DIFFERENTIAL GEOMETRY AND SYMMETRIC SPACESPDF电子书下载

CHAPTER Ⅰ Elementary Differential Geometry 1

1．Manifolds 2

2．Tensor Fields 8

1．Vector Fields ard I-Forms 8

2．The Tensor Algebra 13

3．The Grassmann 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

Exercises 82

Notes 85

CHAPTER Ⅱ Lie Groups and Lie Algebras 87

1．The Exponential Mapping 88

1．The Lie Algebra of a Lie Group 88

2．The Universal Enveloping Algebra 90

3．Left Invariant Affine Connections 92

4．Taylor＇s Formula and Applications 94

2．Lie Subgroups and Subalgebras 102

3．Lie Transformation Groups 110

4．Coset Spaces and Homogeneous Spaces 113

5．The Adjoint Group 116

6．Semisimple Lie Groups 121

Exercises 125

Notes 128

CHAPTER Ⅲ Structure of Semisimple Lie Algebras 130

1．Preliminaries 130

2．Theorems of Lie and Engel 133

3．Cartan Subalgebras 137

4．Root Space Decomposition 140

5．Significance of the Root Pattern 146

6．Real Forms 152

7．Caftan Decompositions 156

Exercises 160

Notes 161

CHAPTER Ⅳ Symmetric Spaces 162

1．Affine Locally Symmetric Spaces 163

2．Groups of Isometries 166

3．Riemannian Globally Symmetric Spaces 170

4．The Exponential Mapping and the Curvature 179

5．Locally and Globally Symmetric Spaces 183

6．Compact Lie Groups 188

7．Totally Geodesic Submanifolds．Lie Triple Systems 189

Exercises 191

Notes 191

CHAPTER Ⅴ Decomposition of Symmetric Spaces 193

1．Orthogonal Symmetric Lie Algebras 193

2．The Duality 199

3．Sectional Curvature of Symmetric Spaces 205

4．Symmetric Spaces with Semisimple Groups of Isometries 207

5．Notational Conventions 208

6．Rank of Symmetric Spaces 209

Exercises 213

Notes 213

CHAPTER Ⅵ Symmetric Spaces of the Noncompact Type 214

1．Decomposition of a Semisimple Lie Group 214

2．Maximal Compact Subgroups and Their Conjugacy 218

3．The Iwasawa Decomposition 219

4．Nilpotent Lie Groups 225

5．Global Decompositions 234

6．The Complex Case 237

Exercises 239

Notes 240

CHAPTER Ⅶ Symmetric Spaces of the Compact Type 241

1．The Contrast between the Compact Type and the Noncompact Type 241

2．The Weyl Group 243

3．Conjugate Points．Singular Points．The Diagram 250

4．Applications to Compact Groups 254

5．Control over the Singular Set 260

6．The Fundamental Group and the Center 264

7．Application to the Symmetric Space U/K 271

8．Classification of Locally Isometric Spaces 273

9．Appendix．Results from Dimension Theory 275

Exercises 278

Notes 280

CHAPTER Ⅷ Hermitian Symmetric Spaces 281

1．Almost Complex Manifolds 281

2．Complex Tensor Fields．The Ricci Curvature 285

3．Bounded Domains．The Kernel Function 293

4．Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type 301

5．Irreducible Orthogonal Symmetric Lie Algebras 306

6．Irreducible Hermitian Symmetric Spaces 310

7．Bounded Symmetric Domains 311

Exercises 322

Notes 325

CHAPTER Ⅸ On the Classification of Symmetric Spaces 326

1．Reduction of the Problem 326

2．Automorphisms 331

3．Involutive Automorphisms 334

4．E．Cartan＇s List of Irreducible Riemannian Globally Symmetric Spaces 339

1．Some Matrix Groups and Their Lie Algebras 339

2．The Simple Lie Algebras over C and Their Compact Real Forms．The Irreducible Riemannian Globally Symmetric Spaces of Type Ⅱ and Type Ⅳ 346

3．The Involutive Automorphisms of Simple Compact Lie Algebras．The Irreducible Globally Symmetric Spaces of Type Ⅰ and Type Ⅲ 347

4．Irreducible Hermitian Symmetric Spaces 354

5．Two-Point Homogeneous Spaces．Symmetric Spaces of Rank One．Closed Geodesics 355

Exercises 358

Notes 359

CHAPTER Ⅹ Functions on Symmetric Spaces 360

1．Integral Formulas 361

1．Generalities 361

2．Invariant Measures on Coset Spaces 367

3．Some Integral Formulas for Semisimple Lie Groups 372

4．Integral Formulas for the Cartan Decomposition 379

5．The Compact Case 382

2．Invariant Differential Operators 385

1．Generalities．The Laplace-Behrami Operator 385

2．Invariant Differential Operators on Reductive Coset Spaces 389

3．The Case of a Symmetric Space 396

3．Spherical Functions．Definition and Examples 398

4．Elementary Properties of Spherical Functions 408

5．Some Algebraic Tools 518

6．The Formula for the Spherical Function 422

1．The Euclidean Type 422

2．The Compact Type 423

3．The Noncompact Type 427

7．Mean Value Theorems 435

1．The Mean Value Operators 435

2．Approximations by Analytic Functions 440

3．The Darboux Equation in a Symmetric Space 442

4．Poisson＇s Equation in a Two-Point Homogeneous Space 444

Exercises 449

Notes 454

BIBLIOGRAPHY 457

LIST OF NOTATIONAL CONVENTIONS 473

SYMBOLS FREQUENTLY USED 476

AUTHOR INDEX 479

SUBJECT INDEX 482