SPACES CURVATURE OF CONSTANT JOSEPH A. WOLF SIXTH EDITION. AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island

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1 SPACES OF CONSTANT CURVATURE SIXTH EDITION JOSEPH A. WOLF AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island

2 SpaceS of constant curvature Sixth edition

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4 SpaceS of constant curvature Sixth edition JoSeph a. Wolf AMS CHELSEA PUBLISHING American Mathematical Society Providence, Rhode Island AMERICAN MATHEMATICAL ΑΓΕΩΜΕ ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ SOCIETY FOUNDED 1888

5 2000 Mathematics Subject Classification. Primary 53 02, 53C21, 53C30, 53C35, 53C50, 20C05, 22C05; Secondary 14L35, 17B45, 20D99. For additional information and updates on this book, visit Library of Congress Cataloging-in-Publication Data Wolf, Joseph Albert, 1936 Spaces of constant curvature / Joseph A. Wolf. 6th ed. p. cm. Includes bibliographical references and index. ISBN (alk. paper) 1. Spaces of constant curvature. 2. Geometry, Riemannian. 3. Riemannian manifolds. 4. Symmetric spaces. I. Title. QA649.W dc Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island USA. Requests can also be made by to reprint-permission@ams.org. c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at

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11 PREFACE TO THE SIXTH EDITION Since publication of the fifth (1984) edition of this book there has been a tremendous amount of activity in discrete subgroups of Lie groups and algebraic groups. This activity had emphasis in several areas, especially differential geometry, harmonic analysis, algebraic geometry and number theory. It also had applications via Fourier transform theory to signal processing and other areas. Most of the new material in this sixth (2010) edition represents an attempt to indicate some of these developments. Much of this is done in Chapter 3 and in the Appendix to Chapter 12. Chapter 3 has some new results and an indication of updates in the section on flat homogeneous pseudo riemannian manifolds. The Appendix to Chapter 12 sketches some background and a brief description (sometimes just consisting of current references) of the more recent work on discrete subgroups of real Lie groups. There the emphasis is on application to pseudo riemannian geometry and pseudo riemannian quotient manifolds, including of course the riemannian case. There has also been an enormous amount of work on spaces of functions on those quotients, but that is well beyond the scope of this book. I thank Oliver Baues for his generous advice and updates concerning the revision of Chapter 3. Thanks also to Jonathan Wahl in connection with the change in the Remark on page 170. I was tempted to modernize the finite group theory in Chapter 6, but that could have made it inaccessible to many differential geometers, and I thank James Milgram and C. T. C. (Terry) Wall for convincing me not to do it. Finally, my thanks to Hillel Furstenberg, David Kazhdan and Toshiyuki Kobayashi for updates and references in the Appendix to Chapter 12. In this new material, note that citations not in the 1986 References section are in Additional References just after. As ever, special thanks are due to my wife Lois for her support while I was preparing this new edition. Berkeley, July 2010 J. A. W. x

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14 Prefaces.. vii, x

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16 Appendix to Chapter References Additional References Index

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26 ADDITIONAL REFERENCES L. AUSLANDER [3] Discrete uniform subgroups of solvable Lie groups, Trans. Amer. Math. Society, vol. 99 (1961) pp A. BAKLOUTI [1] Deformation of discontinuous subgroups acting on some nilpotent homogeneous spaces, Proc. Japan Acad. Math. Sci., vol. 85 (2009), pp [2] WITH I. KÉDEM On the deformation space of Clifford Klein forms of some exponential homogeneous spaces, Internat. J. Math., vol. 20 (2009), pp O. BAUES [1] Prehomogeneous affine representations and flat pseudo riemannian manifolds, to appear in Handbook of pseudo-riemannian geometry and supersymmetry, ed. by V. Cortés, IRMA Lectures in Math. and Theoretical Physics 16, European Math. Society Publishing House, Zürich, Archive: arxiv v1 [mathdg] 4 September A. BOREL [4] Les fonctions automorphes de plusieurs variables complexes, Bull. Soc. Math. France, vol. 80, (1952). pp [5] with HARISH-CHANDRA, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) vol. 75 (1962), pp [6] Compact Clifford-Klein forms of symmetric space, Topology, vol. 2 (1963), pp K. CORLETTE [1] Archimedean superrigidity and hyperbolic geometry, Annals Math., vol. 135 (1992), pp D. C. DUNCAN & E. C. IHRIG [1] Homogeneous spacetimes of zero curvature, Proceedings American Math. Society, vol. 107 (1989), pp [2] Flat pseudo riemannian manifolds with a nilpotent transitive group of isometries, Annals of Global Analysis and Geometry, vol. 10 (1992), pp [3] Translationally isotropic flat homogeneous manifolds with metric signature.n; 2/, Annals of Global Analysis and Geometry, vol. 11 (1993), pp D. FRIED, W. M. GOLDMAN & M. W. HIRSCH [1] Affine manifolds with nilpotent holonomy, Commentarii Math. Helvetici, vol. 56 (1981), pp W. M. GOLDMAN & M. W. HIRSCH [1] Affine manifolds and orbits of algebraic groups, Transactions Amer. Math. Society, vol. 295 (1986), pp

27 ADDITIONAL REFERENCES 409 M. GROMOV & I. PIATETSKI SHAPIRO [1] Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math., vol. 66 (1988), pp F. KASSEL [1] Quotients compacts d espaces homogènes réels ou p-adiques, thèse, l Université Paris Sud XI, November [2] Deformation of proper actions on reductive homogeneous spaces, to appear, arxiv: v1, November T. KOBAYASHI [1] Proper action on a homogeneous space of reductive type, Math. Ann., vol. 285 (1989), pp [2] A necessary condition for the existence of compact Clifford Klein forms of homogeneous spaces of reductive type, Duke Math. J., vol. 67, (1992), pp [3] On discontinuous groups on homogeneous spaces with noncompact isotropy subgroups, J. Geometry and Physics, vol. 12 (1993), pp [4] Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie Theory, vol. 6 (1996), pp [5] Discontinuous groups and Clifford Klein forms of pseudo riemannian homogeneous manifolds, in Algebraic and Analytic Methods in Representation Theory, Schlichtkrull & Ørsted eds., Academic Press (1996), pp [6] Deformation of compact Clifford Klein forms of indefinite Riemannian homogeneous manifolds, Math. Ann., vol. 310 (1998), pp [7] with T. YOSHINO, Compact Clifford Klein forms of symmetric spaces revisited, Pure and Appl. Math. Quarterly, vol. 1 (2005), pp [8] On discontinuous group actions on non riemannian homogeneous spaces, Sugaku Expositions, vol. 22 (2009), Amer. Math. Soc., pp R. L. LIPSMAN [1] Proper actions and a compactness condition, J. Lie Theory, vol. 5 (1995), pp V. S. MAKAROV [1] On a certain class of discrete groups of Lobačevskiǐ space having an infinite fundamental region of finite measure (in Russian), Dokl. Akad. Nauk SSSR, vol. 167 (1966), pp G. S. MARGULIS [1] Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1, Invent. Math., vol. 76 (1984), pp N. MOK [1] Aspects of Kähler geometry on arithmetic varieties, Proc. Symposia Pure Math., vol 52 (1991), Part 2, pp G. D. MOSTOW [1] Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies, vol. 78, Princeton Unov. Press, [2] On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math., vol. 86 (1980), pp [3] Discrete subgroups of Lie groups, in Élie Cartan et les Mathématiques d ajourd hui, Astérisque, hors série (1985), pp

28 410 ADDITIONAL REFERENCES A. SELBERG [1] On discontinuous groups in higher dimensional symmetric spaces, in Contributions to Function Theory, Tata Institute, Bombay, 1960, pp E. B. VINBERG [1] Discrete groups generated by reflections in Lobačevskiǐ spaces (in Russian), Mat. Sbornik, vol. 72 (1967), pp ; correction, ibid., vol. 73 (1967), p J. A. WOLF [17] On the geometry and classification of absolute parallelisms, I, Journal of Differential Geometry, vol. 6 (1972), pp [18] Flat homogeneous pseudo-riemannian manifolds, Geometriae Dedicata, vol. 57 (1995), pp T. YOSHINO [1] On Lipsmans Conjecture, in Harmonische Analysis und Darstellungstheorie Topologischer Gruppen, Mathematisches Forschungsinstitut Oberwolfach Report No. 49/2007, 2007, pp Available as pdf.

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