Elie Cartan (1869-1951) PDF
Preview Elie Cartan (1869-1951)
Recent Titles in This Series 123 122 121 M. A. Akivis and B. A. Rosenfeld, Elie Car;tan (1869-1951), 1993 Zhang Guan-Hou, Theory of entire and meromorphic functions: Deficient and asymptotic values and singular directions, 1993 I. B. Fesenko and S. V. Vostokov, Local fields and their extensions: A constructive approach, 1993 120 119 Takeyuki Hida and Masuyuki Hltsuda, M. V. Karasev and V. P. Maslov, Gaussian processes, 199-3 Nonlinear Poisson brackets. Geometry and quantization, 1993 118 117 116 115 Kenkichi lwasawa, Boris Zilber, Algebraic functions, 1993 Uncountably categorical theories, 1993 G. M. Fel'dman, Arithmetic of probability distributions, and characterization problems on abelian groups, 1993 Nikolai V. Ivanov, Subgroups of Teichmiiller modular groups, 1992 114 Seize Ito, Diffusion equations, 1992 113 Michail Zhitomirskii, Typical singularities of differential I-forms and Pfaffian equations, 1992 112 S. A. Lomov, Introduction to the general theory of singular perturbations, 1992 111 Simon Gindikin, Tube domains and the Cauchy problem, 1992 110 B. V. Shabat, Introduction to complex analysis Part II. Functions of several variables, 1992 109 Isao Miyadera, Nonlinear semigroups, 1992 108 Takeo Yokonuma, Tensor spaces and exterior algebra, 1992 107 B. M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov, Selected problems in real analysis, 1992 106 G.-C. Wen, Conformal mappings and boundary value problems, 1992 105 D. R. Yafaev, Mathematical scattering theory : General theory, 1992 104 R. L. Dobrushin, R. Kotecky, and S. Shlosman, Wulff construction: A global shape from local interaction, 1992 103 102 101 A. K. Tsikh, Multidimensional residues and their applications, 1992 A. M. II'in, Matching of asymptotic expansions of solutions of boundary value problems, 1992 Zhang Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Dong Zhen-xi, Qualitative theory of differential equations, 1992 Groups, generators, syzygies, and orbits in invariant theory, 1992 Partial differential operators of elliptic type, 1992 100 V. L. Popov, 99 Norio Shimakura, 98 V. A. Vassiliev, Complements of discriminants of smooth maps: Topology and applications, 1992 Topology of foliations: An introduction, 1992 Introduction to the classical theory of Abelian functions, 1992 Nonlinear partial differential equations of second order, 1991 93 97 ltiro Tamura, 96 A. I. Markushevich, 95 Guangchang Dong, 94 Yu. S. II'yashenko, Finiteness theorems for limit cycles, 1991 A. T. Fomenko and A. A. Tuzhilin, E lements of the geometry and topology of minimal surfaces in three-dimensional space, 1991 92 E. M. Nikishin and V. N. Sorokin, Rational approximations and orthogonality, 1991 91 Mamoru Mimura and Hirosi Toda, Topology of Lie groups, I and II, 1991 90 S. L. Sobolev, Some applications of functional analysis in mathematical physics, third edition, 1991 (Continued in the back of this publication) Elie Cartan (1869-1951) E LIE CARTAN April 9, 1 869-May 6, 1 9 5 1 Translations of MATHEMATICAL MONOGRAPHS Volume 123 Elie Cartan (1869-1951) M.A. Akivis B. A. Rosenfeld 9JIH KAPTAH (1869-1951) M.A. AKHBHC E. A. PoseaclieJihA 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 century, 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) [ E lie Kartan (1869-1951). E nglish] 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 0065-9282; v. 123) 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. Rozenfel1 d, B. A. (Boris Abramovich) II. Title. III. Series 93-6932 QA29.C355A6613 1993 CIP 516.31761092-dc20 Copyright © 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 � idelines 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. 10 9 8 7 6 5 4 3 2 I 97 96 95 94 93 Contents 46 5 1 5 5 60 73 78 82 Preface XI Chapter 1 . The Life and Work of E . Cartan § 1 . 1 . Parents' home § 1 .2. Student at a school and a lycee § 1 . 3. University student § 1 .4. Doctor of Science § 1 . 5. Professor § 1 .6. Academician § 1 . 7. The Cartan family § 1 . 8. Cartan and the mathematicians of the world 1 2 4 6 8 1 7 24 27 Chapter 2. Lie Groups and Algebras §2. 1 . Groups §2.2. Lie group s and Lie algebras §2. 3. Killing's paper §2.4. Cartan's thesis §2.5. Roots of the classical simple Lie groups §2.6. Isomorphisms of complex simple Lie groups §2. 7. Roots of exceptional complex simple Lie groups §2.8. The Cartan matrices §2.9. The Weyl groups §2. 1 0. The Weyl affine groups §2. 1 1 . Associative and alternative algebras §2. 1 2. Cartan's works on algebras §2. 1 3. Linear representations of simple Lie groups §2. 1 4. Real simple Lie groups §2. 1 5. Isomorphisms of real simple Lie groups §2. 1 6. Reductive and quasireductive Lie groups §2. 1 7. Simple Chevalley groups §2. 1 8. Quasigroups and loops 33 33 37 42 vii viii CONTENTS Chapter 3. Projective Spaces and Projective Metrics § 3 . 1 . Real spaces §3.2. Complex spaces §3.3. Quaternion spaces § 3.4. Octave planes § 3 .5. Degenerate geometries §3.6. Equivalent geometries § 3 . 7 . Multidimensional generalizations o f the Hesse transfer principle §3.8. Fundamental elements §3.9. The duality and triality principles §3. 1 0. Spaces over algebras with zero divisors § 3. 1 1 . Spaces over tensor products of algebras § 3 . 1 2. Degenerate geometries over algebras §3. 1 3. Finite geometries Chapter 4. Lie Pseudogroups and Pfaffi.an Equations §4. 1 . Lie pseudogroups §4.2. The Kac-Moody algebras §4. 3 . Pfaffi.an equations §4.4. Completely integrable Pfaffi.an systems §4. 5 . Pfaffi.an systems in involution §4.6. The algebra of exterior forms §4. 7. Application of the theory of systems in involution §4. 8 . Multiple integrals, integral invariants, and integral geometry §4.9. Differential forms and the Betti numbers §4. 1 0. New methods in the theory of partial differential equations Chapter 5. §5. 1 . §5.2. §5.3. § 5.4. §5.5. §5.6. § 5 .7. §5.8. §5.9. §5. 1 0. § 5. 1 1 . The Method of Moving Frames and Differential Geometry Moving trihedra of Frenet and Darboux Moving tetrahedra and pentaspheres of Demoulin Cartan's moving frames The derivational formulas The structure equations Applications of the method of moving frames Some geometric examples Multidimensional manifolds in Euclidean space Minimal manifolds "Isotropic surfaces" Deformation and projective theory of multidimensional manifolds 87 8 7 93 9 5 96 97 1 0 1 1 07 1 09 1 1 3 1 1 6 1 1 8 1 2 1 1 23 1 25 1 25 1 27 1 29 1 30 1 32 1 34 1 3 5 1 36 1 39 1 42 1 45 1 45 1 47 1 48 1 50 1 52 1 53 1 54 1 58 1 60 1 62 1 66 CONTENTS § 5 . 1 2. § 5. 1 3. Invariant normalization of manifolds "Pseudo-conformal geometry of hypersurfaces" Chapter 6. Riemannian Manifolds. Symmetric Spaces §6. 1 . Riemannian manifolds §6.2. Pseudo-Riemannian manifolds §6.3. Parallel displacement of vectors §6.4. Riemannian geometry in an orthogonal frame §6.5. The problem of embedding a Riemannian manifold into a Euclidean space §6.6. Riemannian manifolds satisfying "the axiom of plane" §6.7. Symmetric Riemannian spaces §6.8. Hermitian spaces as symmetric spaces §6.9. Elements of symmetry §6. 1 0. The isotropy groups and orbits § 6. 1 1 . Absolutes of symmetric spaces §6. 1 2. Geometry of the Cartan subgroups §6. 1 3. The Cartan submanifolds of symmetric spaces §6. 1 4. Antipodal manifolds of symmetric spaces §6. 1 5. Orthogonal systems of functions on symmetric spaces §6. 1 6. Unitary representations of noncompact Lie groups §6. 1 7. The topology of symmetric spaces §6. 1 8. Homological algebra Chapter 7. Generalized Spaces §7. 1 . "Affine connections" and Weyl's "metric manifolds" §7.2. Spaces with affine connection §7.3. Spaces with a Euclidean, isotropic, and metric connection §7 .4. Affine connections in Lie groups and symmetric spaces with an affine connection §7.5. Spaces with a projective connection §7.6. Spaces with a conformal con nection §7.7. Spaces with a symplectic connection §7.8. The relativity theory and the unified field theory §7.9. Finsler spaces §7. 1 0. Metric spaces based on the notion of area § 7. 1 1 . Generalized spaces over algebras §7. 1 2. The equivalence problem and G-structures §7. 1 3. Multidimensional webs ix 1 70 1 74 1 77 1 77 1 8 1 1 8 1 1 83 1 84 1 8 5 1 86 1 9 1 1 93 1 96 1 98 1 99 200 201 202 204 207 209 2 1 1 2 1 1 2 1 2 2 1 5 2 1 6 2 1 9 220 22 1 222 223 225 226 228 23 1 Conclusion 235 Dates of Cartan's Life and Activities 239 List of Publications of Elie Cartan 241
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