Interdependence of the Chapters and the Sections xi

Chapter I

Differentiable Manifolds

1. Differentiable manifolds 1

2. Tensor algebras 17

3. Tensor fields 26

4. Lie groups 38

5. Fibre bundles 50

Chapter II

Theory of Connections

1. Connections in a principle fibre bundle 63

2. Existence and extension of connections 67

3. Parallelism 68

4. Holonomy groups 71

5. Curvature for and structure equation 75

6. Mappings of connections 79

7. Reduction theorem 83

8. Holonomy theorem 89

9. Flat connections 92

10. Local and infinitesimal holonomy groups 94

11. Invariant connections 103

Chapter III

Linear and Affine Connections

1. Connections in a vector bundle 113

2. Linear connections 118

3. Affine connections 125

4. Developments 130

5. Curvature and torsion tensors 132

6. Geodesics 138

7. Expressions in local coordinate systems 140

8. Normal coordinates 146

9. Linear infitesimal holonomy groups 151

Chapter IV

Riemannian Connections

1. Riemannian metrics 154

2. Riemannian connections 158

3. Normal coordinates and convex neighborhoods 162

4. Completeness 172

5. Holonomy groups 179

6. The decomposition theorem of de Rham 187

7. Affine holonomy groups

Chapter V

Curvature and Space Forms

1. Algebraic preliminaries 198

2. Sectional curvature

3. Spaces of constant curvature 204

4. Flat affine and Riemannian connections 209

Chapter VI

Transformations

1. Affine mappings and affine transformations 225

2. Infinitesimal affine transformations 229

3. Isometries and infinitesimal isometries 236

4. Holonomy and infinitesimal isometries 244

5. Ricci tensor and infinitesimal isometries 248

6. Extension of local isomorphisms 252

7. Equivalence problem 256

Appendices

1. Ordinary linear differential equations 267

2. A connected, locally compact metric space is separable 269

3. Partition of unity 272

4. On an arcwise connected subgroup of a Lie group 275

5. Irreducible subgroups of O(n) 277

6. Green's theorem 281

7. Factorization lemma 284

Notes

1. Connections and holonomy groups 287

2. Complete affine and Riemannian connections 291

3. Ricci tensor and scalar curvature 292

4. Spaces of constant positive curvature 294

5. Flat Riemannian manifolds 297

6. Parallel displacement of curvature 300

7. Symmetric spaces 300

8. Linear connections with recurrent curvature 304

9. The automorphism group of a geometric structure 306

10. Groups of isometries and affine transformations with maximum dimensions 308

11. Conformal transformations of a Riemannian manifold 309

Summary of Basic Notations 313

Bibliography 315

Index 325

Errata for Foundations of Differential Geometry, Volume I 330

Errata for Foundations of Differential Geometry, Volume II 331

VOLUME II

Chapter VII

Submanifolds

1. Frame bundles of a submanifold 1

2. The Gauss map 6

3. Covariant differentiation and second fundamental form 10

4. Equations of Gauss and Codazzi 22

5. Hypersurfaces in a Euclidean space 29

6. Type number and rigidity 42

7. Fundamental theorem for hypersurfaces 47

8. Auto-parallel submanifolds and totally geodesic submanifolds 53

Chapter VIII

Variations of the Length Integral

1. Jacobi fields 63

2. Jacobi fields in a Rimannian manifold 68

3. Conjugate points 71

4. Comparison theorem 76

5. The first and second variations of the length integral 79

6. Index theorem of Morse 88

7. Cut loci 96

8. Spaces of non-positive curvature 102

9. Center of gravity and fixed points of isometries 108

Chapter IX

Complex Manifolds

1. Algebraic preliminaries 114

2. Almost complex manifolds and complex manifolds 121

3. Connections in almost complex manifolds 141

4. Hermitian metrics and Kaehler metrics 146

5. Kaehler metrics in local coordinate systems 155

6. Examples of Kaehler manifolds 159

7. Holomorphic sectional curvature 165

8. De Rham decomposition of Kaehler manifolds 171

9. Curvature of Kaehler submanifolds 175

10. Hermitian connections in Hermitian vector bundles 178

Chapter X

Homogeneous Spaces

1. Invariant affine connections 186

2. Invariant connections on reductive homogeneous spaces 190

3. Invariant indefinite Riemannian metrics 200

4. Holonomy groups of invariant connections 204

5. The de Rham decomposition and irreducibility 210

6. Invariant almost complex structures 216

Chapter XI

Symmetric Spaces

1. Affine locally symmetric spaces 222

2. Symmetric spaces 225

3. The canonical connection on symmetric space 230

4. Totally geodesic submanifolds 234

5. Structure of symmetric Lie algebras 238

6. Riemannian symmetric spaces 243

7. Structure of orthogonal symmetric Lie algebras 246

8. Duality 253

9. Hermitian symmetric spaces 259

10. Examples 264

11. An outline of the classification theory

Chapter XII

Characteristic Classes

1. Weil homomorphism 293

2. Invaraint polynomials 298

3. Chern classes 305

4. Pontrjagin classes 312

5. Euler classes 314

Appendices

8. Integrable real analytic almost complex structures 321

9. Some definitions and facts on Lie algebras 325

Notes

12. Connections and holonomy groups (Supplement to Note 1) 331

13. The automorphism group of geometric structure (Supplement to Note 9) 332

14. The Laplacian 337

15. Surafces of constant curvature in R3 343

16. Index of nullity 347

17. Type number and rigidity of imbedding 349

18. Isometric imbeddings 354

19. Equivalence problems for Riemannian manifolds 357

20. Gauss-Bonnet theorem 358

21. Total curvature 361

22. Topology of Riemannian manifolds with positive curvature 364

23. Topology of Kaehler manifolds with positive curvature 368

24. Structure theorems on homogeneous complex manifols 373

25. Invariant connections on homogeneous spaces 375

26. Complex submanifolds 378

27. Minimal submanifolds 379

28. Contact structure and related structures 381

Bibliography 387

Summary of Basic Notations 455

Index for Volumes I and II 459

Errata for Foundations of Differential Geometry, Volume I 469

Errata for Foundations of Differential Geometry, Volume II 470