Differential Geometry and Symmetric Spaces
Differential Geometry and Symmetric Spaces
Published on 1. January 1962
485 pages
978-0-08-087324-4 (ISBN)
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Differential Geometry and Symmetric Spaces
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Sigurdur Helgason
Differential Geometry and Symmetric Spaces
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12/1962
Academic Press
€34.11
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Content
- Front Cover
- Differential Geometry and Symmetric Spaces
- Copyright Page
- Contents
- Preface
- Suggestions to the Reader
- Chapter I. Elementary Differential Geometry
- 1. Manifolds
- 2. Tensor Fields
- 3. Mappings
- 4. Affine Connections
- 5. Parallelism
- 6. The Exponential Mapping
- 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
- Exercises
- 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
- Exercises
- 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
- Exercises
- 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
- 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
- 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
- Notes
- Chapter VII. Symmetric Spaces of the Compact Type
- 1. The Contrast between the Compact Type and the Noncompact Type
- 2. The Weyl Group
- 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. Application to the Symmetric Space U/K
- 8. Classification of Locally Isometric Spaces
- 9. Appendix. Results from Dimension Theory
- Exercises
- Notes
- Chapter VIII. 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
- Notes
- Chapter IX. On the Classification of Symmetric Spaces
- 1. Reduction of the Problem
- 2. Automorphisms
- 3. Involutive Automorphisms
- 4. É. Cartan's List of Irreducible Riemannian Globally Symmetric Spaces
- 5. Two-Point Homogeneous Spaces. Symmetric Spaces of Rank One. Closed Geodesics
- Exercises
- Notes
- Chapter X. Functions on Symmetric Spaces
- 1. Integral Formulas
- 2. Invariant Differential Operators
- 3. Spherical Functions. Definition and Examples
- 4. Elementary Properties of Spherical Functions
- 5. Some Algebraic Tools
- 6. The Formula for the Spherical Function
- 7. Mean Value Theorems
- Exercises
- Notes
- Bibliography
- List of Notational Conventions
- Symbols Frequently
- Author Index
- Subject Index
