Differential Geometry, Lie Groups, and Symmetric Spaces Sigurdur Helgason Graduate Studies in Mathematics Volume 34 nsffvjl American Mathematical Society l Providence, Rhode Island
PREFACE PREFACE TO THE 2001 PRINTING SUGGESTIONS TO THE READER SEQUEL TO THE PRESENT VOLUME GROUPS AND GEOMETRIC ANALYSIS GEOMETRIC ANALYSIS ON SYMMETRIC SPACES xiii xvii xix xxi xxiii xxv CHAPTER I Elementary Differential Geometry 1. Manifolds 2 2. Tensor Fields 8 1. Vector Fields and 1-Forms 8 2. Tensor Algebra 13 3. The Grossman Algebra 17 4. Exterior Differentiation........... 19 3. Mappings 22 1. The Interpretation of the Jacobian 22 2. Transformation of Vector Fields......... 24 3. Effect on Differential Forms 25 4. Affine Connections 26 5. Parallelism 28 6. The Exponential Mapping 32 7. Covariant Differentiation 40 8. The Structural Equations 43 9. The Riemannian Connection 47 10. Complete Riemannian Manifolds.......... 55 11. Isometries 60 12. Sectional Curvature 64 13. Riemannian Manifolds of Negative Curvature....... 70 14. Totally Geodesic Submanifolds 78 15. Appendix............... 82 1. Topology 82 2. Mappings of Constant Rank 86 Exercises and Further Results 88 Notes 95 CHAPTER II Lie Groups and Lie Algebras 1. The Exponential Mapping 98 /. The Lie Algebra of a Lie Group 98
X 2. The Universal Enveloping Algebra 100 3. Left Invariant Affine Connections 102 4. Taylor's Formula and the Differential of the Exponential Mapping..104 2. Lie Subgroups and Subalgebras 112 3. Lie Transformation Groups 120 4. Coset Spaces and Homogeneous Spaces 123 5. The Adjoint Group 126 6. Semisimple Lie Groups............131 7. Invariant Differential Forms 135 8. Perspectives 144 Exercises and Further Results 147 Notes 153 CHAPTER III Structure of Semisimple Lie Algebras 1. Preliminaries 155 2. Theorems of Lie and Engel 158 3. Cartan Subalgebras 162 4. Root Space Decomposition 165 5. Significance of the Root Pattern 171 6. Real Forms 178 7. Cartan Decompositions 182 8. Examples. The Complex Classical Lie Algebras 186 Exercises and Further Results..........191 Notes 196 CHAPTER IV Symmetric Spaces 1. Affine Locally Symmetric Spaces 198 2. Groups of Isometries 201 3. Riemannian Globally Symmetric Spaces........ 205 4. The Exponential Mapping and the Curvature 214 5. Locally and Globally Symmetric Spaces 218 6. Compact Lie Groups 223 7. Totally Geodesic Submanifolds. Lie Triple Systems 224 Exercises and Further Results 226 Notes 227 CHAPTER V Decomposition of Symmetric Spaces 1. Orthogonal Symmetric Lie Algebras......... 229 2. The Duality 235 3. Sectional Curvature of Symmetric Spaces 241 4. Symmetric Spaces with Semisimple Groups of Isometries 243
XI 5. Notational Conventions 244 6. Rank of Symmetric Spaces 245 Exercises and Further Results 249 Notes 251 CHAPTER VI Symmetric Spaces of the Noncompact Type 1. Decomposition of a Semisimple Lie Group 252 2. Maximal Compact Subgroups and Their Conjugacy 256 3. The Iwasawa Decomposition 257 4. Nilpotent Lie Groups 264 5. Global Decompositions 270 6. The Complex Case 273 Exercises and Further Results 275 Notes 279 CHAPTER VII Symmetric Spaces of the Compact Type 1. The Contrast between the Compact Type and the Noncompact Type..281 2. The Weyl Group and the Restricted Roots 283 3. Conjugate Points. Singular Points. The Diagram 293 4. Applications to Compact Groups 297 5. Control over the Singular Set 303 6. The Fundamental Group and the Center 307 7. The Affine Weyl Group 314 8. Application to the Symmetric Space UjK 318 9. Classification of Locally Isometric Spaces 325 10. Geometry of U/K. Symmetric Spaces of Rank One 327 11. Shortest Geodesies and Minimal Totally Geodesic Spheres....334 12. Appendix. Results from Dimension Theory........ 344 Exercises and Further Results 347 Notes 350 CHAPTER VIII Hermitian Symmetric Spaces 1. Almost Complex Manifolds 352 2. Complex Tensor Fields. The Ricci Curvature 356 3. Bounded Domains. The Kernel Function 364 4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type 372 5. Irreducible Orthogonal Symmetric Lie Algebras 377 6. Irreducible Hermitian Symmetric Spaces 381 7. Bounded Symmetric Domains 382 Exercises and Further Results 396 Notes 400
XU CHAPTER IX Structure of Semisimple Lie Groups 1. Cartan, Iwasawa, and Bruhat Decompositions....... 401 2. The Rank-One Reduction 407 3. The SU(2, 1) Reduction 409 4. Cartan Subalgebras 418 5. Automorphisms............. 421 6. The Multiplicities 428 7. Jordan Decompositions 430 Exercises and Further Results 434 Notes 436 CHAPTER X The Classification of Simple Lie Algebras and of Symmetric Spaces 1. Reduction of the P r o b l e m 438 2. T h e Classical G r o u p s and Their Cartan Involutions...... 444 /. Some Matrix Groups and Their Lie Algebras 444 2. Connectivity Properties 447 3. The Involutive Automorphisms of the Classical Compact Lie Algebras. 451 3. Root Systems 455 1. Generalities 455 2. Reduced Root Systems 459 3. Classification of Reduced Root Systems. Coxeter Graphs and Dynkin Diagrams 461 4. The Nonreduced Root Systems 474 5. The Highest Root 475 6. Outer Automorphisms and the Covering Index 478 4. The Classification of Simple Lie Algebras over C 481 5. Automorphisms of Finite Order of Semisimple Lie Algebras.... 490 6. The Classifications 515 1. The Simple Lie Algebras over C and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV 515 2. The Real Forms of Simple Lie Algebras over C. Irreducible Riemannian Globally Symmetric Spaces of Type I and Type IV 517 3. Irreducible Hermitian Symmetric Spaces.......518 4. Coincidences between Different Classes. Special Isomorphisms...518 Exercises and Further Results 520 Notes 535 SOLUTIONS TO EXERCISES 538 SOME DETAILS 586 BIBLIOGRAPHY 599 LIST OF NOTATIONAL CONVENTIONS 629 SYMBOLS FREQUENTLY USED 632 INDEX 635 REVIEWS FOR THE FIRST EDITION 641