the groups this means that N(k~, k2) and N(k~, ~:~) are isometric if and only if (k~, k~) 6 {(k~, k2), (k2, k~)}. Thus, if for example m -- 3, there exist two non-isometric eleven-dimensional generalized Heisenberg groups with three-dimensional center.
3.1.3
Algebraic
features
Let a be a generalized Heisenberg algebra. From the construction follows immediately that ~ is the center of ,. Further, since [n,n] = ~ and [3,n] = 0 we see that n is two-step nilpotent. Thus P r o p o s i t i o n Every generalized Heisenberg algebra is a two-step nilpotent Lie algebra with center 3. 23
It is well-known that every non-singular skew-symmetric bilinear m a p / 3 : ~ x --* IR has a matrix representation
(o,
with respect to a suitable basis of 0. So, if the dimension of the center is one, n is isomorphic to a Heisenberg algebra in the classical sense. Explicitly, if n = 2k for some k E IN, consider the Lie algebra of all matrices of the form 0
X I
. . .
X k
Z
Yl
:
0
(x, y ~ Yd, z C IR)
yk 0
with Lie bracket given by the usual commutator of matrices. This Lie algebra is the (n + 1)-dimensional Heisenberg algebra. From now on U, V, W will always denote vectors in v and X, Y, Z vectors in 3. Vectors in n will always be written in the form U + X. We define
1u + xl
:=,/
.
Recall that we have the defining relations J1" = -IXl2idv
and < J x g , V > = <[u, v ] , x >
,
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] - 2
JxV>X,
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 = <[V, U], Z > = <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
:~
<[Vi, Vii, Yk>
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<[u,Y,l,Y~> 1
= O~(x)- ~t<&v,,u> = , ) ~ ( 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)
=
<(o)
=
z(o.
o
k
k
0
1
Similarily, Y,(p)
= Lp.,Yi(e) =
Lm
t ~ expn(tY~))(O
=
0 ~ ( t ~ LAe~p.(tY,)))(O)
=
3,(0) 27
with
fl,(t) = expn(U + X + tY,) . Here we have (vk o Z,)(t) = ~ ( u )
, (v~ o Z,)(t) = O d x )
+ ~,~t ,
and hence
(vk o ~,)'(0) = 0 , (u, o ~,)'(0) = 6,~. This implies
Y,(;) = ~,(0) = ~o( ; ) . Thus the assertion is proved. []
3.1.6
Levi Civita connection
Let V be the Levi Civita connection of a generalized Heisenberg group (N, g). The connection V is completely determined by its values on the left-invariant vector fields in n. Since 9 is left-invariant, we may eompute V by
29(vv+y(v + x ) , w + z ) = g([v, u], z ) - 9([u, w], Y) + g([w, v], x ) and obtain
Vv+,,(u + x ) = - ~ - s x v - ! j , , u _ 2[u, v ] .
2
3.1.7
2
Curvature
Let R be the Pdemannian curvature tensor, Q and p the Ricci tensor of type (1, 1) and (0, 2), respectively, and v the scalar curvature of a generalized Heisenberg group (N,g). All these objects are completely determined by its values on the tangent space T~N ~ n of N at the identity e. By a straightforward computation one gets
R(U + X, V + Y)(W + Z) l 1 W ~J[v.vclU+ ~J[u, wlV + -~J[u,v]
-
~JxJY
+ 1-<X'2 Y > W 1 +~[V, JxW] - ~[U, JvW] - ~[U, JzV] + ~Z, Q(U + x)
-
(Vv+vQ)(U + X) T
2~U + 24 X ' 2m + n(JxV - [U, V])
~--
8
--
1 4
--WWt
.
28
Further, if U + X, V + Y E n are orthonormal and a is the span of U + X and V + Y, then the sectional curvature K(a) of N with respect to a is 3
K(~) = 4(IV121Yl= + IWl21Xl~) + < X , Y > < 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
"
=
<[9, J x J ~ 9 ] , Z > = < 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 : = ~<[V,Y],V> = O,
1 V X E ~ : = - ~ < 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 = 0 for U + X being one of the eigenvectors V + Y, JyV and -[Y[2V + ]VI2Y. On p the assertion holds obviously since Rv+y ]P = O. t
Finally, to verify the assertion on q, let X be a unit vector in some Lr Then
R'v+rX = - (~]V]2 + ]YI~) J x V , Rv +x,lx V = 3]VJ~IYIJKxV - ~]V[2Jx JrV - ]VI 2 (1]V]2 + j y ] 2 ) X , !
t
Rv+y JKxV
= 3,VJ~JyI#yJxV-~,VJ~JKxJyV - IV[ 2 ( l j v [ 2 +
R'v +xJx Jr V --
Ivl'lrl'Jx v -
IYI~) K X ,
lvl lrl J,,x Jy v
-,vl lrl ( lvr + lY, )
.
First suppose that j = k and #k = - 1 . From the preceding formulae we see that
Rv +r(span{X, JKxV} ) C s p a n { K X , J x Y } _l_span{X, JKxV} , from which the assertion follows. Next, if j = 0 and #0 = 0, then
Rv+ydxV _k J x V 43
and
Rv+y(span{X, Jx JyV} ) C span{ JxV} _L span{X, Jx JyV} , which implies again the assertion. Finally, in the remaining cases we have R v + g ( s p a n { X , Jgx V, Jx Jv Y } ) C
s p a n { K X , JxV, JKxJrV}
_L span{X, JKxV, JxJyV} , which also implies the required result. [] According to Proposition 3.1.7, none of the generalized Heisenberg groups is a locally s y m m e t r i c space. As a C-space, which is in addition also a N-space, is necessarily locally symmetric (see 2.8), we get
C o r o l l a r y None of the generalized Heisenberg groups is a ~3-space.
3.5
Rotation
of the eigenspaces
along geodesics
In this section we concentrate on the behavior of the eigenspaces of the Jacobi o p e r a t o r along geodesics in generalized Heisenberg groups. T h e tensor field T~ constructed in this section will be of fundamental i m p o r t a n c e for the c o m p u t a t i o n of Jacobi fields in 3.7. Let V + Y E n be a unit vector and q := span{Y • Jy~.V, Jy•
.
Thus q=
kerad(V) Y•
@3
, if Y = 0 , if V = 0 ,
a n d q is as in T h e o r e m 3.3 if V 76 0 # Y. Further, kerad(V) o@IRY na|
q1 =
, ifY=O , ifV=O , ifV760CY
with n3 a n d p as in T h e o r e m 3.3. Next, let 3' : ]R ---* N be the geodesic in N with 3'(0) = e a n d ;/(0) = Y + Y. Then
+(t)=
V(t)+Y , if Y 7 6 0 V ,ifY=O
with
- 1
V(t) = cos(]Y[t)Y + I@l sin(]YJt)JvY In the following c o m p u t a t i o n s we will use the equation d
~/(t ~ v(t)) = j~v(t). 44
o
If not stated otherwise, different tangent spaces along 7 will be identified by left translation. We now define a tensor field T~ along 7 by
{
T,(t)(U + X)
3jvU z
+
z
1jxV(t)+-~[U,V(t)] 1
-~J~U+~SxV(t)
+
9
, if U + X
~[v, v(t)] , if u + x 9 ,•
L e m m a 1 T~ is a skew-symmetric parallel tensor field along 7.
Proof. If Y = O, then T,(t)(U + X ) = ~JxV + [U,V]. T7 is clearly skew-symmetric and
Tr
+ X) = Vv(T,(t)(V + X)) - T,(t)Vv(V + X) = ~ V v J x V + ~Vv[U, V]+ ~T,(t)JxV+ ~T,(t)[U, V] --~[dxV'V]-~J[u'v]V+i =
[JxV'V] + l
'
O.
If V = 0, then
1
T4t)(U + X) = -~J~U . Also here, T~ is skew-symmetric and Z~(t)(U + X)
=
V~(T4t)(U + X)) - T4t)Vv(U + X)
1
1U__O
---- - ~ U + 4 Now, let V # 0 # Y .
'
IfU+XE,•
1
X = {--y--~2<x,Y>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 <(t)(u + x)
= Vv(,)+r(T4t)(U + X)) - T~(t)Vv(,)+r(U + X ) =
- ~1v ~ ( , ) + r z y u +
~Vv(o+rJxV(t)
+ ~1V ~(,)+r[u,v(t)]
+lT~(t)JrU + ~T~(t)JxV(t) + ~T~(t)[U, V(t)] : 4J}U+~[JrU, V(t)]+~JxJrV(t)-~JyJxV(t)
- [JxV(O,v(o]+
~J~,v(,)~v(t) i
-~ J~U + ~[JrU, V(t)] - ~ JrJxV(t)
= ~Jx JrV(t)- ~JyJxV(t)+ ~[U, JrV(t)]+-~1 [JyU, V(t)] :
since
O,
1 JxJrV(t) - JrJxV(t) = ]-~-{<X, r>(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 - [yli<[JyU, Y(t)], Y>Y + 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 <(t)(v + x)
=
Vv(o+rZ(O(U + x ) - T.,(OVv(,)+~(U + x ) 1 1 Vv(,)+y YrU + ~Vv(,)+~YxV(t)+ ~v~(,)+~[u,v(t)]
~
+~T~(t)JrU + ~T~(t)JxV(t)~T~(t)[U, + v(t)] 46
-~[JxV(t), V(t)] + ~[U, JyV(t)]-~J[v,v(o]V(t)
v(O] + =
=
O~
for < X , Y > = 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~<[U,J.V(t)], 1 Y > 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) --:----
_
2 .. " /
-
21yI:
JyV(t)>Y
This yields
[P~, T~](t)(U + X) =
3JvV(t)-~[yl~JrU-1
,JrV(t)>y-3[VJ~JxV(t)
+ [Yl~JxV(t) + g V ( t ) -
IV[ JvV(t)>Y 8-~.U,
1 2 3 - g J Y [ JxV(t) + -~V(t) Jyv(
_.
-
~ y , , v ( t )
)>Y + 1
49
3iV[ 2 llyj2jy U 8 - - ~ < U , JrV(t)>Y + -
lvl2jxV(t )
1 = ~v(t) + ~Jyy(t) _ ~lYl2jxy(t)
21y] 2 Y =
!
R~(t)(V
+ X),
and this completes the proof. [] The combination of Lemmas 1 and 2 now provides T h e o r e m Every generalized Heisenberg group is a Eo-space. Proposition 1 in 2.9 shows that
R4t)
= e-'T'(~ o P+(0) o , T , < 0 ) ,
where different tangent spaces along 7 are identified via parallel translation. Thus the rotation of the eigenspaees of P~ is described by the one-parameter subgroup d T'(~ of the orthogonal group O(n). An analogous statement holds also for the higher order Jacobi operators R(,k) (see 2.9).
3.6
Some corollaries
We will now draw some conclusions from the results in the preceding section. First of all, according to Theorem 2 in 3.2, not every generalized Heisenberg group is a Riemannian g.o. space. On the other hand, according to Proposition 2 in 2.9, every Riemannian g.o. space is a r So Theorem 3.5 implies C o r o l l a r y 1 There exist Co-spaces which are not Riemannian g.o. spaces. Next, from Proposition 5 in 2.11 and Theorem 3.5 we may conclude: C o r o l l a r y 2 A generalized Heisenberg group is a ~E-space if and only if it is an | This corollary will be useful in Section 3.9. As any E0-space is also a E-space, Theorem 3.5 implies an alternative proof of C o r o l l a r y 3 Every generalized Heisenberg group is a E-space. According to Theorem 4 in 3.2, every generalized Heisenberg group is a D'Atri space. The proof of this, provided by A. Kaplan, uses harmonic analysis. From Proposition 3 in 2.9 we know that every E0-space is a D'Atri space. Thus we now have an alternative proof of C o r o l l a r y 4 Every generalized Heisenberg group is a D'Atri space. One of the open problems concerning e-spaces is the question whether they are locally homogeneous or not. An obvious way to tackle this problem is to combine all 50
the operators T, in the condition R'~ = [R,, T~], which characterizes C-spaces, and to see whether they provide a homogeneous structure (see 2.1). Now, consider the endomorphism Tv+r defined in 3.5. They always have the property Tv+v ( V + Y ) = O. If all the operators Tv+v would fit together to provide a homogeneous structure, it would by means of Proposition 2 in 2.1 necessarily be a naturally reductive one. But according to Theorem 1 in 3.2 this is possible only if the dimension of the center is one or three. Thus, we conclude that the endomorphisms Tv+v do in general not fit together to provide a homogeneous structure on a generalized Heisenberg group.
3.7
J a c o b i fields
In order to compute the Jacobi fields on generalized Heisenberg groups, we use the following general idea. L e m m a Let M be a Riemannian manifold with Levi Civita connection V and 7 : I ~ M a geodesic in M parametrized by arc length. We denote by 0 the standard unit tangent field on I. Suppose there exists a XTo-parallel skew-symmetric tensor field T7 along 7 such that the Jacobi operator P~ along 7 satisfies V o I ~ := R'~ = [P~, T,]. Then define a new covariant derivative Vo := Vo + T, , and put
/ ~ := R~ + T~ . Then R~, [~ and T~ are Vo-parallel along 7 and the ]acobi equation along ~ is V o V o B - 2T~VaB + [ ~ B = 0 .
Proof. For any vector field B along 7 we have (~:oR,)B
=
%R,B-
R,%B
=
V o I L B + T~R~B - P ~ V o B - P ~ T , B
=
(VoR, - [R,,T,])B
----- 0 ,
(VoT,)B
=
VoT, B -
T, V o B
=
VoT, B + T : B - T, V o B - T•B
=
(VoT,)B
~-
O,
Hence P~ and T~ are ~'o-parallel. This implies that a l s o / ~ = P~ + T~ is ~'o-parallel. Using the fact that T~ is ~7o-parallel we now compute for any Jacobi field B along 7: 0 =
VoVoB +R,B
=
V o V o B - T v V o B - VoT, B + T : B + P ~ B
=
V o V o B - 2T, V o B + [ L B 9
51
This proves the assertion. [] This l e m m a is of interest as soon as one wishes to c o m p u t e explicitly Jacobi fields in n o n - s y m m e t r i c R i e m a n n i a n manifolds. The advantage of introducing ~'o is t h a t with respect to this covariant derivative the Jacobi equation becomes a second order linear equation with constant coefficients. Of course, in general, a V0-parallel tensor field T~ as in the lemma does not exist. But, as follows from the very definition, on any C0-space (and hence on any generalized Heisenberg group) the Jacobi equation can be transformed in such a way. The m e t h o d s of solving the Jacobi equation on normal homogeneous Riemannian manifolds used by I. Chavel [Cha] or, more generally, on n a t u r a l l y reductive R i e m a n n i a n homogeneous spaces by W. Ziller [Zill], who use the canonical connection on these spaces, are a special case of the above procedure. We shall now a p p l y this m e t h o d to compute explicitly some p a r t i c u l a r Jacobi fields on generalized Heisenberg groups. T h e o r e m Let V + Y be a unit vector in n and 7 : IR --~ N the geodesic in N with 7(0) = e and ;y(O) = V + Y. For U + X 6 n we denote by Bu+x the Jacobi field along 7 with initial values Bu+x(O) = 0 and B~+x(O ) = V + X . We have
(i) Y = O. Then 1
B (t) = tU + t2[U, V], Bx(t)
=
(t-
12 ~ta) x + ~t J x V
for alI U 6 ~ and X 6 a. (ii) V = O. Then Bu(t)
=
sin(t)U + (1 - c o s ( t ) ) J v U ,
Bx(t)
=
tX
for alI U 6 ~ and X 6 a. (iii) Y ?~ 0 # Y. Then ~/(t) = Y ( t ) + Y with
v(t)
:= c o s ( I Y I t ) v +
1 sin(lYlt)jyy
and the span of V + Y, - ] Y I 2 V + IV[2Y, JyV, p := ker ad(V) M k e r a d ( J y V ) ,
YL Jy v, Jy.J,.V is n. Any two of these vectors or vector spaces are perpendicular to each other except J v • and J v • (unless Kv.v = 0). For U 6 p and X 6 Y• we then have Bv+v(t)
=
tV(t) + tY , 52
(Iw,-
B-IYI2V+lVpy(t)
,(,)
1 + ~-~-~ (1 - cos(]Ylt)) J , V ( t )
+'v'~ ( ~-~ ~ sin(IYlt)-
)
IVl~t Y ,
(1 - cos(iYIt) )V(t ) + ~y~ sin(IYIt)Jyy( t )
IYl2 +~-(cos(lYlt)1)Y, Bu(t)
=
Bx(t)
=
1
1 sin(IYIt)U + ~-~(1 - cos(IYIt))JvU ,
3 IVI~ k2]y] ( sin(IYIt)(cos(IYIt) - 2) +
1 ~-~y--~ + ]Yi ~t ) X
+ 2--~-~(co41Ytt)- 1)2KX 1
+ ~ - ~ c~ +~ B~xv(t)
-
1
- cos(lYlt))JxY(t )
sin(IYIt)(cos(iYlt) - 1)Jx JyV(t) ,
IV[~ cos(JYIt )( cos( lYJt ) - 1)x +~
IYI sin(IYIt)(1 -
cos(IYIt))KX
1 + F ~ sin(lY[t)(2 cos(lYJt) - 1)JxV(t) 1 +~--~(cos(IY]t) - 1)(2cos([Y[t) + 1 ) J x J y V ( t ) , B~j.v(t)
-
[V[~ sin([YIt )( cos( [Y]t ) - 1)X
JYI
+ ]V[~ cos([Y[t)(cos([Y[t) - 1 ) I ( X +(1 - cos(IYlt))(2 cos(lYIt) + 1)ZxV(t) 1 +~-~ sin(IYlt)(2 cos(lYlt) - 1 ) J x J v V ( t ) . Note that K X = O i f and only i f X C Lo , J x J y V ( t ) = ]YIJKxV(t) i f Z ELk and #k = - 1 . Proo/. (i) Y = 0. By means of Theorem 3.1.9 we have "~ = V. We start with
53
U C ker ad(V). Then
-I
Vvg =-2[u,v]
= 0,
whence U is parallel along 7- By Theorem 3.3 we have R~U = R v U = O. This shows that the Jacobi field Bu is given by Bv(t) = tU. /
!
According to 3.5 we know.that R~ = [R~, T~] and T~ = 0 with J x V +-~I[U, V] 9
T~(U + X ) =
Let U C k e r a d ( V ) • = J3V and X := [U,V]. Then J x V = - U and 1
VoU = VoU + T~U = - ~ [U, Y] + ~ [U, V] = 0 VoX = VoX + T,X =Thus U a n d X
JxV +~JxV =O.
are Vo-parallel. Further, we have
1 T~X = --~U, ~ T~U = -~X,
7:u = - !4u , T:x = - i4X
'
and hence, using Theorem 3 . 3 ( i ) , R , u = R , U + T:U = - 34- U - ~U 1 = -U,
ft~X = I ~ X + T:X = l-x - 1 X = O . 4
4
Thus, in this special situation, the Jacobi equation is
(0 z2
+(:
1
0
z2
0
z2
'
with initial values
(), () () zl
0
0) =
1
22
zl z2
o
for Bu
0
0) =
0 1
for B x
'
where Bu and B x are equal to zxU + z2X, respectively. The above equations are It
!
Z 1 + Z 2 -- Zl II
Z2
=
I
--
Z1
0 , 0
m
Integrating the second equation gives z; = zl + 2'2(0) .
Inserting this into the first one yields Z'l' -~ z/2(0) = 0 , 54
,
a n d therefore 1 t 2 zl(t) = --~z~(o)t + z'dO)t
For z2 we then get 1 , z2(t) =-~z~(O)t
3
1 , 2 + ~zl(O)t + z;(O)t .
Finally, taking account of the initial values then gives the expressions for B u and B x as s t a t e d in (i). (ii) V = 0. T h e n ~ = Y. For all X E 3 we have ~ r X = 0, whence X is parallel along 7. This implies B x ( t ) = t X . According to 3.5 we have R'~ = [R~, T~] and T~ = 0 with T,(U + X) = -~JyU
.
Z
Then (7oU = V o U + TvU = - ~ J v U - ~ J r U = - J y U VoJyU = -J~U rTofToU = - U
,
= g ,
,
V o ~7oJvU = - J r U
,
R , u = P~U + T:U= -~~U + ~J:~U = o , ~JrU
= 0.
Thus, B u is of the form By = zlU + zflvU
.
Then VoBu
=
z ' l U - z l J v U + z ' f l r U + z2U ,
VoVoBu
=
z','U - 2Z'lJyU - z l U + z ~ J y V + 2z'~U - Z f l y U .
So, 0 = V o V o B v - 2 T ~ V o B u + [l~Bu = (z'l'+ z'2)U + (z~ - z ' l ) J r U .
This implies the system of differential equations II
Z1 +
I
Z2 =
It
0
~
l
Z 2 -- Z 1 =
0
with initial conditions z,(0) = 0 , z',(0) = 1 , z~(0) = 0 , z;(0) = 0 . T h e solutions of it are z~(t) = s i n ( t )
,
z~(t)
by which (ii) is proved. 55
= 1 -cos(t),
(iii) V ~ 0 ~ Y. T h e first statements in (iii) have been proved in 3.1.9 and 3.3. T h e formula for Bv+y is a consequence of ~(t) = V(t) + Y. F r o m 3.5 we know t h a t R~i = [P~, T~] and T' = 0 on ,3 : = span{V, JrV, Y } with
T~(t)(U + X ) = - ~ J y U +
+
V(t)] .
For the sake of brevity we put El(t):=
IYIv"
IYl"
Z~(t).-
1
J,Y(t).
E~ a n d E2 are o r t h o n o r m a l and, using Theorem 3.3(iii)(1), we have 1
1
T~E~=-~E2
, T~E~=-~E~,
[~EI=O
, R, E2=-JVJ2E2,
VoEI = O , ( 7 0 E 2 = 0 . Thus BE,(O) is of the form zlE1 + z2E~ and the Jacobi equation is
( ) ( o)( )( )() () () () () z~ z2
_
0 1
1
z~ z2
z~ z~
(0) =
0 0
'
zl z2
(0) =
0 0
'
+
0 0
0 -[Vl 2
z~ z2
-- 0 ,
z~ z2
O) =
1 0
for B~(o) ,
z~ z2
0) =
0 1
for B~(0) 9
with initial values
Explicitly, we o b t a i n
Z1" + z ' ~ "
Z2
'-IVl2z2
--
Z 1
=
0,
=
0
Integrating the first equation gives z'l = - z 2 + z'~(0), a n d inserting this into the second one implies
zh' + IYl~z~ - z'~(O) = O . Here we get the solutions 1 zl(t)-[y[~(~y~sin([Y,t)-[V[2t)
z2(t)
-
1 i y p ( X - cos(lYlt)) ,
zl(t)
-
i y l i ( c o s ( l Y l t ) - l) ,
z~(t)
=
/l'Y---Tsin([Ylt)'
1
1
56
for BE,(0), i = 1,2, respectively. Multiplication of BE,(0) and B~,(0) with ]VIIY[ then gives the expressions as stated in the theorem. Further, according to 3.5, we have R'v = I/L,, T,] and 2r~ = 0 on p with
TW = -~J,,V. Then, using Theorem 3.3(iii)(2), we obtain for U E p,
VoU=-JvU VoVoU = - I Y I W
, VoJvU=IYI~U, , VoVoJvU = - [ Y I ~ J v U ,
&u
, R,J~U=O.
=o
Thus, By is of the form
By = zlU + z f l v U 9 Then
~oB~ = z'tU - z ~ J , U + z'=dvU + z 2 l Y l = V , VoVoBu
=
zi'U - 2z',J,,U - z, lYl=U + z $ J v U + 2z'~lYl=U - z = l Y i = d v U .
So,
0 = V o V o B v - 2Z,~7oBv + R, Bu = (z,I I +
Igl
2
l z~)U +
(z2
It -
z't)JrS 9
This implies the system of differential equations z',' + IYl~;'= = o
,
z~t! - z ,I =
0
with initial conditions z,(O) = O, z',(O) = 1 , z~(O) = O, z;(O) = O. The solutions of it are
zl(t) =
sin(lYft) , z2(t) -- i--~(1 - cos(Jglt)),
by which the formula for Bu is proved. It remains to compute the Jacobi fields for U + X E q. From 3.5 we know that R ; = [R,, T.~] and T', = 0 on q for
1
T,(t)(U + X ) = ~ J r V + ~ J x V ( t ) + 2[V,V(t)l . We will use the notations as in 3.3. We first consider the case #~ ~ {0, - 1 } . Let X be a unit vector in L~. Then
T,(t)X
= ~JxV(t), 57
T~(t)KX = ~J~xV(t), T~(t)J~V(t) = - ~ J x J y V ( t ) - ~ , V l 2 X , T~(t)J,~V(t) T4t)JxJ.V(t) = ~]rl2J~.r(t)- [IV['IYIKX , T~(t)JKxJrV(t) -- ~lYl%~v(t)- ~t,jlVl~lYlX . Therefore, qj(X) := span{X, KX, JxV(t), Js.xV(t), Jx JyV(t), JKx JYV(t)} is invariant under the action of T~(t). Further, we easily verify that
X , KX , JxV(t), J~xV(t), JxJrV(t), JKxJrV(t) are V0-parallel. Unfortunately, the vector fields JxV(t) and JKxJyV(t) (and also gKxV(t) and JxJyV(t), respectively) are not orthogonal to each other, since
<JxV(t), YKxJyV(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 ( / ) / ~ ) + / ~ / ) / ~
= (
Therefore Y is the Jacobi field along 7 with initial values r(o) = 0 and
( r o Y ) ( 0 ) --- Vo(/)E)(0) = ~'o(D/~)(0) - Tv(/)/~)(0) = (Vo/))/~(0) = / ~ ( 0 ) . Using this fact we have, at 7(r), A D E = A Y = - V r ~ = - V o Y = - V o Y + T~Y = - ( ( Y o D ) E + T ~ D E ,
and therefore A D = - V o D + T~D .
Since D ( r ) is non-singular (otherwise 7(r) would be a conjugate point along 7, in which case G , ( r ) could not be a hypersurface), we may multiply the last equation from the right w i t h / ) - 1 , by which the assertion follows. [] 69
We will apply this lemma now to deduce Theorem
1 Every generalized Heisenberg group is an ~r
Proof. We produce the situation described in the preceding lemma with M as a generalized Heisenberg group N, p = e and "~(0) = V + Y E n. Our aim is to show that the eigenvalue functions of A(r), considered as functions of r, are odd functions. Since - A ( - r ) is the shape operator of Ge(r) at 7 ( - r ) , this then proves the assertion. We shall frequently use the computations carried out in the proof of Theorem 3.7. Once again we consider the three cases.
(i) Y = 0. For U 6 ker ad(Y) N y x we have T~U = O , ~7oU = O , D ( r ) U = rU ,
whence A(r)U = - 1 - U . r
Next, let U E ker ad(V) • be a unit vector. Then X := [U, V] E 3 is a unit vector and we have U = - J x V . Both U and X are V0-parallel and with respect to U , X we have, with a ( U ) : = span{U,X},
r(6 -3r) 1(0 1) 1(2 2r)
f)(r)la(U) = g
and
3r
T~(r)la(U) = -~
Then we get
(Vob)(r)lo(U) b-'(r)lo(U) (V0D)(r)l~(U) o Z)-I(r)I~(U)
=
6- r~
1 0
2
2r
"
2 - r2
'
2 ( 6-r2 r ( 1 2 + r 2) -3r
_
3r ) 6 '
1 ~ 12 + 4r 2 - 6 r r ( 1 2 + r 2) ~ 6 r + r 3 12
_
,
and therefore, A(r)[a(U) -
1 ( 2 4 + 8r2 r3 ) 2r(12 + r 2) r3 24 '
It follows that t r A [ a ( U ) is an odd function of r and tr A2[a(U) is an even function of r. From the characteristic equation for the eigenvalues we conclude that the eigenfunctions of A[a(U) are odd. Summing up, we see that all eigenfunctions of A are odd functions. (ii) V = 0. Then ; / = Y and R,! = Ry0 = 0. So we may apply the L e m m a with T~ = 0 (hence, V0 = V0 and consequently, we drop the bar in the following). For X E y l we have V y X = 0 and D ( r ) X = r X , and therefore A(r)X = -1X r
70
.
Next, let U E v be a unit vector. Then
Vv(cos(t/2)U + sin(t/2)JvU) = O, which means that
Eu(t) := cos(t/2)U + sin(t/2)JvV is the V0-parallel vector field along 3' with initial value Eu(0) = U. According to Theorem 3.7 we have
By(t) = 2 sin(t/2)Ev(t) , and hence
D(r)Eu(r) = 2 sin(r /2)Eu(r) 9 This implies
A(r)l~ = - 2 cot(r/2)id~ . We again conclude that the eigenvalue functions of A are odd functions. (iii) Y ~ 0 ~ Y. It was proved in 3.7 that
El(t):= ,-:=.,IYl + M y -IV[ y(t)
1 JyV(t) [V[[Y[
,
[Y[
is a ~r0-parallel orthonormal frame field of a3 := "3 n (V + Y)• , and that, with respect to El, E~,
and D(r)laz = ~
1 ( sin(lYlr)-lVl~]Yir [g[(1 - cos(IYlr))
[Yl(cos(lYlr)- 1) ) IYI 2 sin(lglr) "
From this we obtain at once
1 (cos('Y[r)-'V'2 (voD)(r)la3 = ~
IYI sin(IYlr)
-'Y'sin(lY[r)) IVl ~cos(lYlr)
'
1 act O(r)]a3 = ~ v ( r ) with
u(r) := [-~[(1 - c o s ( l Y l r ) ) - [Yl2rsin([Ylr), and further, 1 D-'(r)la~ = ~
[Y[(l-cos([Ylr)) ) [Yl(cos(IYIr)- 1) sin(lYlr)-IVI21YI r "
([Y,~sin(lY[r)
71
Therefore,
A(r)]~3 = 1 2~(r)
(
21gl2sin(lylr)
2lYl(1-cos(lYlr))-v(r) ) 21VI2JYl~cos(IYlO "
2 1 Y I 0 - c o s ( I Y I 0 ) - ~(r) 2sin(IYIr)-
This shows t h a t tr Alia is an o d d function of r and tr A~]ia is an even function of r. Thus the two eigenvalue functions of A]n3 are odd functions. Next, let U E p be a unit vector. Then, as above, it follows t h a t
Ev(t) : = cos(IYIt/2)U + ~y] sin(IYIt/2)JrU is the V0-parallel vector field along 7 with initial value Ev(O) = U. According to 3.7 we have 2
Bu( t ) = -~l sin( lY lt /2 )Eu( t ) '
from which we readily compute
A( r )Ev( r ) = - ~ - cot( Ig lr /2 )Ev( r ) . As p is invariant under V0-parallel translation, it follows t h a t
A(r)lp =
- y]Y] cot(]Y]r/2)idp .
A l t h o u g h all the necessary material is available to write down explicitly the shape o p e r a t o r on q, we omit this tedious work and restrict to the features of A which are needed to prove the required result. F i r s t we consider the case #s 9( { 0 , - 1 } . Let X E L~ be a unit vector, E l , . . . ,E6 a n d q~(X) as in 3.7. The space q~(X) is invariant under T~ a n d / 5 a n d E I , . . . , E 6 are V o - p a r a l M , which shows t h a t / ) ( r ) and hence also A ( r ) m a p s qj(X) into itself. According to 3.7 we have
Olq~(X) ~ f~ :=
0
e
e
0
0
e
e
0
0
e
e
0
e
0
0
e
e
0
0
e
e
0
0
e
0
e
e
0
0
e
e
0
0
e
e
0
which means t h a t with respect to E ~ , . . . , E6 the tensor field /)]q~(X) is a m a t r i x function f~ for which each coefficient function is even (e) or o d d (o). Note t h a t if a coefficient function is zero we may choose e or o in the way we like. Also from 3.7 (see the m a t r i x P there) we know that T~Iqj(X) ~ K~ . 72
From this we also obtain d e t D I q i ( X ) ,~ e , (7oblqj(X)
~
oft,
D-1Iq~(X)
~
~,
V o D o D-~h~(X)
~
oft 2 ~ ft ,
and therefore, A[q~(X) ~ ~ .
A straightforward computation now gives trA[qj(X)~o trA41q1(X)~e
, trA2]qj(Z)~e , trAS[qj(X)~o
, trAalqj(X)~o ; trA6[q~(X)~e
from which we conclude that the eigenvalue functions of AIq~(X ) are odd functions. Further, if #0 = 0 and X E Lo is a unit vector, we consider (as in 3.7) the vector fields El, Ea, E5 and get Dlq0(X ) ~ f ~ : =
(o o) e o e
O
e
, T, l q 0 ( X ) ~ f t ,
o
which, by a straightforward computation, yields Aiq0(Z) ~ ft.
Thus trA]q0(X) -~ o , trA2[q0(X) ~ e , trAa[q0(X) ~ o , which shows that the eigenvalue functions of A]q0(X ) are odd. Finally, if p~ = - 1 and X E L k is a unit vector, we consider (as in 3.7) the vector fields El, E~, Ea, E4 to get 0
DI qk(X) ~ ft :=
e
e
0
~ o o c e
0
0
e
0
e
e
0
, T~lqk(X) ~ f~,
and, by a straightforward computation, we now obtain Alqk(X ) ~ Q . Hence, trAIq~(X) ~ o , trA~lqdX) ~ ~, t r A % ( X )
~ o , tr A % ( X )
~ ~,
which implies that the eigenvalue functions of A]qk(X ) are again odd functions of r. As Tv(r)G,(r) can be decomposed orthogonally into ha, p and suitable spaces q~(X), we conclude that all eigenvalue functions of A(r) are odd functions of r. This proves the assertion also in the last case V # 0 # Y. [] 73
The preceding Theorem and Corollary 2 in 3.6 then implies Theorem
2. Every generalized Heisenberg group is a T*-space.
As any ~r or | is a D'Atri space we now have an alternative proof of Theorem 4 in 3.2 stating that every generalized Heisenberg group is a D'Atri space. Contracting twice the Gauss equation of second order, and using the constancy of the spectrum of the Jacobi operator along a geodesic % we obtain that the scalar curvature of G,(r) at 7(r) is given by 1 --rnn4 - 2tr Rv+y + (tr A~(,)) 2 - tr A~(,)2 . As a consequence of Theorem 1 we therefore get
Corollary The scalar curvature of any geodesic sphere in a generalized Heisenberg group is the same at antipodal points.
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 coordinates
Let (M, g) be a k-dimensional Riemannian manifold, p C M, e l , . . . , eL an orthonormal basis of TpM and x l , . . . , xk the induced normal coordinates on an open neighborhood of p. Further, let (g,j) be the matrix-valued map defined by gij : :
g
,
9
So, (gij) is the metric tensor of M with respect to the normal coordinates x l , . . . , xk. An elementary, but crucial, observation is the following L e m m a If x l , . . . ,xk and k l , . . . , x k are two systems of normal coordinates centered at p, then (gij) and (~j) differ only by conjugation with an orthogonal transformation. In particular, the eigenvalues of (gij) are independent of the choice of the normal coordinates centered at p. It was proved by J.E. D'Atri [Dat] that the eigenvalues of (gii) are the same at antipodal points (with respect to the center of the normal coordinates) if M is a naturally reductive Riemannian homogeneous space. This result was generalized by O. Kowalski and the third author [KoVa4] to the class of Riemannian g.o. spaces. Furthermore, they proved in [KoVa2] that this property of the metric also holds on commutative spaces. Up to now it was an open problem whether this geometric property of the metric characterizes Riemannian g.o. spaces or not. The following theorem shows that this is not the case. T h e o r e m On every generalized Heisenberg group the eigenvalues of the metric with respect to normal coordinates are the same at antipodal points (with respect to the center of the normal coordinates). 74
Proof. Clearly, by the homogeneity of a generalized Heisenberg group N , it suffices to s t u d y the metric with respect to normal coordinates centered at the identity e. Let 7 : IR --~ N be a geodesic in M p a r a m e t r i z e d by arc length a n d with 7(0) = e. We define V + Y := ;/(0), choose an o r t h o n o r m a l basis e l , . . . , e,+m of T , N ~- n such t h a t el = V + Y, and denote the resulting n o r m a l coordinates by X l , . . . ,x,+m. Let Be, denote the Jacobi field along 7 with initial values B,,(0) = 0 and B',,(0) = ei. From the definition of normal coordinates it follows t h a t
+(r) =~-O(7(r)) and B~,(r) = rb~,,x(7(r)) , i = 2 , . . . , n + m . Now let V0 a n d / ) be as in L e m m a 3.9. Further, let /~ be the ~'0-parallel vector field along 7 w i t h / ~ i ( 0 ) = ei. According to L e m m a 3.7 we then have Be.=DEi, Thus, for i, j = 2 , . . . , n + m, we obtain
i=2,...,n+m.
(00)
9,~(7(r)) = g ~ , O~j (7(r)) = -~g(DE,,bE,)(r) = - ~ g ( 9 ~ D L , Ejl(r), and, as a consequence of the Gauss Lemma, glj(7(r))
-~ O .
As 7 is p a r a m e t r i z e d by arc length, we also have
gl,(7(r)) = 9(+,+)(r) = 1. Therefore, the metric (9ii) along 7 with respect to the n o r m a l coordinates is given
by 1
r2 0 0
-.-
0
0 with respect to/~1,. 9 9 So, in order to prove the assertion, it suffices to show that L)TL)(r) a n d D T D ( - r ) have the same eigenvalues for small r E IR+. T h e p r o o f of this will be done along the same lines as the proof of T h e o r e m 1 in 3.9, from which we also take the p a r t i c u l a r form o f / ) in the three cases we s t u d y now. (i) Y = 0. If U G ker ad(V) N V • then B y ( r ) = rU and hence
DTD(r)U = r2U . 75
Next, let U C k e r a d ( V ) • be a unit vector. Then we have U = - d x V with X := [U, V] E 3. Both U and X are V0-parallel, and with respect to U, X we obtain, with a ( U ) := s p a n { U , X } ,
~(g) ~
e o tr (DTD)2I~(U)are
which implies that tr DTDla(U) and even functions of r. From this we derive that the eigenvalues of DTD(r)[a(U) and D T D ( - r ) ] a ( U ) are the same. As the orthogonal complement of IRV in n can be spanned by vectors and two-planes of the above form, the assertion in this case follows. (ii) V = 0. In this case we have
D ( r ) I Y l = r idy• and D(r)I~ = 2 sin(r/2)idD , from which the assertion also follows. (iii) V r 0 # Y. We use the same notations as in the proof of Theorem 1 in 3.9. W i t h respect to El, E2 we have
C O and hence
Dr D[r,3 ~
0
C
Therefore, tr DT/)I~3 ,~ e , tr(DrD)2l~3 ~ e , which shows that the eigenvalues of DTD]~a are even functions of r. Next, as 2 b ( r ) l o = ~ sin( [Vlr /2 )id o ,
the eigenvalues of DTDIo are even functions of r. If #j ~ { 0 , - 1 } and X E L j is a unit vector, then
DJq,(X)
O~OOe ~OOCCO CO0~eO occooe oecoo~ eooeco
76
/
with respect to E l , . . . , E6, and hence
bLblqAX)
e o o e e o
~ e ~ o o ~
1 7 6 1 7 6/ c o o e c o o c o e e o o e c o o o e
This implies
trDrDIqj(X) ~ e , tr(DrD)2lqj(X ) ~ e , tr(Dr/))alqj(X ) ~ e , t r ( D T D ) 4 ] q j ( X ) ~ e , tr(/)TD)51qj(X ) ~ e , tr(/)T/))61qj(X ) ~ e , from which we conclude that the eigenvalue functions of tions 9
DrDIq~(X)are
even func-
(o o)
If #0 = 0 and X E L0 is a unit vector, then
Dlqo(X)~
o
e
O e
e
O
with respect to El, Ea, E~. This implies
/)~blq0(X)~
o
~
o
e
o
e
.
As above we conclude that the eigenvalue functions of DTDIqo(X) are even functions. If #k = - 1 and X E L k is a unit vector, we have
DIq~(X)
0 ~ e O
~ 0 O ~
0 0 ~ 0 ~ e O
with respect to E l , E~, Ea, E4, and therefore, e o o
DTDlqk(X)
o e ~ o
o e e o
c o o e
Also here we conclude easily that the eigenvalue functions of .DTJDIqk(X ) are even functions. As the orthogonal complement of V + Y in , can be decomposed orthogonally into ha, P and suitable spaces qj(X), we obtain that all eigenvalue functions of D T D are even functions of r. This proves the assertion also in the case V # 0 # Y. [] Note t h a t the p r o p e r t y for the eigenvalues of the metric tensor with respect to n o r m a l coordinates provides, once again, a proof for the D ' A t r i p r o p e r t y of generalized Heisenberg groups. 77
Chapter 4 D a m e k - R i c c i spaces In Section 1 we present basic material about the Damek-Ricci spaces. Most of the contents of this section is due to [Bog], [CoDoKoRi], [Dam1] and [Dam2]. T h e s t u d y of the geometry of certain left-invariant distributions in 4.1.10 a n d the proof for the (non-)existence of invariant nearly K/ihler structures on Damek-Ricei spaces in 4.1.13 are new. In Section 2 we determine the complete s p e c t r u m of the Jaeobi o p e r a t o r and the corresponding eigenspaces. The s p e c t r u m has also been w r i t t e n down in [Sza2] without providing a proof. We finish Section 2 with a detailed discussion of the sectional curvature. In Section 3 we provide two alternative proofs for the fact t h a t a Damek-Ricci space is a C-space if and only if it is a s y m m e t r i c space. T h e explicit knowledge of the eigenspaces of the Jacobi o p e r a t o r enables us also to prove t h a t a Damek-Ricci space is a N-space if and only if it is symmetric. In Section 4 we provide an alternative proof for the harmonicity of Damek-Ricei spaces by showing t h a t the distance function to the identity is isoparametric. In Section 5 we discuss several consequences of the results of the previous sections. Most of t h e m are new geometrical characterizations of the symmetric manifolds among the Damek-Ricci spaces.
4.1 4.1.1
Basic concepts Definition
Let n be a generalized Heisenberg algebra, a a one-dimensional real vector space and A a non-zero vector in a. We denote the inner product and the Lie bracket on n by <., .>n a n d [., .Ira respectively, and define a new vector space 5"~n~rt
as the direct sum of n and a. Each vector in ~ can be written in a unique way in the form V + Y + s A with some V E ~, Y E 3 and s E IR. T h r o u g h o u t this c h a p t e r vectors in ~ will be written in this form, and we always use the symbols U, V, W for vectors in v, X, Y, Z for vectors in 3 and r, s, t for real numbers. 78
We now define an inner product <., .> and a Lie bracket [., .] on ~ by
:= < U + X , V+Y>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 := <[Vii. ~ ] , Y k > .
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<[U, V,], Y,~>
=
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 +~<
1
1
- ~( + ~ t ) u + ~(
84
W> +
~t)v
'
I [U, Jz V] - ~ [U, Jr W] + ~ [V, Jx W] + ~ t[U, V] + ~ s[U, w] - lr[v, w] 2 - < V + Y + s A , W + Z + t A > X + Y 1 V 1 1 + ~ < , W > X - ~Y + ~Z 1 + {--~<JzU, V> - ~<JyU, W> + ~<JxV, W> - r (~ + ) + s (~ + < X , Z > ) } A ,
In p a r t i c u l a r , the formula for the Ricci tensor shows t h a t Proposition
1
Every Damek-Ricci space is an Einstein manifold.
Further, let a be a two-dimensional subspace of o r t h o n o r m a l vectors U + X and V + Y + s A in ~ so t h a t a = span{U+X, For the sectional
TeS ~- ~. T h e n there exist
V+Y+sA} .
K(a) of S with respect to a we then have
h'(~)=-3lsX-[U, Vll2-~<x,g> 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
~
-3[U, JyV] + ~s[U, V] - ( 1 - ~,V,2) X + Y + {3 - r (~JV]2 + IY,2) + s (~ + < X , Y > ) } 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 > = <[(JA)0, V], 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 < ~ < - ~ <_ ~ < - ~ < ~ < 0 , where a, = x2 ~
j = O , to=O,
s = O , IVI ~ =
2 I
The eigenvalues are the solutions of
(~+1) (~+
=
~[V]41YI2(1 + ~j)
and satisfy the relation ~1 + a2 + a3 Furthermore, we have (a) (if j = O, #o = 0 and s = O) the eigenvalue~ and eigenspaces of Rv+v]qo are
- ~ ( I + 3 I V I 2) , J t o V ; I
~(31VI2-5+3]YI~)
, {(4~+l)X+3JxJyV]X
(b) (otherwise) the eigenspace of
Rv+Y§
E L0};
with respect to ,~, is
{(4ai+l)(4x,+l+3[V]2)X + 3(4ai+l+3Wl2)Jx(JvV-sV) -91Vi2JiYiKX_,xV I x E L j } . Proof. As R v + y + , a ( V + Y + s A ) = 0, we will always assume that U + X + r A is orthogonal to V + Y + s A . Then we have Rv+y+,A(U+X+rA) :
+
+
We now consider six cases. 98
E
(i) V = 0 ,
Y=0.
Then t~,A(U § X ) : _Iu
_ x
4
and the assertion follows. (ii) V = 0, s = 0. Then
R y ( U + X + r A ) = -1-U - X - rA 4 and the s t a t e m e n t follows. (iii) Y = O, s = O. Then
R v ( U + X + r A ) = ~Jtu.v]V - 1-U - 1-X 4 4
-
I_ A
4
If U + X + r A E ( k e r a d ( V ) N V ' ) 9 3 G a, then it is an eigenvector of Rv with corresponding eigenvalue - 1 / 4 . If U C ker ad(V) • then there exists a vector Z C so t h a t U = J z V and we get
3
R v U = ~ J[Jzv, v]
4
4
"
This proves the assertion. (iv) V = O,Y ~ O,s ~ O. Then < X , Y > = - r s , IYI ~ + s 2 = 1, and hence Ry+,A(UWX§
1
= -~U
- X -
rA
,
which proves the assertion. (v) V ~ 0, Y = 0, s ~ 0. Since = - r s and IWl = + s = = 1, we get
RV+,A(U+X+rA) = ~Jtu, v ] V - ~ s J x V - 1 U
+ ~s[U, V] - I (x + 3s~)X - l r A .
F i r s t suppose t h a t U + r A C ker ad(V) • a. Then RV+sA(U-[-rA) = -I(u§
9
Next, if 0 r X E ~, then
RV+~AX = - ~ s J x V - ~ ( l + 3 s ~ ) X ,
Thus Rv+,A m a p s span{X, JxV} into itself. Let ~ be an eigenvalue of Rv+,A restricted to this 2-plane. Then there exist a, ~ C IR so that a X + / ~ J x V is a corresponding eigenvector. As j3 = 0 is impossible (since X is not an eigenvector of Rv+,a), we may p u t ~ = - 3 s / 4 . Then the eigenvector equation
Rv+,A ( a X - ~ s J x V ) = ~ a X - ~ s J x V 99
yields
= -5(1+3s=)~+
~
=
s~lVI~,
~-l+~J.
From the second equation we get a : g + 1 - ~s ~ , and inserting this into the first equation then implies
Thus ~ e { - 1 / 4 , - 1 } . If ~ = - 1 / 4 , then a = 3 ( 1 - s ~ ) / 4 = 3[VI2/4; and for n = - 1 we get a = - 3 s 2 / 4 . From this the assertion finally follows. (vi) V • 0, Y ?t 0. Clearly, R v + y + , A ( V + Y + s A ) = 0. A straightforward computation shows that Rv+Y+,A(JyV § s Y -
[Yl2A) =
-(JyY
§ s Y - [Y[2A)
and Rv+y+sA((aS § 31Yl2)V + (Ds - a ) J y Y - / 3 1 V I 2 Y - alVI2A) =
-4((as
+ •lY]2)Y + (Zs - a ) J y V - / ~ I V I 2 Y - alY]2A) .
Thus the statement about the eigenvalues and eigenspaces of Rv+Y+,A on ~4 is proved. Next, for all U E p one easily derives 1
Rv+r+,aU = - ~ U
.
We continue by proving that each qj is invariant under Rv+Y+sA. For X E L~with #5 ~ - 1 we obtain Rv+r+~AX
=
Rv+Y+,aJxV
-
Rv+Y+,AJxJyV
=
Ifj
= k a n d #k =
3jxJyV-~sJxV-
(1-~,V[
1
,~(1 + 31Vl~)JxV -
--41Vl~lglJ~:xr -
~)X,
lvl trlA.x
-
slVl x
,
JxJrV+ Irl~lgJ~x - s l V J ~ I Y I K X .
- 1 , then J x J y V = JIYIKxV and hence
Rv+v+,AX Rv+Y+,AJxV
=
-34 J i Y i K x V - ~ s J x V - (1 - ~,VI 2) X ,
:
- 51( 1 + 31Vl2)JxY - ~ [ V I 2 I Y I K X - ~slV]~ X . 100
So we see t h a t q~ is invariant under
.il~V+Y+,A
.
We continue with the case #k = - 1 and put 1? : =
JvV
-
~V
.
T h e n we get for each non-zero X E L k
Rv+~,+,,~dx f/
= ~[Vl~(~ ~ + Irl~)x - ~(1 + 3[V[2)dx fI .
Thus s p a n { X , J x V } is invariant under Rv+r+,A. Hence there exist non-zero c~,/3 6 IR and ~ 6 119.such t h a t
Rv+y+,A(aX + ~JxV) = ~ a X + ~ J x f z . C o m p a r i n g coefficients leads to the equations
We m a y normalize ;9 by p u t t i n g r :=
3/4.
Then the second equation yields
a=~+~+
VI ~ .
Inserting a and ;3 into the first equation then gives
Thus ~ 6 { - 1 , - 1 / 4 } is
is an eigenvalue of Rv+y+,A Iqk. The corresponding eigenspace
{]VI 2X + J x V I X e Lk} , i f ~ = - 1 / 4 , {(IVl ~ - 1)X + J x V I X ~
Zk} ,
if ~ = - 1 .
Next, we consider the case #j # - 1 . We s t a r t with the subcase j = 0, p0 = 0 and s = 0. For X E L0 we have
Rv+YJxV = - ~ ( 1 + 3lVl2)JxV . Thus - ( 1 + 31V1~)/4 is an eigenvalue of Rv+y and JLoV is contained in the corresponding eigenspace. Further, we know that
Rv,yX
= ~JxJvV-
]01
(I-~[V[2) X,
Thus, span{X, J x J v V } is invariant under Rv+Y. Hence there exist some non-zero a, ~ E ]R so that a X + j 3 J x J r V is an eigenvector of Rv+y, say R,+,(ox + 9yxJ, V) = ~ x + ,~ZJxyyV .
Then we get the equations
,~ = ~--~.
1
The second equation then gives
and normalizing ~ by ~ := 3/4, we deduce o! =
1
tO+ ~ .
Thus the first equation implies
The solutions of this quadratic equation are ~(3IV[ 2 - 5•
•
and the corresponding eigenspaces are given by {(4~ + 1)X + 3 S x J v Y I X e L0} . It is easy to check that all three eigenyalues satisfy the equation
Moreover, two of these eigenvalues coincide, namely - ( 1 + 31Vt2)/4 and (31vI 2 - 5 31Yl~/1 + 31vt=)/s, precisely if IVI ~ = 2/3. We now treat the remaining cases. Let X be a non-zero vector in L~ and put I~ := J r V -
sV .
Then we obtain Rv+r+,AX
Rv+v+,AJx~" Rv +Y+sAJIr IgX-sX V
=
1 ~lYl2(s 2 -,~lYb~)X - ~(1 + 3]Yl2)JiriKX_,xY. 102
Now in the case we m'e considering, the vectors X , J x V and JIYIKX-,xV a r e linearly independent. We also see that the span of these three vectors is invariant under Rv+y+,A. In order to compute the eigenvalues and eigenvectors we define a nonzero vector E by E := a X +/3JxCd + *JlYIt:X-,xV 9 Then the eigenvector equation Rv+Y+sA E = tee
yields the following system of equations:
1
The second equation yields
As /3 # 0 (otherwise also a - 0 and thus, by the first equation, s = 0 = IY] or (~ = 0, which is impossible in this case), we may normalize/3 by/3 := 3/4 and get a=~+~
1
9
The third equation implies
,~+E+
IVl ~ a = -
iVl~Z=-
ivl ~,
which yields
,sr162
IVl~.
31v1~)/4and obtain _ ~lVl,lYi2(l+~j)
Eventually, we multiply the first equation with tc + (1 + 0
=
1 ~ 3 + ~ 3to2 + ~9 s + ~
=
(~ + 1 ) ( ~ + ~ ) 2 _ 271VI'IYI2(l+64 #1)-
Thus each eigenvalue ~ of Rv+y+,AIqj is a solution of (~ + 1) (to +
=
JVF~lrl2(1 + , ~ ) ,
and the corresponding eigenspace is given by {(4tc+l)(4~+l+3]Y12)X + 3(4~r
- 9]V]2JrvltCX_,xV l X E Lj} . 103
It remains to investigate whether some of these eigenvalues can be equal. Therefore we define the functions
f : IR---~ IR , x ~--~(x W1) (x +
,
27
g: e ~ ~ ,
(x,y) ~ ~ ( 1 + ~,~)z~ ~
h : ~ ,
27 xH~(l+~)x~(1-x
~),
where
G := {(x,y) E ]R21x,y > O , x 2 + y ~ <
1}.
The eigenvalues t~ of Rv+Y+sAIqj are the solutions of f ( ~ ) = g(IV[,]YI) 9 The function f has two zeroes, namely at - 1 tiplicity two. The only local m a x i m u m of f only local m i m i m u m is at - 1 / 4 with value 0. interior of G and g(x, y) = 0 if x = 0 or y = in the set
and - 1 / 4 , the second one is of mulis at - 3 / 4 with value 1/16, a n d the Clearly, g has no critical points in the 0. Hence the m a x i m u m of g must lie
{(x,y) e ~ I x,y > 0, x ~ + y ~ = 1}. In order to find it we differentiate h and obtain
h'(x) = ~27( 1 + tL~)z3(2 - 3x 2) . Thus the critical points of h are at 0 and • We have h(0) = 0 and h ( ~ ) = (1 + # j ) / 1 6 < 1/16, and equality if and only if #j = 0. So the m a x i m a l value of g is (1 + p i ) / 1 6 and is attained for ( x , y ) = ( V ~ , vfl-/3). Combining this with the above discussion of the graph of f we conclude that all eigenvalues tr ~2, tr are distinct unless #i -= 0, s = 0 and IV[ 2 = 2/3. [] T h e T h e o r e m shows that all eigenvalues of the Jacobi o p e r a t o r lie between - 1 and 0. As the m i n i m a l and m a x i m a l value of the sectional curvature K ( a ) of S arise as eigenvalues of the Jacobi operator we conclude t h a t - 1 < K ( a ) < 0. F u r t h e r m o r e , - 1 always arises as an eigenvalue, whence - 1 is the m i n i m a l value of the sectional curvature of S. It can also be seen immediately t h a t the m a x i m a l value of K ( a ) is - 1 / 4 provided that a satisfies the J2-condition, t h a t is (see 4.1.9), if S is a s y m m e t r i c space. If n does not satisfy the J2-condition, the m a x i m a l value of the sectional curvature of S is not known to us in general. J. Boggino [Bog] s t a t e d t h a t every n o n - s y m m e t r i c Damek-Ricci space admits a two-plane c~ for which K ( a ) vanishes. But a careful check shows that the sectional curvature of S with respect to the plane a written down by Boggino in the proof of T h e o r e m 2 does not vanish except in the situation described in the following. The preceding T h e o r e m shows t h a t zero occurs as a value of K(a) if and only if there exists a unit vector V + Y E a with IVI 2 = 2/3 such t h a t zero is an eigenvalue of the o p e r a t o r K~.y. Recall t h a t zero is an eigenvalue of K~,y means that there exists a non-zero vector X C Y• so
104
t h a t J x J y V is orthogonal to J3V. Let V + Y E n be a unit vector with V 7~ 0 ~ Y. Then we m a y decompose ~ orthogonally into = IRY |
|
L, 0-.-
0 Lk 9
According to the Theorem, the dimension of L~ is even if #~ 7~ 0. So if rn, the dimension of 6, is even, L0 must be odd-dimensional and hence zero arises as an eigenvalue of K~,y. As a is one-dimensional and , is even-dimensional, it follows t h a t m is even precisely if S is odd-dimensionaL Summing up, we conclude P r o p o s i t i o n The sectional curvatures K ( a ) of a Damek-Ricci space S satisfy - 1 <_ K ( a ) <_ 0 and - 1 is attained. If S is symmetric then - 1 <_ K ( a ) <_ - 1 / 4 and - 1 / 4 is attained. S attains zero as a value of the sectional curvature if and only if there exist a unit vector V + Y E , with IVI 2 = 2/3 and a non-zero vector X E Y Jso that J x J y V is orthogonal to J~V. If S is odd-dimensional, then zero is attained as a v a l u e
of K(,~).
For the even-dimensional non-symmetric Damek-Ricci spaces we do not know whether zero arises as a value for the sectional curvature except in some special cases t r e a t e d by E. Damek [Daml], where also another detailed account on the sectional curvature of these spaces can be found.
4.3
Eigenvalues and eigenvectors along geodesics
In this section we will prove that the eigenvalues of the Jacobi o p e r a t o r of a DamekRicci space S are constant along geodesics precisely if S is symmetric. We will also show t h a t S is s y m m e t r i c if and only if the eigenspaces of the Jacobi o p e r a t o r along geodesics are invariant under parallel translation. Theorem space.
1 A Damek-Ricci space is a e-space if and only if it is a symmetric
We will provide two alternative proofs of Theorem 1, one using the eigenvalues and the other one using eigenvectors of the Jacobi operator.
First Proof. If S is a symmetric Damek-Ricci space, then it is obviously a espace. So, let S be a non-symmetric Damek-Ricci space. T h e n the corresponding generalized Heisenberg algebra n does not satisfy the J2-condition, which means t h a t there exists a unit vector V + Y E n with V 5~ 0 r Y such t h a t not all eigenvalues of K~,y are equal to - 1 . Let ff : IR--~ S be the geodesic in S w i t h 3'(0) = e and ;/(0) = V + Y. According to Theorem 2 in 4.1.11 we have ~/(t) = V(t) + Y ( t ) + A(t) with
v(t) - ~/h~.l
IYl~)(t)y + 20(t)J.V) 105
Y(t) A(t)
= h(t)Y, (lnh)'(t)A ,
=
O(t) =
tanh(t/2),
x(t)
:
h(t)
=
+ Ir{'0'(0,
1 - 02(t) -
-
Clearly, Y(t) l = Y z for all t E IR. Thus the operators Kv(t),ro) are well-defined on Y• for all t E IR. A straightforward computation shows that IV(t), JxJyV(t)] = h(t)[V, J x J y V l , and therefore, using Y(t) = h(t)Y, [V(t)l ~ = IV[2h(t) and [Y(t)l = ]Ylh(t),
I,'v(o,~r
=
IV(t)l'llv(t)l
[v(t), Jx Jy(,)v(r
=
K v 'r X "
"2 F r o m this we conclude that the eigenvalues of Iivu)y(o do not d e p e n d on t. Now, let # r - 1 be an eigenvalue of K V,Y' 2 According to Theorem 4.2 there are some eigenvalues ~(t) of the Jacobi operator R~(t) = Rv(o+v(t)+a(Osatisfying the relation
(
(~(~) + 1) ~(~) +
=
IV(0141r(t)l~(1 + . / =
27,11,
~.,V,"y?ht0(:o~ +.).
As 1 + # r 0 and h is non-constant, the zeroes of this t h i r d order equation cannot be constant. This shows that R v has non-constant eigenvalues and hence S is not a ~-spaee. rn
Second Proof. If S is a symmetric Damek-Ricci space, then it is a C-space. F r o m now on we assume that S is a non-symmetric Damek-Ricci space. T h e n the corresponding generalized Heisenberg algebra , does not satisfy the J~-condition. This implies that there exists a unit vector V + Y + sA 6 n with V # 0 # Y a n d s # 0 such t h a t not all eigenvalues of K := K2vv are equal to - 1 . Let # be such an eigenvalue. According to Theorem 4.2 there exists an eigenvalue x of Rv+v+,a such that Rv+r+,aE = RE with E : = ( 4 ~ + 1 ) ( 4 ~ + l + 3 1 V I 2 ) X + 3(4~+l+31Vl2)Yx(JvV-sV) - 9lVl~JIvlKx-,xV, where X is some unit eigenvector of K 2 with respect to #. Using the formula for R~+v+,a in 4.1.8 we o b t a i n 2
,
-~Rv+y+,AE =
-31VI21YI2(4~ + 1 + 31VI2(1 + # ) ) J x V -3lgl2[Yls(4,~ + 1)JKxg +31v[2s(4~ + 1 ) J x J r V -3lvl21Yl(4~ + 1 ) J u x Z v V . 106
We write E also in this form, that is, E
=
-3s(4~ + 1)JxV
-91yl~lYIy.xy + 3 ( 4 x + 1 + 31VI2)JxJyY +(4t~ + 1)(4~ + 1 + 31Y]2)X . T h e vectors
JxV,
J~:xV, JxdrV , JKxJvV , X
are pairwise orthogonal to each other except in the cases when
< JK x V, Jx Jv V >
=
-[vI2IYI~,
<JxV, J,,xS,,V>
=
IvI~IY[~.
Considering this, a straightforward computation then yields 2 , -~ = ISIV["IYI~s(I + #)(4~ + 1)(4tr + 1 + 3[VI~) . By the a s s u m p t i o n IV], IY], s, and 1 + p are non-zero. As we are in the case (vi)(B) of T h e o r e m 4.2, tr cannot be equal to - 1 / 4 , whence 4x + 1 ~ 0. If 4~ + 1 +3IV[ 2 = 0, then JIYIgX-,xV would be an eigenvector of Rv+Y+,n. But we have
Rv+Y+,AJIYIKX-,xV = 31Vl2(s~ - # l Y r ) X - 1(1 + 3[Vl2)JtYtKX-,xV , a n d since # <_ 0, the X - c o m p o n e n t does not vanish. So we conclude t h a t
7t 0 , which shows t h a t condition (iii) in Proposition 2 of 2.8 does not hold. cannot be a E-space. []
Hence S
T h e second m a i n result of this section is Theorem space.
2 A Damek-Ricci space S is a ql-space if and only if it is a symmetric
Proof. If S is symmetric, then it is also a q3-space. Suppose now t h a t S is a N-space. Let V + Y + s A C s be a unit vector with V # 0 r Y and s 7t O, K := Kv,y, and 0 7~ X E Y ' . According to Proposition 3 in 2.8 we have [Rv+Y+,A, R~+y+,A] = 0 .
Since Rv+y+,AX = 0 ,
Rv+r+.aRv+v+.~X =
~[V[=(S(JxJrV - JwtKxV) - [ Y l ( J x x J v V + lO7
IYIJxV)),
and J x d r V - JIvIKxV , J K x J r V + I Y l J x V are orthogonal to each other, we conclude that JxJvV=JIvlKxV
for a l l X E Y •
Thus the corresponding generalized Heisenberg algebra n satisfies the J2-condition and hence S is a symmetric space. []
4.4
Harmonicity
In this section we provide an alternative proof of the result by E. Damek and F. Rieci [DaRil] stating that S is always a harmonic space. We start with a Lemma providing a formula for the Laplacian A = div grad of S in terms of the global coordinates on S introduced in 4.1.5. L e m m a The Laplace operator A of S i~ given by 0
2
(
1
O
)
le~ ~.j.k
0202 0 0 Or, Oyk
Proof. Using the notations in 4.1.5 we have A
= EV~+EY~2+A2-y~Vv, i
i
i
i
G-EVr,
i
-
Y~-VAA
i
m+
A.
Inserting here the expressions for I~, :t'] and A according to Lemma 4.1.5 yields the assertion by a lengthy but straightforward computation. [] The next step is to derive a formula for the differentiable function f~e
1-d2 :=
2
e
on S, where d,(p) is the distance from p C S to c, in terms of these coordinates. P r o p o s i t i o n 1 We have
,~
= ,I, o ~ ,
where ~ i~ the diffeomorphism 9 :[4, ec[---,[0, ee[, t H 2 A r t a n h 2 ( 1 ~ - t 4 - ) 108
and
((
k I ' : = e -~
vi+e ~
1+~
+
y~ : S --~ [4, o0[ .
Proof. Let p : = (expn(U + X ) , r ) E S be arbitrary, but not equal to e. As S is complete, simply connected, and of non-positive curvature, there is a unique geodesic 3' : IR ~ S in S which is p a r a m e t r i z e d by arc length and satisfies 3'(0) = e and 3"(t) = p for some t 9 IR+. Clearly, we have de(p) = t. If V + Y + sA : = ~/(0), then, by means of T h e o r e m 1 in 4.1.11, we obtain the relations U
=
20(1 -
X x
=
sO)(t) v + 202(t)JyV , X
2~
X - -
(t),
with
O(t) = t a n h ( t / 2 ) , x(t) = (1 - sO)2(t) +
IYl~02(t
This implies
IUI ~ -- 4t'~
IXl ~ -- i~-~[(t)lYI 2 0 ~ ( t )
X
= 1
-
x(t)e"
X"
and therefore, ~2
2
x(t) = (1 - ~o)=(t) + IYl~e-'(O = (1 - ~o)~(t) + T I X I and
1 = IVI ~ + IYI -~ + ~ = ~ x ( t ) l U l ~ + ~ o (t)lXl~ + ~ .
02(t) and comparing it with the preceding
Multiplication of the last equation with one yields
~((t) (1 + ~lgl~)= We now replace
1-
2sO(t)+O2(t).
02(t) by 1 - x(t)e" and obtain x(t) (l + ~lUI2 + e') = 2 ( 1 - sO)(t) .
Squaring this equation, then replacing (1 -.~O)~(t) by dividing by x(t) yields x(t)
1+
IUI ~+ ~"
(X - x21XI2/4)(t), a n d then
+ IXI ~ = 4 .
Inserting the resulting expression for x(t) in
do(p) = t = 2Artanh(0(t)) = 2 a r t a n h 109
(J
1-
x(t)e"
)
then yields the required formula for Q,. The fact that ,I~ is a diffeomorphism from [4, oo[ onto [0, oo[, and the statement that ~ maps S onto [4, c~[, can be proved easily. [] Before we come to harmonicity we recall the notion of an isoparamatric function. A differentiable function f : M ---* ]R on a Riemannian manifold M is called isoparametric if IIgradf]l 2 = a o f and z S f = / 3 o f with some continuous functions a,/3. The theory of isoparametric hypersurfaces (see 4.1.10) originates from a result by E. Caftan [Car] stating that a function on a space of constant curvature is isoparametric if and only if its level hypersurfaces have constant principal curvatures. Cartan's result is not true in more general spaces as was shown by Q.M. Wang [Wanl] by providing an isoparametric function on complex projective space whose level hypersurfaces have non-constant principal curvatures. The geometric meaning of ]]grad fll s = a o f is that locally the level hypersurfaces of f are equidistant (see for instance [Wan2] for a proof). Functions satisfying only this condition are also called transnormal. For a transnormal function f the equation /kf = / 3 o f is equivalent to the geometrical property that the level hypersurfaces of f have constant mean curvature (see for instance [Nom]). P r o p o s i t i o n 2 The function f~e is an isoparametric function on S. Proof. As g r a d ( ~ o ~) = ( r
~)grad ~
and A(~I, o ~) = ( ~ " o @)]]grader 2 + (r
o ~)AtIs,
it suffices to prove that ~ is an isoparametric function; and this can be checked without difficulty by using the expression for A in the preceding Lemma. Explicitly, it turns out that IIgrad ~l["
=
A~
=
~2_4~, l+~+m
~-2(m+l).
So the assertion is proved. [] The level hypersurfaces of ~ are the geodesic spheres in S centered at e. The preceding Proposition 2, together with the homogeneity of S, therefore implies that the geodesic spheres in S have constant mean curvature. We now apply characterization (iv) for harmonic spaces in Proposition 1 of 2.6 and eventually obtain Theorem
[DaRil] Every Damek-Ricci space is a harmonic space.
By a look at the table for the dimensions of generalized Heisenberg groups in 3.1.2 it follows that the lowest dimension of a non-symmetric Damek-Ricci space is seven. It is still an open problem whether there exist non-symmetric harmonic spaces in dimensions 5 , 6 , 8 , 9 , 10, 17, 18, 2 6 , . . . 110
and all other higher dimensions in which there are no non-symmetric Damek-Ricci spaces.
4.5
Geometrical
consequences
We will now draw some conclusions from the results obtained in 4.3 and 4.4. According to B.Y. Chen and the third author [ChVa, Theorem 6.22] every non-flat harmonic space is irreducible as a Riemannian manifold. So a consequence of the harmonicity of the Damek-Ricci spaces, and the fact that these spaces are non-flat, is Theorem
1 Every Damek-Ricci space is irreducible as a Riemannian manifold.
Several of the classes of Riemannian manifolds introduced in Section 2 form a subclass of the C-spaces. So, as a consequence of Theorem 1 in 4.3 we obtain T h e o r e m 2 For a Damek-Ricci space S the following statements are equivalent: (i) S is a symmetric space; (ii) S is a naturally reductive Riemannian homogeneous space; (iii) S is a Riemannian g.o. space; (iv) S is a weakly symmetric space; (v) S is a commutative space; (vi) S is a Co-space; (vii) S is a ~E-space; (viii) S is an ~E-space; (ix) S is a globally Osserman space. Using the irreducibilty of Damek-Ricci spaces the equivalence of (i), (ii), (iii) and (v) can also be obtained from more general results about homogeneous Hadamard manifolds. For (iii), and hence also (ii), we refer to [Wo12], and for (v) see the remark at the end of [Woll]. Another consequence of the harmonicity of Damek-Ricci spaces is Theorem
3 Every Damek-Ricci space is
(i) a probabilistic commutative space; (ii) a D'Atri space.
111
So, all the Damek-Ricci spaces belonging to the various classes i n t r o d u c e d in Section 2 have now been completely determined. As consequences of the preceding two theorems we o b t a i n C o r o l l a r y 1 Every non-symmetric Damek-Ricci space is a D'Atri space which is not a E-space. C o r o l l a r y 2 Every non-symmetric Damek-Ricci space is a probabilistic commutative space which is not commutative. Two our knowledge these two corollaries provide the first examples showing t h a t the classes of r and D ' A t r i spaces, and the classes of commutative and probabilistie c o m m u t a t i v e spaces, respectively, do not coincide. It is still an open p r o b l e m whether any C-space is a D ' A t r i space or not. Of great interest is also the question whether there exist compact quotients of Damek-Ricci spaces. In this regard a general result of R. Azencott and E. Wilson [AzWi] implies T h e o r e m 4 A Damek-Ricei space admits a quotient of finite volume if and only if it is a symmetric space. We recall t h a t a Riemannian manifold M is called semi-symmetric if its curvature tensor R satisfies R ( X , Y) 9 R = 0 for all vector fields X, Y on M , where R ( X , Y ) acts on R as a derivation. Clearly, every symmetric space is also semi-symmetric. According to E. Boeckx [Boel], every semi-symmetric R i e m a n n i a n manifold whose Ricci tensor is a Killing tensor is locally symmetric. Now, every Damek-Ricci space is an Einstein manifold and hence its Ricci tensor is obviously a Killing tensor. Thus we derive Theorem metric.
5 A Damek-Ricci space is semi-symmetric if and only if it is sym-
A nice geometrical problem, whose treatment has been developed by B.Y. Chen and the t h i r d a u t h o r in [ChVa], is to study the geometry of R i e m a n n i a n manifolds in terms of properties of their small geodesic spheres. Let us recall t h a t a R i e m a n n i a n manifold M is called curvature-homogeneous if at any two points p, q C M the Riem a n n i a n curvature tensors coincide via a linear isometry from TvM onto TqM. In [BePrVa2] it was proved that a Riemannian manifold, whose small geodesic spheres are curvature-homogeneous, is a harmonic globally Osserman space. As geodesic spheres in the symmetric Damek-Ricci spaces are even homogeneous, this a n d Theorem 2 i m p l y T h e o r e m 6 All geodesic spheres in a Damek-Ricci space S are curvature-homogeneous if and only if S is symmetric. T h e h a r m o n i c i t y of S just means that the geodesic spheres in S have constant m e a n curvature. Assume that the geodesic spheres in S are isoparametric. Then, obviously, the principal curvatures of geodesic spheres in S are the same at a n t i p o d a l points, whence S is an Gr Furthermore, the two-point homogeneity of the
112
symmetric Damek-Ricci spaces implies immediately that their geodesic spheres are isoparametric. Thus we conclude T h e o r e m 7 [TrVa3] All geodesic spheres in a Damek-Ricci space S are isoparametric if and only if S is symmetric. So the distance function f~e (see 4.4) on a non-symmetric Damek-Ricci space provides another example of an isoparametric function whose level hypersurfaces do not have constant principal curvatures. The long-standing conjecture about harmonic spaces, reformulated in terms of geodesic spheres, stated that a Riemannian manifold is locally isometric to a twopoint homogeneous space if and only if all its small geodesic spheres have constant mean curvature. The non-symmetric Damek-Ricci spaces provide counterexamples to this conjecture, but the preceding theorem shows that the principal curvatures of small geodesic spheres in these spaces are non-constant. This motivated the second and third author to fommlate the C o n j e c t u r e [TrVa3] A Riemannian manifold is locally isometric to a two-point homogeneous space if and only if all its small geodesic spheres are isoparametric. As any space, whose small geodesic spheres are isoparametric, is a globally Osserman space [GiSwVa], the conjecture is known to be true in many dimensions and cases (see 2.12 for details). For a further discussion of this conjecture we refer to
[TrVa3]. For a treatment of an intrinsic version of the above conjecture by using the Ricci tensor of the small geodesic spheres instead of their shape operator, see [GiSwVa]. Moreover, Theorem 4.4 also yields that the Damek-Ricci spaces provide counterexamples for other conjectures concerning harmonic spaces, more precisely, concerning k-harmonic spaces (see [TrVa3], [TrVa4], [Van2], [Wil]). Although the Damek-Ricci examples, with their rich and interesting geometry, provide a negative answer to the fundamental conjecture about harmonic manifolds, they are not at the end of a long story. On the contrary, they give rise to many yet unsolved questions and one may hope and believe that the research about these problems will lead to beautiful geometrical results. We finish this section by mentioning just a few of them: - Do there exist harmonic spaces which are not locally symmetric and not locally isometric to a Damek-Ricci space? Find explicit examples. - Are harmonic spaces always locally homogeneous? - Classify the harmonic Hadamard manifolds, homogeneous or not. - Classify the harmonic spaces with strictly negative sectional curvature. - Do there exist non-flat Ricci flat harmonic spaces? - Classify the harmonic spaces and study their geometry in more detail. 113
Some research about these questions has already been started. For example, in [Heb] the author started the study of Hadamard manifolds which are D'Atri spaces and obtained several results relating to the theory about harmonic spaces. See also [BeCoGa] concerning the fourth problem in the compact case.
114
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122
D'Atri space, 13,17,18,19,36,50,74,77, 111,112,114 of dimension three, 13 of dimension four, 14 distance function on Damek-Ricci spaces, 108
Index
q-umbilical hypersurface, 7 Euclidean Laplacian, 11 extrinsic sphere, 87
Alekseevskii space, 96 analyticity of e-spaces, 16 of D'Atri spaces, 13 asymptotically harmonic, 12
first mean value operator, 12 fundamental conjecture about harmonic manifolds, 12 generalized Heisenberg algebra, 22 generalized Heisenberg group, 22 geodesics on Damek-Ricci spaces, 93 on generalized Heisenberg groups, 31 geodesic flow, 16 geodesic spheres, 6,7,8,9,12,13,16, 18,69,112,113 geodesic spray, 16 global coordinates on Damek-Ricci spaces, 82 on generalized Heisenberg groups, 26
basic Jacobi vector field, 16 Bergman metric, 91 e-space, 14,18,19,42,50,105,111 of dimension two, 14 of dimension three, 14 r 16,18,19,50,111 of dimension three, 17 Cayley transform, 91 classical Heisenberg algebra,, 24,79 Clifford algebra, 22 Clifford module, 22-23 commutative space, 10,12,13,15,17, 18,19,35,74,111,112 of dimension < 5, 10 conjugate points on generalized Heisenberg groups, 66 constant mean curvature, 12,110,113 constant scalar curvature of C-spaces, 16 of D'Atri spaces, 13 of geodesic spheres, 12 curvature tensor of Damek-Ricci spaces, 84-85 of generalized Heisenberg groups, 28 curvature-adapted geodesic spheres, 16 curvature-homogeneous manifold, 112
Hadamard manifold, 85,113,114 harmonic function, 11,12 harmonic space, 11,13,36,110,112, 113,114 of dimension < 4, 12 harmonicity of Damek-Ricci spaces, 108 Heisenberg group of dimension three, 6 of dimension five, 7 homogeneous structure, 5,51 horosphere, 6,9,88 horosphere foliation, 88 integrability of subbundles on Damek-Ricci spaces, 87 on generalized Heisenberg groups, 32
Damek-Ricci space, 79
123
irreducibility of Damek-Ricci spaces, 111 of generalized Heisenberg groups, 32 isometry group of Damek-Ricci spaces, 94 of generalized Heisenberg groups, 34 isoparametric function, 110 hypersurface, 88,110 geodesic spheres, 113 Iwasawa decomposition, 25,79
Lie exponential map for Damek-Ricci spaces, 80 for generalized Heisenberg groups, 26 Liouville surface, 14 mean curvature of geodesic spheres, 13 metric tensor in normal coordiantes on generalized Heisenberg groups, 74 ~70-operator, 51 naturally reductive Riemannian homogeneous space, 4,8,9,10, 11,13,14,17,18,19,35,74,111 of dimension two, 5 of dimension three, 6 of dimension four, 6 of dimension five, 7 natural torsion-free connection, 4-5 nearly Kghler structure, 95 on Damek-Ricci spaces, 95
J2-condition, 25,33,39,86,104 Jz-operator, 22 Jacobi equation, 51 Jacobi operator, 14,19 on Damek-Ricci spaces, 85,96 on generalized Heisenberg groups, 29,36 Jacobi operators of higher order, 17 Jacobi vector field, 16 on generalized Heisenberg groups, 52
Osserman conjecture, 19 Osserman space, 19 globally, 19,111,112,113 pointwise, 20,42
k-harmonic spaces, 113 k-symmetric spaces, 4 K-operator, 33 Kiihler structure on Damek-Ricci spaces, 96 on generalized Heisenberg groups, 34 Killing tensor, 14,16,29,112 Killing-t ransversally symmetric space, 8
~l-space, 14,18,44,107 of dimension two, 14 of dimension three, 14-15 Q-symmetric space, 7-8 principal curvatures of geodesic spheres, 18,113 on generalized Heisenberg groups, 69 principal curvature spaces of geodesic spheres, 16 probabilistic commutative space, 10,12,13,111,112 of dimension three, 11
Laplace equation, 11 Laplace operator on Damek-Ricci spaces, 108 Ledger tensor, 15 Levi Civita connection on Damek-Ricci spaces, 84 on generalized Heisenberg groups, 28
quaternionic K~hler structure on Damek-Ricci spaces, 96 quotients of finite volume of Damek-Ricci spaces, 112
124
ray symmetric space, 9 reductive Lie algebra, 4 Ricci tensor of Damek-Ricci spaces, 85 of generalized Heisenberg groups, 28 of geodesic spheres, 113 Ricci flat harmonic spaces, 113 Riemannian foliation, 87 Riemannian g.o. space, 8,10,13,15, 17,18,35,50,74,111 of dimension _< 5, 8 of dimension six, 8 Pdemannian submersion, 87
transnormal function, 110 tubes, 6,9 two-step nilpotent Lie algebra, 23 two-step nilpotent Lie group, 8,26 volume density function, 11 volume-preserving geodesic symmetry, 13 warped product, 14 weakly symmetric space, 9,10,18,19, 36,111 of dimension three, 9 of dimension four, 9
@~-space, 18,50,70,111 of dimension < 3, 19 Sasaki metric, 16 Sasakian space form, 18,19 scalar curvature of Damek-Ricci spaces, 85 of generalized Heisenberg groups, 28 of geodesic spheres, 74 second mean value operator, 10 sectional curvature of Damek-Ricci spaces, 85,104-105 of generalized Heisenberg groups, 29 self-dual Einstein space of dimension four, 20 semi-symmetric manifold, 112 shape operator of geodesic spheres, 16,18 solvable Lie group, 80 special curvature tensor, 17 spherical submanifold, 87 symmetric Damek-Ricci spaces, 86,105,107,111,112,113 symmetric-like Riemannian manifolds, 4 ff~-space, 17,19,50,74,111 of dimension _< 3, 18 ff~-space, 18 Tv-tensor, 45
125
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Vol. 1553: J.-L- Colliot-Th61~ne, J. Kato, P. Vojta. Arithmetic Algebraic Geometry. Trento, 1991. Editor: E. Ballico. VII, 223 pages. 1993.
Vol. 1578: J. Bernstein, V. Lunts, Equivariant Sheaves and Functors. V, 139 pages. 1994.
Vol. 1554: A. K. Lenstra, H. W. Lenstra, Jr. (Eds.), The Development of the Number Field Sieve. VII1, 131 pages. 1993. Vol. 1555: O. Liess, Conical Refraction and Higher Microlocalization. X, 389 pages. 1993. Vol. 1556: S. B. Kuksin, Nearly Integrable InfiniteDimensional Hamiltonian Systems. XXVII, 101 pages. 1993. Vol. 1557: J. Az4ma, P. A. Meyer, M. Yor (Eds.), S6minaire de Probabilit4s XXVII. VI, 327 pages. 1993. Vol. 1558: T. J. Bridges, J. E. Furter, Singularity Theory and Equivariant Symplectic Maps. VI, 226 pages. 1993. Vol. 1559: V. G. Sprindluk, Classical Diophantine Equations. XII, 228 pages. 1993. Vol. 1560: T. Bartsch, Topological Methods for Variational Problems with Symmetries. X, 152 pages. 1993. Vol. 1561 : 1. S. Molchanov, Limit Theorems for Unions of Random Closed Sets. X, 157 pages. 1993. Vol. 1562: G. Harder, Eisensteinkohomologie und die Konstruktion gemischter Motive. XX, 184 pages. 1993. Vol. 1563: E. Fabes, M. Fukushima, L. Gross, C. Kenig, M. ROckner, D. W. Stroock, Dirichtet Forms. Varenna. 1992. Editors: G. Dell'Antonio, U. Mosco. VII, 245 pages. 1993. Vol. 1564: J. Jorgenson, S. Lang, Basic Analysis of Regularized Series and Products. IX, 122 pages. 1993. Vol. 1565: L. Boutet de Monvel, C. De Concini, C. Procesi, P. Schapira, M. Vergne. D-modules, Representation Theory, and Quantum Groups. Venezia, 1992. Editors: G. Zampieri, A. D'Agnolo. VII, 217 pages, i993. Vol. 1566: B. Edixhoven, J.-H. Evertse (Eds.), Diophantine Approximation and Abelian Varieties. Xlll, 127 pages. 1993. Vol. 1567: R. L. Dobrushin, S. Kusuoka, Statistical Mechanics and Fractals. VII, 98 pages. 1993. Vol. 1568: F. Weisz, Martingale Hardy Spaces and their Application in Fourier Analysis. VIII, 217 pages, t994.
Vol. 1576: K. Kitahara, Spaces of Approximating Functions with Haar-Like Conditions. X, 110 pages. 1994.
Vol. 1579: N. Kazamaki, Continuous Exponential Martingales and BMO. VII, 91 pages. 1994. Vol. 1580: M. Milman, Extrapolation and Optimal Decompositions with Applications to Analysis. XI, 161 pages. 1994. Vol. 1581: D. Bakry, R. D. Gill, S. A. Molchanov, Lectures on Probability Theory. Editor: P. Bernard. VIII, 420 pages. 1994. Vol. 1582: W. Balser, From Divergent Power Series to Analytic Functions. X, 108 pages. 1994. Vol. 1583: J. Az6ma, P. A. Meyer, M. Yor (Eds.), S6minaire de Probabilit6s XXVIII. VI, 334 pages. 1994. Vol. 1584: M. Brokate, N. Kenmochi, I. Miiller, J. F. Rodriguez, C. Verdi, Phase Transitions and Hysteresis. Montecatini Terme, 1993. Editor: A. Visintin. VII. 291 pages. 1994. Vol. 1585: G. Frey (Ed.), On Artin's Conjecture for Odd 2dimensional Representations. VIII, 148 pages. 1994. Vol. 1586: R. Nillsen, Difference Spaces and Invariant Linear Forms. XII, 186 pages. 1994. Vol. 1587: N. Xi~ Representations of Affine Hecke Algebras. VIII, 137 pages. 1994. Vol. 1588: C. Scheiderer, Real and l~taIe Cohomology. XXIV, 273 pages. 1994. Vol. 1589: J. Bellissard, M. Degli Esposti, G. Forni, S. Graffi, S. lsola, J. N. Mather, Transition to Chaos in Classical and Quantum Mechanics. Montecatini, 1991. Editor: S. Graffi. VII, 192 pages. 1994. Vol. 1590: P. M. Soardi, Potential Theory on Infinite Networks. Vlll, 187 pages. 1994. Vol. 1591 : M. Abate, G. Patrizio, Finsler Metrics - A Global Approach. IX, 180 pages. 1994. Vol. 1592: K. W. Breitung, Asymptotic Approximations for Probability Integrals. IX, 146 pages. 1994. Vol. 1593: J. Jorgenson & S. Lang, D. Ooldfeld, Explicit Formulas for Regularized Products and Series. VIII, 154 pages. 1994.
Vol. 1569: V. Totik, Weighted Approximation with Varying Weight. VI, 117 pages. 1994.
Vol. 1594: M. Green, J. Murre, C. Voisin, Algebraic Cycles and Hodge Theory. Torino, 1993. Editors: A. Albano, F. Bardelli. Vll, 275 pages. 1994.
Vol. 1570: R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations. XV, 234 pages. 1994.
Vol. 1595: R.D.M. Accola, Topics in the Theory of Riemann Surfaces. IX, 105 pages. 1994.
Vol. 157I: S. Yu. Pilyugin, The Space of Dynamical Systems with the C~ X, 188 pages. 1994.
Vol. 1596: L. HeindorL L. B. Shapiro, Nearly Projective Boolean Algebras. X, 202 pages. 1994.
Vot. 1572: L. G6ttsche, Hilbert Schemes of ZeroDimensional Subscbemes of Smooth Varieties. IX, 196 pages. 1994.
Vol. 1597: B. Herzog, Kodaira-Spencer Maps in Local Algebra. XVII, 176 pages. 1994.
Vol. 1573: V. P. Havin, N. K. Nikolski (Eds.), Linear and Complex Analysis - Problem Book 3 - Part 1. XXII, 489 pages. 1994. Vol. 1574: V. P. Havin, N. K. Nikolski (Eds.), Linear and Complex Analysis - Problem Book 3 - Part II. XXII, 507 pages. 1994.
Vol. 1598: J. Berndt, F. Tricerri, L Vanhecke, Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces. VIII, 125 pages. 1995.