Differential Geometry and Symmetric Spaces (Pure & Applied Mathematics)

✍ Scribed by Sigurdur Helgason

• Publisher: Academic Press
• Year: 1962
• Tongue: English
• Leaves: 501
• Category: Library

✦ Synopsis

Sigurdur Helgason's Differential Geometry and Symmetric Spaces was quickly recognized as a remarkable and important book. For many years, it was the standard text both for Riemannian geometry and for the analysis and geometry of symmetric spaces. Several generations of mathematicians relied on it for its clarity and careful attention to detail. Although much has happened in the field since the publication of this book, as demonstrated by Helgason's own three-volume expansion of the original work, this single volume is still an excellent overview of the subjects. For instance, even though there are now many competing texts, the chapters on differential geometry and Lie groups continue to be among the best treatments of the subjects available. There is also a well-developed treatment of Cartan's classification and structure theory of symmetric spaces. The last chapter, on functions on symmetric spaces, remains an excellent introduction to the study of spherical functions, the theory of invariant differential operators, and other topics in harmonic analysis. This text is rightly called a classic. Sigurdur Helgason was awarded the Steele Prize for Groups and Geometric Analysis and the companion volume, Differential Geometry, Lie Groups and Symmetric Spaces.

✦ Table of Contents

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