Differential Geometry, Lie Groups, and Symmetric Spaces

Transcription

1 Differential Geometry, Lie Groups, and Symmetric Spaces Sigurdur Helgason Graduate Studies in Mathematics Volume 34 nsffvjl American Mathematical Society l Providence, Rhode Island

2 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 The Grossman Algebra Exterior Differentiation Mappings The Interpretation of the Jacobian Transformation of Vector Fields Effect on Differential Forms Affine Connections Parallelism The Exponential Mapping Covariant Differentiation The Structural Equations The Riemannian Connection Complete Riemannian Manifolds Isometries Sectional Curvature Riemannian Manifolds of Negative Curvature Totally Geodesic Submanifolds Appendix Topology 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

3 X 2. The Universal Enveloping Algebra Left Invariant Affine Connections Taylor's Formula and the Differential of the Exponential Mapping Lie Subgroups and Subalgebras Lie Transformation Groups Coset Spaces and Homogeneous Spaces The Adjoint Group Semisimple Lie Groups Invariant Differential Forms Perspectives 144 Exercises and Further Results 147 Notes 153 CHAPTER III Structure of Semisimple Lie Algebras 1. Preliminaries Theorems of Lie and Engel Cartan Subalgebras Root Space Decomposition Significance of the Root Pattern Real Forms Cartan Decompositions Examples. The Complex Classical Lie Algebras 186 Exercises and Further Results Notes 196 CHAPTER IV Symmetric Spaces 1. Affine Locally Symmetric Spaces Groups of Isometries Riemannian Globally Symmetric Spaces The Exponential Mapping and the Curvature Locally and Globally Symmetric Spaces Compact Lie Groups 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 The Duality Sectional Curvature of Symmetric Spaces Symmetric Spaces with Semisimple Groups of Isometries 243

4 XI 5. Notational Conventions 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 Maximal Compact Subgroups and Their Conjugacy The Iwasawa Decomposition Nilpotent Lie Groups Global Decompositions 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 The Weyl Group and the Restricted Roots Conjugate Points. Singular Points. The Diagram Applications to Compact Groups Control over the Singular Set The Fundamental Group and the Center The Affine Weyl Group Application to the Symmetric Space UjK Classification of Locally Isometric Spaces Geometry of U/K. Symmetric Spaces of Rank One Shortest Geodesies and Minimal Totally Geodesic Spheres Appendix. Results from Dimension Theory Exercises and Further Results 347 Notes 350 CHAPTER VIII Hermitian Symmetric Spaces 1. Almost Complex Manifolds Complex Tensor Fields. The Ricci Curvature Bounded Domains. The Kernel Function Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type Irreducible Orthogonal Symmetric Lie Algebras Irreducible Hermitian Symmetric Spaces Bounded Symmetric Domains 382 Exercises and Further Results 396 Notes 400

5 XU CHAPTER IX Structure of Semisimple Lie Groups 1. Cartan, Iwasawa, and Bruhat Decompositions The Rank-One Reduction The SU(2, 1) Reduction Cartan Subalgebras Automorphisms The Multiplicities 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 T h e Classical G r o u p s and Their Cartan Involutions /. Some Matrix Groups and Their Lie Algebras Connectivity Properties The Involutive Automorphisms of the Classical Compact Lie Algebras Root Systems Generalities Reduced Root Systems Classification of Reduced Root Systems. Coxeter Graphs and Dynkin Diagrams The Nonreduced Root Systems The Highest Root Outer Automorphisms and the Covering Index The Classification of Simple Lie Algebras over C Automorphisms of Finite Order of Semisimple Lie Algebras The Classifications The Simple Lie Algebras over C and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV The Real Forms of Simple Lie Algebras over C. Irreducible Riemannian Globally Symmetric Spaces of Type I and Type IV Irreducible Hermitian Symmetric Spaces Coincidences between Different Classes. Special Isomorphisms 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