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Elie Cartan (1869-1951)

Author: M A Akivis; B A Rozenfelʹd
Publisher: Providence, R.I. : American Mathematical Society, ©1993.
Series: Translations of mathematical monographs, v. 123.
Edition/Format:   Book : Biography : EnglishView all editions and formats
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Genre/Form: Biography
Biographies
Named Person: Elie Cartan; Elie Cartan; Elie Cartan
Material Type: Biography
Document Type: Book
All Authors / Contributors: M A Akivis; B A Rozenfelʹd
ISBN: 082184587X 9780821845875
OCLC Number: 27679819
Credits: Translated by V.V. Goldberg ; translation edited by Simeon Ivanov.
Description: xii, 317 p. : ill. ; 27 cm.
Contents: Ch. 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 --
Ch. 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 --
Ch. 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 --
Ch. 4. Lie Pseudogroups and Pfaffian Equations. 4.1. Lie pseudogroups. 4.2. The Kac-Moody algebras. 4.3. Pfaffian 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 --
Ch. 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" --
Ch. 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. 6.10. 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 --
Ch. 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 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 --
Dates of Cartan's Life and Activities --
List of Publications of Elie Cartan --
Appendix A. Rapport sur les Travaux de M. Cartan / H. Poincare --
Appendix B. Sur une degenerescence de la geometrie euclidienne / E. Cartan --
Appendix C. Allocution de M. Elie Cartan --
Appendix D. The Influence of France in the Development of Mathematics.
Series Title: Translations of mathematical monographs, v. 123.
Other Titles: Ėli Kartan (1869-1951).
Responsibility: M.A. Akivis, B.A. Rosenfeld ; [translated by V.V. Goldberg ; translation edited by Simeon Ivanov].

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schema:description"Ch. 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 -- Ch. 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 -- Ch. 3. Projective Spaces and Projective Metrics. 3.1. Real spaces."@en
schema:description"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" -- Ch. 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."@en
schema:description"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 -- Ch. 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 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."@en
schema:description"7.12. The equivalence problem and G-structures. 7.13. Multidimensional webs -- Dates of Cartan's Life and Activities -- List of Publications of Elie Cartan -- Appendix A. Rapport sur les Travaux de M. Cartan / H. Poincare -- Appendix B. Sur une degenerescence de la geometrie euclidienne / E. Cartan -- Appendix C. Allocution de M. Elie Cartan -- Appendix D. The Influence of France in the Development of Mathematics."@en
schema:description"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 -- Ch. 4. Lie Pseudogroups and Pfaffian Equations. 4.1. Lie pseudogroups. 4.2. The Kac-Moody algebras. 4.3. Pfaffian 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 -- Ch. 5. The Method of Moving Frames and Differential Geometry."@en
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