Differential Geometry, Lie Groups, and Symmetric Spaces (eBook)

Sigurdur Helgason (Autor)

eBook Download: PDF

1979 | 1. Auflage
628 Seiten
Elsevier Science (Verlag)
978-0-08-087396-1 (ISBN)

Infos

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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The present book is intended as a textbook and reference work on three topics in the title. Together with a volume in progress on "e;Groups and Geometric Analysis"e; it supersedes my "e;Differential Geometry and Symmetric Spaces,"e; 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.

Inhalt

Front Cover 1
Differential Geometry, Lie Groups, and Symmetric Spaces 4
Copyright Page 5
Contents 6
Preface 10
Tentative Contents of the Sequel 16
Chapter I. Elementary Differential Geometry 18
1. Manifolds 19
2. Tensor Fields 25
3. Mappings 39
4. Affine Connections 43
5. Parallelism 45
6. The Exponential Mappping 49
7. Covariant Differentiation 57
8. The Structural Equations 60
9. The Riemannian Connection 64
10. Complete Riemannian Manifolds 72
11. Isometries 77
12. Sectional Curvature 81
13. Riemannian Manifolds of Negative Curvature 87
14. Totally Geodesic Submanifolds 95
15. Appendix 99
Exercises and Further Results 105
Notes 112
Chapter II. Lie Groups and Lie Algebras 114
1. The Exponential Mapping 115
2. Lie Subgroups and Subalgebras 129
3. Lie Transformation Groups 137
4. Coset Spaces and Homogeneous Spaces 140
5. The Adjoint Group 143
6. Semisimple Lie Groups 148
7. Invariant Differential Forms 152
8. Perspectives 161
Exercises and Further Results 164
Notes 170
Chapter III. Structure of Semisimple Lie Algebras 172
1. Preliminaries 172
2. Theorems of Lie and Engel 175
3. Cartan Subalgebras 179
4. Root Space Decomposition 182
5. Significance of the Root Pattern 188
6. Real Forms 195
7. Cartan Decompositions 199
8. Examples. The Complex Classical Lie Algebras 203
Exercises and Further Results 208
Notes 213
Chapter IV. Symmetric Spaces 214
1. Affine Locally Symmetric Spaces 215
2. Groups of Isometries 218
3. Riemannian Globally Symmetric Spaces 222
4. The Exponential Mapping and the Curvature 231
5. Locally and Globally Symmetric Spaces 235
6. Compact Lie Groups 240
7. Totally Geodesic Submanifolds. Lie Triple Systems 241
Exercises and Further Results 243
Notes 244
Chapter V. Decomposition of Symmetric Spaces 246
1. Orthogonal Symmetric Lie Algebras 246
2. The Duality 252
3. Sectional Curvature of Symmetric Spaces 258
4. Symmetric Spaces with Semisimple Groups of Isometries 260
5. Notational Conventions 261
6. Rank of Symmetric Spaces 262
Exercises and Further Results 266
Notes 268
Chapter VI. Symmetric Spaces of the Noncompact Type 269
1. Decomposition of a Semisimple Lie Group 269
2. Maximal Compact Subgroups and Their Conjugacy 273
3. The Iwasawa Decomposition 274
4. Nilpotent Lie Groups 281
5. Global Decompositions 287
6. The Complex Case 290
Exercises and Further Results 292
Notes 296
Chapter VII. Symmetric Spaces of the Compact Type 298
1. The Contrast between the Compact Type and the Noncompact Type 298
2. The Weyl Group and the Restricted Roots 300
3. Conjugate Points. Singular Points . The Diagram 310
4. Applications to Compact Groups 314
5. Control over the Singular Set 320
6. The Fundamental Group and the Center 324
7. The Affine Weyl Group 331
8. Application to the Symmetric Space U/K 335
9. Classification of Locally Isometric Spaces 342
10. Geometry of U/K. Symmetric Spaces of Rank One 344
11. Shortest Geodesics and Minimal Totally Geodesic Spheres 351
12. Appendix. Results from Dimension Theory 361
Exercises and Further Results 364
Notes 367
Chapter Vlll. Hermitian Symmetric Spaces 369
1. Almost Complex Manifolds 369
2. Complex Tensor Fields. The Ricci Curvature 373
3. Bounded Domains. The Kernel Function 381
4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type 389
5. Irreducible Orthogonal Symmetric Lie Algebras 394
6. Irreducible Hermitian Symmetric Spaces 398
7. Bounded Symmetric Domains 399
Exercises and Further Results 413
Notes 417
Chapter IX. Structure of Semisimple Lie Groups 418
1. Cartan, Iwasawa, and Bruhat Decompositions 418
2. The Rank-One Reduction 424
3. The SU(2, 1) Reduction 426
4. Cartan Subalgebras 435
5. Automorphisms 438
6. The Multiplicities 445
7. Jordan Decompositions 447
Exercises and Further Results 451
Notes 453
Chapter X. The Classification of Simple Lie Algebras and of Symmetric Spaces 455
1. Reduction of the Problem 455
2. The Classical Groups and Their Cartan Involutions 461
3. Root Systems 472
4. The Classification of Simple Lie Algebras over C 498
5. Automorphisms of Finite Order of Semisimple Lie Algebras 507
6. The Classifications 532
Exercises and Further Results 537
Notes 552
Bibliography 604
Solutions to Exercises 555
List of Notational Conventions 634
Symbols Frequently Used 637
Index 640
Pure and Applied Mathematics 646

Produkt-Details

• Erscheint lt. Verlag: 9.2.1979
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-087396-0 / 0080873960
• ISBN-13: 978-0-08-087396-1 / 9780080873961