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CHAPTER Ⅰ Elementary Differential Geometry1

1．Manifolds2

2．Tensor Fields8

1．Vector Fields ard I-Forms8

2．The Tensor Algebra13

3．The Grassmann Algebra17

4．Exterior Differentiation19

3．Mappings22

1．The Interpretation of the Jacobian22

2．Transformation of Vector Fields24

3．Effect on Differential Forms25

4．Affine Connections26

5．Parallelism28

6．The Exponential Mapping32

7．Covariant Differentiation40

8．The Structural Equations43

9．The Riemannian Connection47

10．Complete Riemannian Manifolds55

11．Isometries60

12．Sectional Curvature64

13．Riemannian Manifolds of Negative Curvature70

14．Totally Geodesic Submanifolds78

Exercises82

Notes85

CHAPTER Ⅱ Lie Groups and Lie Algebras87

1．The Exponential Mapping88

1．The Lie Algebra of a Lie Group88

2．The Universal Enveloping Algebra90

3．Left Invariant Affine Connections92

4．Taylor＇s Formula and Applications94

2．Lie Subgroups and Subalgebras102

3．Lie Transformation Groups110

4．Coset Spaces and Homogeneous Spaces113

5．The Adjoint Group116

6．Semisimple Lie Groups121

Exercises125

Notes128

CHAPTER Ⅲ Structure of Semisimple Lie Algebras130

1．Preliminaries130

2．Theorems of Lie and Engel133

3．Cartan Subalgebras137

4．Root Space Decomposition140

5．Significance of the Root Pattern146

6．Real Forms152

7．Caftan Decompositions156

Exercises160

Notes161

CHAPTER Ⅳ Symmetric Spaces162

1．Affine Locally Symmetric Spaces163

2．Groups of Isometries166

3．Riemannian Globally Symmetric Spaces170

4．The Exponential Mapping and the Curvature179

5．Locally and Globally Symmetric Spaces183

6．Compact Lie Groups188

7．Totally Geodesic Submanifolds．Lie Triple Systems189

Exercises191

Notes191

CHAPTER Ⅴ Decomposition of Symmetric Spaces193

1．Orthogonal Symmetric Lie Algebras193

2．The Duality199

3．Sectional Curvature of Symmetric Spaces205

4．Symmetric Spaces with Semisimple Groups of Isometries207

5．Notational Conventions208

6．Rank of Symmetric Spaces209

Exercises213

Notes213

CHAPTER Ⅵ Symmetric Spaces of the Noncompact Type214

1．Decomposition of a Semisimple Lie Group214

2．Maximal Compact Subgroups and Their Conjugacy218

3．The Iwasawa Decomposition219

4．Nilpotent Lie Groups225

5．Global Decompositions234

6．The Complex Case237

Exercises239

Notes240

CHAPTER Ⅶ Symmetric Spaces of the Compact Type241

1．The Contrast between the Compact Type and the Noncompact Type241

2．The Weyl Group243

3．Conjugate Points．Singular Points．The Diagram250

4．Applications to Compact Groups254

5．Control over the Singular Set260

6．The Fundamental Group and the Center264

7．Application to the Symmetric Space U/K271

8．Classification of Locally Isometric Spaces273

9．Appendix．Results from Dimension Theory275

Exercises278

Notes280

CHAPTER Ⅷ Hermitian Symmetric Spaces281

1．Almost Complex Manifolds281

2．Complex Tensor Fields．The Ricci Curvature285

3．Bounded Domains．The Kernel Function293

4．Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type301

5．Irreducible Orthogonal Symmetric Lie Algebras306

6．Irreducible Hermitian Symmetric Spaces310

7．Bounded Symmetric Domains311

Exercises322

Notes325

CHAPTER Ⅸ On the Classification of Symmetric Spaces326

1．Reduction of the Problem326

2．Automorphisms331

3．Involutive Automorphisms334

4．E．Cartan＇s List of Irreducible Riemannian Globally Symmetric Spaces339

1．Some Matrix Groups and Their Lie Algebras339

2．The Simple Lie Algebras over C and Their Compact Real Forms．The Irreducible Riemannian Globally Symmetric Spaces of Type Ⅱ and Type Ⅳ346

3．The Involutive Automorphisms of Simple Compact Lie Algebras．The Irreducible Globally Symmetric Spaces of Type Ⅰ and Type Ⅲ347

4．Irreducible Hermitian Symmetric Spaces354

5．Two-Point Homogeneous Spaces．Symmetric Spaces of Rank One．Closed Geodesics355

Exercises358

Notes359

CHAPTER Ⅹ Functions on Symmetric Spaces360

1．Integral Formulas361

1．Generalities361

2．Invariant Measures on Coset Spaces367

3．Some Integral Formulas for Semisimple Lie Groups372

4．Integral Formulas for the Cartan Decomposition379

5．The Compact Case382

2．Invariant Differential Operators385

1．Generalities．The Laplace-Behrami Operator385

2．Invariant Differential Operators on Reductive Coset Spaces389

3．The Case of a Symmetric Space396

3．Spherical Functions．Definition and Examples398

4．Elementary Properties of Spherical Functions408

5．Some Algebraic Tools518

6．The Formula for the Spherical Function422

1．The Euclidean Type422

2．The Compact Type423

3．The Noncompact Type427

7．Mean Value Theorems435

1．The Mean Value Operators435

2．Approximations by Analytic Functions440

3．The Darboux Equation in a Symmetric Space442

4．Poisson＇s Equation in a Two-Point Homogeneous Space444

Exercises449

Notes454

BIBLIOGRAPHY457

LIST OF NOTATIONAL CONVENTIONS473

SYMBOLS FREQUENTLY USED476

AUTHOR INDEX479

SUBJECT INDEX482