Spaces of Constant Curvature

Home , Constant curvature

SPACES OF CONSTANT CURVATURE

SIXTH EDITION

JOSEPH A. WOLF

AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island

chel-372-wolf2-cov.indd 1 10/19/10 1:35 PM SpaceS of conStant curvature

Sixth edition

http://dx.doi.org/10.1090/chel/372.H

SpaceS of conStant curvature

Sixth edition

JoSeph a. Wolf

AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island

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F O 8 U 88 NDED 1 2000 Mathematics Subject Classiﬁcation. Primary 53–02, 53C21, 53C30, 53C35, 53C50, 20C05, 22C05; Secondary 14L35, 17B45, 20D99.

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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 978-0-8218-5282-8 (alk. paper) 1. Spaces of constant curvature. 2. Geometry, Riemannian. 3. Riemannian manifolds. 4. Symmetric spaces. I. Title.

QA649.W64 2010 516.373—dc22 2010035675

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PREFACETOTHESIXTHEDITION

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 (some- times 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 revi- sion 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 geome- ters, 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.

Prefaces . . vii, x

Appendix to Chapter 12 . . . . . 396 References ...... 402 Additional References . . . . . 408 Index ...... 413

402 403 404 405 406 407 ADDITIONALREFERENCES

L.AUSLANDER [3] Discrete uniform subgroups of solvable Lie groups, Trans. Amer. Math. Society, vol. 99 (1961) pp. 398–402. A.BAKLOUTI [1] Deformation of discontinuous subgroups acting on some nilpotent homogeneous spaces, Proc. Japan Acad. Math. Sci., vol. 85 (2009), pp. 41–45. [2] WITH I.KEDEM´ On the deformation space of Clifford–Klein forms of some exponen- tial homogeneous spaces, Internat. J. Math., vol. 20 (2009), pp. 817–839. O.BAUES [1] Prehomogeneous affine representations and flat pseudo–riemannian manifolds, to appear in “Handbook of pseudo-Riemannian geometry and supersymmetry”, ed. by V. Cortes,´ IRMA Lectures in Math. and Theoretical Physics 16, European Math. Society Publishing House, Zurich,¨ 2010. Archive: arXiv 0809.24v1 [mathDG] 4 September 2008. A.BOREL [4] Les fonctions automorphes de plusieurs variables complexes, Bull. Soc. Math. France, vol. 80, (1952). pp. 167–182. [5] with HARISH-CHANDRA, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) vol. 75 (1962), pp. 485–535. [6] Compact Clifford-Klein forms of symmetric space, Topology, vol. 2 (1963), pp. 111– 122. K.CORLETTE [1] Archimedean superrigidity and hyperbolic geometry, Annals Math., vol. 135 (1992), pp. 165–182. D.C.DUNCAN &E.C.IHRIG [1] Homogeneous spacetimes of zero curvature, Proceedings American Math. Society, vol. 107 (1989), pp. 785–795. [2] Flat pseudo–riemannian manifolds with a nilpotent transitive group of isometries, Annals of Global Analysis and Geometry, vol. 10 (1992), pp. 87–101. [3] Translationally isotropic flat homogeneous manifolds with metric signature .n; 2/, An- nals of Global Analysis and Geometry, vol. 11 (1993), pp. 3–24. D.FRIED, W. M. GOLDMAN & M. W. HIRSCH [1] Affine manifolds with nilpotent holonomy, Commentarii Math. Helvetici, vol. 56 (1981), pp. 487–523. W. M. GOLDMAN & M. W. HIRSCH [1] Affine manifolds and orbits of algebraic groups, Transactions Amer. Math. Society, vol. 295 (1986), pp. 175–198.

408 ADDITIONALREFERENCES 409

M.GROMOV &I.PIATETSKI–SHAPIRO [1] Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Etudes´ Sci. Publ. Math., vol. 66 (1988), pp. 93-103 F. KASSEL [1] Quotients compacts d’espaces homogenes` reels´ ou p-adiques, these,` l’Universite´ Paris– Sud XI, November 2009. [2] Deformation of proper actions on reductive homogeneous spaces, to appear, arXiv:0911.4247v1, November 2009. T. KOBAYASHI [1] Proper action on a homogeneous space of reductive type, Math. Ann., vol. 285 (1989), pp. 249–263. [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.653-664. [3] On discontinuous groups on homogeneous spaces with noncompact isotropy subgroups, J. Geometry and Physics, vol. 12 (1993), pp. 133-44. [4] Criterion for proper actions on homogeneous spaces of reductive groups, J. Lie The- ory, vol. 6 (1996), pp.147-163. [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. 99–165. [6] Deformation of compact Clifford–Klein forms of indefinite Riemannian homogeneous manifolds, Math. Ann., vol. 310 (1998), pp. 394408. [7] with T. YOSHINO, Compact Clifford–Klein forms of symmetric spaces revisited, Pure and Appl. Math. Quarterly, vol. 1 (2005), pp. 591–663. [8] On discontinuous group actions on non–riemannian homogeneous spaces, Sugaku Expositions, vol. 22 (2009), Amer. Math. Soc., pp. 1–19. R.L.LIPSMAN [1] Proper actions and a compactness condition, J. Lie Theory, vol. 5 (1995), pp. 25-39. V. S. MAKAROV [1] On a certain class of discrete groups of Lobacevskiˇ ˇı space having an infinite funda- mental region of finite measure (in Russian), Dokl. Akad. Nauk SSSR, vol. 167 (1966), pp. 30–33. G.S.MARGULIS [1] Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1, Invent. Math., vol. 76 (1984), pp. 93–120. N.MOK [1] Aspects of Kahler¨ geometry on arithmetic varieties, Proc. Symposia Pure Math., vol 52 (1991), Part 2, pp. 335–396. G.D.MOSTOW [1] “Strong Rigidity of Locally Symmetric Spaces”, Annals of Math. Studies, vol. 78, Princeton Unov. Press, 1973. [2] On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math., vol. 86 (1980), pp. 171–276. [3] Discrete subgroups of Lie groups, in “Elie´ Cartan et les Mathematiques´ d’ajourd’hui”, Asterisque,´ hors serie´ (1985), pp. 328–331. 410 ADDITIONALREFERENCES

A.SELBERG [1] On discontinuous groups in higher–dimensional symmetric spaces, in “Contributions to Function Theory,” Tata Institute, Bombay, 1960, pp. 147–164. E.B.VINBERG [1] Discrete groups generated by reﬂections in Lobacevskiˇ ˇı spaces (in Russian), Mat. Sbornik, vol. 72 (1967), pp. 471–488; correction, ibid., vol. 73 (1967), p. 303. J.A.WOLF [17] On the geometry and classiﬁcation of absolute parallelisms, I, Journal of Differential Geometry, vol. 6 (1972), pp. 317–342. [18] Flat homogeneous pseudo-riemannian manifolds, Geometriae Dedicata, vol. 57 (1995), pp. 111–120. T. YOSHINO [1] On Lipsmans Conjecture, in “Harmonische Analysis und Darstellungstheorie Topolo- gischer Gruppen”, Mathematisches Forschungsinstitut Oberwolfach Report No. 49/2007, 2007, pp. 2917–2919. Available as http://www.mfo.de/programme/schedule/2007/42/OWR 2007 49.pdf.

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CHEL/372.H

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