Differential Geometry and Symmetric Spaces (eBook)

Front Cover 1
Differential Geometry and Symmetric Spaces 4
Copyright Page 5
Contents 12
Preface 8
Suggestions to the Reader 10
Chapter I. Elementary Differential Geometry 16
1. Manifolds 17
2. Tensor Fields 23
3. Mappings 37
4. Affine Connections 41
5. Parallelism 43
6. The Exponential Mapping 47
7. Covariant Differentiation 55
8. The Structural Equations 58
9. The Riemannian Connection 62
10. Complete Riemannian Manifolds 70
11. Isometries 75
12. Sectional Curvature 79
13. Riemannian Manifolds of Negative Curvature 85
14. Totally Geodesic Submanifolds 93
Exercises 97
Notes 100
Chapter II. Lie Groups and Lie Algebras 102
1. The Exponential Mapping 103
2. Lie Subgroups and Subalgebras 117
3. Lie Transformation Groups 125
4. Coset Spaces and Homogeneous Spaces 128
5. The Adjoint Group 131
6. Semisimple Lie Groups 136
Exercises 140
Notes 143
Chapter III. Structure of Semisimple Lie Algebras 145
1. Preliminaries 145
2. Theorems of Lie and Engel 148
3. Cartan Subalgebras 152
4. Root Space Decomposition 155
5. Significance of the Root Pattern 161
6. Real Forms 167
7. Cartan Decompositions 171
Exercises 175
Notes 176
Chapter IV. Symmetric Spaces 177
1. Affine Locally Symmetric Spaces 178
2. Groups of Isometries 181
3. Riemannian Globally Symmetric Spaces 185
4. The Exponential Mapping and the Curvature 194
5. Locally and Globally Symmetric Spaces 198
6. Compact Lie Groups 203
7. Totally Geodesic Submanifolds. Lie Triple Systems 204
Exercises 206
Notes 206
Chapter V. Decomposition of Symmetric Spaces 208
1. Orthogonal Symmetric Lie Algebras 208
2. The Duality 214
3. Sectional Curvature of Symmetric Spaces 220
4. Symmetric Spaces with Semisimple Groups of Isometries 222
5. Notational Conventions 223
6. Rank of Symmetric Spaces 224
Exercises 228
Notes 228
Chapter VI. Symmetric Spaces of the Noncompact Type 229
1. Decomposition of a Semisimple Lie Group 229
2. Maximal Compact Subgroups and Their Conjugacy 233
3. The Iwasawa Decomposition 234
4. Nilpotent Lie Groups 240
5. Global Decompositions 249
6. The Complex Case 252
Exercises 254
Notes 255
Chapter VII. Symmetric Spaces of the Compact Type 256
1. The Contrast between the Compact Type and the Noncompact Type 256
2. The Weyl Group 258
3. Conjugate Points. Singular Points. The Diagram 265
4. Applications to Compact Groups 269
5. Control over the Singular Set 275
6. The Fundamental Group and the Center 279
7. Application to the Symmetric Space U/K 286
8. Classification of Locally Isometric Spaces 288
9. Appendix. Results from Dimension Theory 290
Exercises 293
Notes 295
Chapter VIII. Hermitian Symmetric Spaces 296
1. Almost Complex Manifolds 296
2. Complex Tensor Fields. The Ricci Curvature 300
3. Bounded Domains. The Kernel Function 308
4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type 316
5. Irreducible Orthogonal Symmetric Lie Algebras 321
6. Irreducible Hermitian Symmetric Spaces 325
7. Bounded Symmetric Domains 326
Exercises 337
Notes 340
Chapter IX. On the Classification of Symmetric Spaces 341
1. Reduction of the Problem 341
2. Automorphisms 346
3. Involutive Automorphisms 349
4. É. Cartan’s List of Irreducible Riemannian Globally Symmetric Spaces 354
5. Two-Point Homogeneous Spaces. Symmetric Spaces of Rank One. Closed Geodesics 370
Exercises 373
Notes 374
Chapter X. Functions on Symmetric Spaces 375
1. Integral Formulas 376
2. Invariant Differential Operators 400
3. Spherical Functions. Definition and Examples 413
4. Elementary Properties of Spherical Functions 423
5. Some Algebraic Tools 433
6. The Formula for the Spherical Function 437
7. Mean Value Theorems 450
Exercises 464
Notes 469
Bibliography 472
List of Notational Conventions 488
Symbols Frequently 491
Author Index 494
Subject Index 497