UID:
almahu_9947366441102882
Format:
1 online resource (647 p.)
ISBN:
1-281-76361-6
,
9786611763619
,
0-08-087396-0
Series Statement:
Pure and applied mathematics, a series of monographs and textbooks ; 80
Content:
The present book is intended as a textbook and reference work on three topics in the title. Together with a volume in progress on ""Groups and Geometric Analysis"" it supersedes my ""Differential Geometry and Symmetric Spaces,"" published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful.
Note:
Description based upon print version of record.
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Front Cover; Differential Geometry, Lie Groups, and Symmetric Spaces; Copyright Page; Contents; Preface; Tentative Contents of the Sequel; Chapter I. Elementary Differential Geometry; 1. Manifolds; 2. Tensor Fields; 3. Mappings; 4. Affine Connections; 5. Parallelism; 6. The Exponential Mappping; 7. Covariant Differentiation; 8. The Structural Equations; 9. The Riemannian Connection; 10. Complete Riemannian Manifolds; 11. Isometries; 12. Sectional Curvature; 13. Riemannian Manifolds of Negative Curvature; 14. Totally Geodesic Submanifolds; 15. Appendix; Exercises and Further Results; Notes
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Chapter II. Lie Groups and Lie Algebras1. The Exponential Mapping; 2. Lie Subgroups and Subalgebras; 3. Lie Transformation Groups; 4. Coset Spaces and Homogeneous Spaces; 5. The Adjoint Group; 6. Semisimple Lie Groups; 7. Invariant Differential Forms; 8. Perspectives; Exercises and Further Results; Notes; Chapter III. Structure of Semisimple Lie Algebras; 1. Preliminaries; 2. Theorems of Lie and Engel; 3. Cartan Subalgebras; 4. Root Space Decomposition; 5. Significance of the Root Pattern; 6. Real Forms; 7. Cartan Decompositions; 8. Examples. The Complex Classical Lie Algebras
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Exercises and Further ResultsNotes; Chapter IV. Symmetric Spaces; 1. Affine Locally Symmetric Spaces; 2. Groups of Isometries; 3. Riemannian Globally Symmetric Spaces; 4. The Exponential Mapping and the Curvature; 5. Locally and Globally Symmetric Spaces; 6. Compact Lie Groups; 7. Totally Geodesic Submanifolds. Lie Triple Systems; Exercises and Further Results; Notes; Chapter V. Decomposition of Symmetric Spaces; 1. Orthogonal Symmetric Lie Algebras; 2. The Duality; 3. Sectional Curvature of Symmetric Spaces; 4. Symmetric Spaces with Semisimple Groups of Isometries; 5. Notational Conventions
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6. Rank of Symmetric SpacesExercises and Further Results; Notes; Chapter VI. Symmetric Spaces of the Noncompact Type; 1. Decomposition of a Semisimple Lie Group; 2. Maximal Compact Subgroups and Their Conjugacy; 3. The Iwasawa Decomposition; 4. Nilpotent Lie Groups; 5. Global Decompositions; 6. The Complex Case; Exercises and Further Results; Notes; Chapter VII. Symmetric Spaces of the Compact Type; 1. The Contrast between the Compact Type and the Noncompact Type; 2. The Weyl Group and the Restricted Roots; 3. Conjugate Points. Singular Points . The Diagram; 4. Applications to Compact Groups
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5. Control over the Singular Set6. The Fundamental Group and the Center; 7. The Affine Weyl Group; 8. Application to the Symmetric Space U/K; 9. Classification of Locally Isometric Spaces; 10. Geometry of U/K. Symmetric Spaces of Rank One; 11. Shortest Geodesics and Minimal Totally Geodesic Spheres; 12. Appendix. Results from Dimension Theory; Exercises and Further Results; Notes; Chapter Vlll. Hermitian Symmetric Spaces; 1. Almost Complex Manifolds; 2. Complex Tensor Fields. The Ricci Curvature; 3. Bounded Domains. The Kernel Function
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4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type
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English
Additional Edition:
ISBN 0-12-338460-5
Language:
English
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