This book describes the life and achievements of the great French mathematician, Elie Cartan. Here readers will find detailed descriptions of Cartan's discoveries in Lie groups and algebras, associative algebras, differential equations, and differential geometry, as well of later developments stemming from his ideas. There is also a biographical sketch of Cartan's life. A monumental tribute to a towering figure in the history of mathematics, this book will appeal to mathematicians and historians alike.
Readership: Graduate students, mathematicians, and historians.
Author(s): M. A. Akivis and B. A. Rosenfeld
Series: Translations of Mathematical Monographs, MMONO/123
Publisher: American Mathematical Society
Year: 1993
Language: English
Pages: C+xii+317+B
Front Matter
Cover
Translations of Mathematical Monographs 123
S Title
Photo of ELIE CARTAN
Elie Cartan (1869-1951)
Copyright (C) 1993 by the American Mathematical Society.
ISBN 0-8218-4587-X
ISBN 978-0-8218-4587-5
QA29.C355A6613 1993 16.3'76' 092-dc20
LCCN 93-6932 CIP
Contents
Preface
CHAPTER1 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
The CARTAN FAMILY
§1.8. Cartan and the mathematicians of the world
CHAPTER 2 Lie Groups and Algebras
§2.1. Groups
§2.2. Lie groups 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
§2.10. The Weyl affine groups
§2.11. Associative and alternative algebras
§2.12. Cartan's works on algebras
§2.13. Linear representations of simple Lie groups
§2.14. Real simple Lie groups
§2.15. Isomorphisms of real simple Lie groups
§2.16. Reductive and quasireductive Lie groups
§2.17. Simple Chevalley groups
§2.18. Quasigroups and loops
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 of the Hesse transfer principle
§3.8. Fundamental elements
§3.9. The duality and triality principles
§3.10. Spaces over algebras with zero divisors
§3.11. Spaces over tensor products of algebras
§3.12. Degenerate geometries over algebras
§3.13. Finite geometries
CHAPTER 4 Lie Pseudogroups and Pfaffian Equations
§4.1. Lie pseudogroups
§4.2. The Kac-Moody algebras
§4.3. Pfaflian equations
§4.4. Completely integrable Pfaffian systems
§4.5. Pfaffian 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
§4.10. New methods in the theory of partial differential equations
CHAPTER 5 The Method of Moving Frames and Differential Geometry
§5.1. Moving trihedra of Frenet and Darboux
§5.2. Moving tetrahedra and pentaspheres of Demoulin
§5.3. Cartan's moving frames
§5.4. The derivational formulas
§5.5. The structure equations
§5.6. Applications of the method of moving frames
§5.7. Some geometric examples
§5.8. Multidimensional manifolds in Euclidean space
§5.9. Minimal manifolds
§5.10, "Isotropic surfaces"
§5.11. Deformation and projective theory of multidimensional manifolds
§5.12. Invariant normalization of manifolds
§5.13. "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 manifoldin to 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
§6.10. The isotropy groups and orbits
§6.11. Absolutes of symmetric spaces
§6.12. Geometry of the Cartan subgroups
§6.13. The Cartan submanifolds of symmetric spaces
§6.14. Antipodal manifolds of symmetric spaces
§6.15. Orthogonal systems of functions on symmetric spaces
§6.16. Unitary representations of noncompact Lie groups
§6.17. The topology of symmetric spaces
§6.18. Homological algebra
CHAPTER 7 Generalized Spaces
§7.1. "Affine connections" and Weyl's "metric manifolds"
§7.2. Spaces with af'ine connection
§7.3. Spaces with a Euclidean, isotropic, and metric connection
§7.4. Afllne connections in Lie groups and symmetric spaces with an af'ine connection
§7.5. Spaces with a projective connection
§7.6. Spaces with a conformal connection
§7.7. Spaces with a symplectic connection
§7.8. The relativity theory and the unified field theory
§7.9. Finsler spaces
§7.10. Metric spaces based on the notion of area
§7.11. Generalized spaces over algebras
§7.12. The equivalence problem and G-structures
§7.13. Multidimensional webs
Conclusion
Back Matter
Dates of Cartan's Life and Activities
List of Publications of 1lie Cartan
APPENDIX A Rapport sur les Travaux de M. Cartan
Groupes continus et finis
Groupes discontinus et finis
Groupes continus et infinis
Equations aux derivees partielles
Conclusions
APPENDIX B Sur une degenerscence de la geometrie euclidienne
APPENDIX C Allocution de M. Elie Cartan
APPENDIX D The Influence of France in the Development of Mathematics
Bibliography
Back Cover
