SPACESOF CONSTANT CURVATURE

SIXTH EDITION

JOSEPH A. WOLF

AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island

chel-372-wolf2-cov.indd 1 10/19/10 1:35 PM

SpaceS of conStant curvature

Sixth edition

SpaceS of conStant curvature

Sixth edition

JoSeph a. Wolf

AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island

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

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.

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2010035675

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1

PREFACE TO THE SIXTH EDITION

Since publication of the fifth (1984) edition of this book there has been atremendous amount of activity in discrete subgroups of Lie groups and algebraicgroups. This activity had emphasis in several areas, especially differential ge-ometry, harmonic analysis, algebraic geometry and number theory. It also hadapplications via Fourier transform theory to signal processing and other areas.

Most of the new material in this sixth (2010) edition represents an attempt toindicate some of these developments. Much of this is done in Chapter 3 and inthe Appendix to Chapter 12. Chapter 3 has some new results and an indication ofupdates in the section on flat homogeneous pseudo–riemannian manifolds. TheAppendix to Chapter 12 sketches some background and a brief description (some-times just consisting of current references) of the more recent work on discretesubgroups of real Lie groups. There the emphasis is on application to pseudo–riemannian geometry and pseudo–riemannian quotient manifolds, including ofcourse the riemannian case. There has also been an enormous amount of work onspaces of functions on those quotients, but that is well beyond the scope of thisbook.

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 changein the Remark on page 170. I was tempted to modernize the finite group theoryin 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 notto do it. Finally, my thanks to Hillel Furstenberg, David Kazhdan and ToshiyukiKobayashi for updates and references in the Appendix to Chapter 12.

In this new material, note that citations not in the 1986 “References” sectionare in “Additional References” just after.

As ever, special thanks are due to my wife Lois for her support while I waspreparing this new edition.

Berkeley, July 2010 J. A. W.

x

Prefaces ii, x v..

Appendix to Chapter 12 . . . . 396.

References . . . . . . . 402Additional References . . . . . 408

. . . . . . . 413 Index

402

403

404

405

406

407

ADDITIONAL REFERENCES

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 ap-pear in “Handbook of pseudo-Riemannian geometry and supersymmetry”, ed. by V.Cortes, IRMA Lectures in Math. and Theoretical Physics 16, European Math. SocietyPublishing House, Zurich, 2010. Archive: arXiv 0809.24v1 [mathDG] 4 September2008.

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

ADDITIONAL REFERENCES 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 homoge-neous spaces of reductive type, Duke Math. J., vol. 67, (1992), pp.653-664.

[3] On discontinuous groups on homogeneous spaces with noncompact isotropy sub-groups, 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 homogeneousmanifolds, 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 homogeneousmanifolds, Math. Ann., vol. 310 (1998), pp. 394408.

[7] with T. YOSHINO, Compact Clifford–Klein forms of symmetric spaces revisited, Pureand Appl. Math. Quarterly, vol. 1 (2005), pp. 591–663.

[8] On discontinuous group actions on non–riemannian homogeneous spaces, SugakuExpositions, 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 Lobacevskiı 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 than1, Invent. Math., vol. 76 (1984), pp. 93–120.

N. MOK

[1] Aspects of Kahler geometry on arithmetic varieties, Proc. Symposia Pure Math., vol52 (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

A. SELBERG

[1] On discontinuous groups in higher–dimensional symmetric spaces, in “Contributionsto Function Theory,” Tata Institute, Bombay, 1960, pp. 147–164.

E. B. VINBERG

[1] Discrete groups generated by reflections in Lobacevskiı 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 DifferentialGeometry, 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 ashttp://www.mfo.de/programme/schedule/2007/42/OWR 2007 49.pdf.

ADDITIONAL REFERENCES

413

414

415

416

417

418

419

420

CHEL/372.H

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