Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces
Generalized Heisenberg groups, or H-type groups, introduced by A. Kaplan, and Damek-Ricci harmonic spaces are particularly nice Lie groups with a vast spectrum of properties and applications. These harmonic spaces are homogeneous Hadamard manifolds contai
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Jiirgen Berndt Franco Tricerri Lieven Vanhecke
Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces
Springer
Authors Jiirgen Berndt Mathematisches Institut Universitat zu Koln Weyertal 86-90 D-50931 Kaln, Germany E-mail: [email protected] Franco Tricerri t formerly: Dipartimento di Matematica "D. Dini" Universita di Firenze Lieven Vanhecke Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200 B B-3001 Leuven, Belgium E-mail: [email protected]
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. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany
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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 Jacobi 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. Szabo 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 a
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