Élie Cartan (1869-1951)

334 Pages·1993·16.17 MB·English

by  M. A. Akivis, B. A. Rosenfeld

Translations of
MATH EMATICAL
MONO G "PHS
Volume 1.23
Elle Cartan.
[1869-1951)
M. A. Akivis
B, A. Rosenfeld
American Mathematical society'
Translations of Mathematical Monographs 123
Elie Cartan
(1869-1951)
ELIE CARTAN
April 9, 1869-May 6, 1951
Translations of
MATHEMATICAL
MONOGRAPHS
Volume 123
Elie Cartan
(1869-1951)
M. A. Akivis
B. A. Rosenfeld
American Mathematical Society
IV Providence, Rhode Island
3JIH KAPTAH (1869-1951)
M. A. AICHBHc
E. A. Po3eH$eni6
Translated by V. V. Goldberg from an original Russian manuscript
Translation edited by Simeon Ivanov
1991 Mathematics Subject Classification. Primary 01A70;
Secondary 01A60, 01A55.
ABSTRACT. The scientific biography of one of the greatest mathematicians of the 20th cen-
tury, Elie Cartan (1869-1951), is presented, as well as the development of Cartan's ideas by
mathematicians of the following generations.
Photo credits: p. iv-Centre National de la Recherche Scientifique; pp. 2, 3, 9, 10, 17, 19, 25,
27, 28, 29-Henri Cartan; p. 31-Department of Geometry, Kazan University, Tatarstan, Russia
Library of Congress Cataloging-in-Publication Data
Akivis, M. A. (Maks Aizikovich)
[Elie Kartan (1869-1951). English]
Elie Cartan (1869-1951)/M. A. Akivis, B. A. Rosenfeld; [translated from the Russian by
V. V. Goldberg; translation edited by Simeon Ivanov].
p. cm.-(Translations of mathematical monographs, ISSN 0065-92 82; v. 12 3)
Includes bibliographical references.
ISBN 0-8218-4587-X (acid-free)
1. Cartan, Elie, 1869-1951. 2. Mathematicians-France-Biography. 3. Lie groups.
4. Geometry, Differential. I. Rozenfel'd, B. A. (Boris Abramovich) II. Title. III. Series
QA29.C355A6613 1993 93-6932
5 16.3' 76' 092-dc20 CIP
Copyright Q 1993 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 uide1ines
established to ensure permanence and durability.
Information on Copying and Reprinting can be found at the back of this volume.
This publication was typeset using AMS-TEX,
the American Mathematical Society's TEX macro system.
109876 5432 1 9796959493
Contents
Preface xi
Chapter 1. The Life and Work of E. Cartan
§ 1.1. Parents' home
§ 1.2. Student at a school and a lycee 2
§ 1.3. University student 4
§ 1.4. Doctor of Science 6
§ 1.5. Professor 8
§ 1.6. Academician 17
§ 1.7. The Cartan family 24
§ 1.8. Cartan and the mathematicians of the world 27
Chapter 2. Lie Groups and Algebras 33
§2.1. Groups 33
§2.2. Lie groups and Lie algebras 37
§2.3. Killing's paper 42
§2.4. Cartan's thesis 45
§2.5. Roots of the classical simple Lie groups 46
§2.6. Isomorphisms of complex simple Lie groups 51
§2.7. Roots of exceptional complex simple Lie groups 51
§2.8. The Cartan matrices 53
§2.9. The Weyl groups 55
§2.10. The Weyl affine groups 60
§2.11. Associative and alternative algebras 63
§2.12. Cartan's works on algebras 67
§2.13. Linear representations of simple Lie groups 69
§2.14. Real simple Lie groups 73
§2.15. Isomorphisms of real simple Lie groups 78
§2.16. Reductive and quasireductive Lie groups 82
§2.17. Simple Chevalley groups 84
§2.18. Quasigroups and loops 85
vii
Viii CONTENTS
Chapter 3. Projective Spaces and Projective Metrics 87
§3. 1. Real spaces 87
§3.2. Complex spaces 93
§3.3. Quaternion spaces 95
§3.4. Octave planes 96
§3.5. Degenerate geometries 97
§3.6. Equivalent geometries 101
§3.7. Multidimensional generalizations of the Hesse transfer
principle 1 07
§3.8. Fundamental elements 1 09
§3.9. The duality and triality principles 113
§3.1 0. Spaces over algebras with zero divisors 116
§3.11. Spaces over tensor products of algebras 118
§3.12. Degenerate geometries over algebras 121
§3.13. Finite geometries 123
Chapter 4. Lie Pseudogroups and Pfaffian Equations 125
§4.1 . Lie pseudogroups 125
§4.2. The Kac-Moody algebras 127
§4.3. Pfaffian equations 129
§4.4. Completely integrable Pfaffian systems 1 30
§4.5. Pfaffian systems in involution 1 32
§4.6. The algebra of exterior forms 134
§4.7. Application of the theory of systems in involution 135
§4.8. Multiple integrals, integral invariants, and integral
geometry 136
§4.9. Differential forms and the Betti numbers 139
§4.10. New methods in the theory of partial differential equations 142
Chapter 5. The Method of Moving Frames and Differential
Geometry 145
§5.1. Moving trihedra of Frenet and Darboux 145
§5.2. Moving tetrahedra and pentaspheres of Demoulin 147
§5.3. Cartan's moving frames 148
§5.4. The derivational formulas 150
§5.5. The structure equations 152
§5.6. Applications of the method of moving frames 153
§5.7. Some geometric examples 1 54
§5.8. Multidimensional manifolds in Euclidean space 158
§5.9. Minimal manifolds 160
§5.10. "Isotropic surfaces" 162
§5.11. Deformation and projective theory of multidimensional
manifolds 1 66
CONTENTS ix
§5.12. Invariant normalization of manifolds 170
§5.13. "Pseudo-conformal geometry of hypersurfaces" 174
Chapter 6. Riemannian Manifolds. Symmetric Spaces 177
§6.1. Riemannian manifolds 177
§6.2. Pseudo-Riemannian manifolds 181
§6.3. Parallel displacement of vectors 181
§6.4. Riemannian geometry in an orthogonal frame 183
§6.5. The problem of embedding a Riemannian manifold into a
Euclidean space 184
§6.6. Riemannian manifolds satisfying "the axiom of plane" 185
§6.7. Symmetric Riemannian spaces 186
§6.8. Hermitian spaces as symmetric spaces 191
§6.9. Elements of symmetry 193
§6.10. The isotropy groups and orbits 196
§6.11. Absolutes of symmetric spaces 198
§6.12. Geometry of the Cartan subgroups 199
§6.13. The Cartan submanifolds of symmetric spaces 200
§6.14. Antipodal manifolds of symmetric spaces 201
§6.15. Orthogonal systems of functions on symmetric spaces 202
§6.16. Unitary representations of noncompact Lie groups 204
§6.17. The topology of symmetric spaces 207
§6.18. Homological algebra 209
Chapter 7. Generalized Spaces 211
§ 7.1. "Affine connections" and Weyl's "metric manifolds" 211
§7.2. Spaces with affine connection 212
§7.3. Spaces with a Euclidean, isotropic, and metric connection 215
§7.4. Affine connections in Lie groups and symmetric spaces
with an affine connection 216
§7.5. Spaces with a projective connection 219
§7.6. Spaces with a conformal connection 220
§7.7. Spaces with a symplectic connection 221
§7.8. The relativity theory and the unified field theory 222
§7.9. Finsler spaces 223
§7.10. Metric spaces based on the notion of area 225
§7.11. Generalized spaces over algebras 226
§7.12. The equivalence problem and G-structures 228
§7.13. Multidimensional webs 231
Conclusion 235
Dates of Cartan's Life and Activities 239
List of Publications of Elie Cartan 241