Differential geometry, Lie groups, and symmetric spaces

Sigurdur Helgason0123384605, 9780123384607, 9780080873961

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.

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Table of contents :
Series ……Page 1
Title page ……Page 2
Date-line ……Page 3
CONTENTS ……Page 4
Preface ……Page 8
Suggestions to the Reader ……Page 12
Tentative Contents of the Sequel ……Page 14
CHAPTER I Elementary Differential Geometry ……Page 16
1. Manifolds ……Page 17
1. Vector Fields and 1-Forms ……Page 23
2. Tensor Algebra ……Page 28
3. The Grassman Algebra ……Page 32
4. Exterior Differentiation ……Page 34
1. The Interpretation of the Jacobian ……Page 37
2. Transformation of Vector Fields ……Page 39
3. Effect on Differential Forms ……Page 40
4. Affine Connections ……Page 41
5. Parallelism ……Page 43
6. The Exponential Mapping ……Page 47
7. Covariant Differentiation ……Page 55
8. The Structural Equations ……Page 58
9. The Riemannian Connection ……Page 62
10. Complete Riemannian Manifolds ……Page 70
11. Isometries ……Page 75
12. Sectional Curvature ……Page 79
13. Riemannian Manifolds of Negative Curvature ……Page 85
14. Totally Geodesic Submanifolds ……Page 93
1. Topology ……Page 97
2. Mappings of Constant Rank ……Page 101
Exercises and Further Results ……Page 103
Notes ……Page 110
CHAPTER II Lie Groups and Lie Algebras ……Page 112
1. The Lie Algebra of a Lie Group ……Page 113
2. The Universal Enveloping Algebra ……Page 115
3. Left Invariant Affine Connections ……Page 117
4. Taylor’s Formula and the Differential of the Exponential Mapping ……Page 119
2. Lie Subgroups and Subalgebras ……Page 127
3. Lie Transformation Groups ……Page 135
4. Coset Spaces and Homogeneous Spaces ……Page 138
5. The Adjoint Group ……Page 141
6. Semisimple Lie Groups ……Page 146
7. Invariant Differential Forms ……Page 150
8. Perspectives ……Page 159
Exercises and Further Results ……Page 162
Notes ……Page 168
1. Preliminaries ……Page 170
2. Theorems of Lie and Engel ……Page 173
3. Cartan Subalgebras ……Page 177
4. Root Space Decomposition ……Page 180
5. Significance of the Root Pattern ……Page 186
6. Real Forms ……Page 193
7. Cartan Decompositions ……Page 197
8. Examples. The Complex Classical Lie Algebras ……Page 201
Exercises and Further Results ……Page 206
Notes ……Page 211
CHAPTER IV Symmetric Spaces ……Page 212
1. Affine Locally Symmetric Spaces ……Page 213
2. Groups of Isometries ……Page 216
3. Riemannian Globally Symmetric Spaces ……Page 220
4. The Exponential Mapping and the Curvature ……Page 229
5. Locally and Globally Symmetric Spaces ……Page 233
6. Compact Lie Groups ……Page 238
7. Totally Geodesic Submanifolds. Lie Triple Systems ……Page 239
Exercises and Further Results ……Page 241
Notes ……Page 242
1. Orthogonal Symmetric Lie Algebras ……Page 244
2. The Duality ……Page 250
3. Sectional Curvature of Symmetric Spaces ……Page 256
4. Symmetric Spaces with Semisimple Groups of Isometries ……Page 258
5. Notational Conventions ……Page 259
6. Rank of Symmetric Spaces ……Page 260
Exercises and Further Results ……Page 264
Notes ……Page 266
1. Decomposition of a Semisimple Lie Group ……Page 267
2. Maximal Compact Subgroups and Their Conjugacy ……Page 271
3. The Iwasawa Decomposition ……Page 272
4. Nilpotent Lie Groups ……Page 279
5. Global Decompositions ……Page 285
6. The Complex Case ……Page 288
Exercises and Further Results ……Page 290
Notes ……Page 294
1. The Contrast between the Compact Type and the Noncompact Type ……Page 296
2. The Weyl Group and the Restricted Roots ……Page 298
3. Conjugate Points. Singular Points. The Diagram ……Page 308
4. Applications to Compact Groups ……Page 312
5. Control over the Singular Set ……Page 318
6. The Fundamental Group and the Center ……Page 322
7. The Affine Weyl Group ……Page 329
8. Application to the Symmetric Space $U/K$ ……Page 333
9. Classification of Locally Isometric Spaces ……Page 340
10. Geometry of $U/K$. Symmetric Spaces of Rank One ……Page 342
11. Shortest Geodesics and Minimal Totally Geodesic Spheres ……Page 349
12. Appendix. Results from Dimension Theory ……Page 359
Exercises and Further Results ……Page 362
Notes ……Page 365
1. Almost Complex Manifolds ……Page 367
2. Complex Tensor Fields. The Ricci Curvature ……Page 371
3. Bounded Domains. The Kernel Function ……Page 379
4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type ……Page 387
5. Irreducible Orthogonal Symmetric Lie Algebras ……Page 392
6. Irreducible Hermitian Symmetric Spaces ……Page 396
7. Bounded Symmetric Domains ……Page 397
Exercises and Further Results ……Page 411
Notes ……Page 415
1. Cartan, Iwasawa, and Bruhat Decompositions ……Page 416
2. The Rank-One Reduction ……Page 422
3. The $SU(2,1)$ Reduction ……Page 424
4. Cartan Subalgebras ……Page 433
5. Automorphisms ……Page 436
6. The Multiplicities ……Page 443
7. Jordan Decompositions ……Page 445
Exercises and Further Results ……Page 449
Notes ……Page 451
1. Reduction of the Problem ……Page 453
1. Some Matrix Groups and Their Lie Algebras ……Page 459
2. Connectivity Properties ……Page 462
3. The Involutive Automorphisms of the Classical Compact Lie Algebras ……Page 466
1. Generalities ……Page 470
2. Reduced Root Systems ……Page 474
3. Classification of Reduced Root Systems. Coxeter Graphs and Dynkin Diagrams ……Page 476
4. The Nonreduced Root Systems ……Page 489
5. The Highest Root ……Page 490
6. Outer Automorphisms and the Covering Index ……Page 493
4. The Classification of Simple Lie Algebras over $C$ ……Page 496
5. Automorphisms of Finite Order of Semisimple Lie Algebras ……Page 505
1. The Simple Lie Algebras over $C$ and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV ……Page 530
2. The Real Forms of Simple Lie Algebras over $C$. Irreducible Riemannian Globally Symmetric Spaces of Type I and Type IV ……Page 532
4. Coincidences between Different Classes. Special Isomorphisms ……Page 533
Exercises and Further Results ……Page 535
Notes ……Page 550
Solutions to Exercises ……Page 553
Some Details ……Page 601
Bibliography ……Page 608
List of Notational Conventions ……Page 638
Symbols Frequently Used ……Page 641
Index ……Page 644