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• 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
• Chapter II. Lie Groups and Lie Algebras
• 1. 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
• Exercises and Further Results
• Notes
• 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
• 6. Rank of Symmetric Spaces
• Exercises 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
• 5. Control over the Singular Set
• 6. 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
• 4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type
• 5. Irreducible Orthogonal Symmetric Lie Algebras
• 6. Irreducible Hermitian Symmetric Spaces
• 7. Bounded Symmetric Domains
• Exercises and Further Results
• Notes
• Chapter IX. Structure of Semisimple Lie Groups
• 1. Cartan, Iwasawa, and Bruhat Decompositions
• 2. The Rank-One Reduction
• 3. The SU(2, 1) Reduction
• 4. Cartan Subalgebras
• 5. Automorphisms
• 6. The Multiplicities
• 7. Jordan Decompositions
• Exercises and Further Results
• Notes
• Chapter X. The Classification of Simple Lie Algebras and of Symmetric Spaces
• 1. Reduction of the Problem
• 2. The Classical Groups and Their Cartan Involutions
• 3. Root Systems
• 4. The Classification of Simple Lie Algebras over C
• 5. Automorphisms of Finite Order of Semisimple Lie Algebras
• 6. The Classifications
• Exercises and Further Results
• Notes
• Bibliography
• Solutions to Exercises
• List of Notational Conventions
• Symbols Frequently Used
• Index
• Pure and Applied Mathematics