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PART Ⅰ General Differential Theory1

CHAPTER Ⅰ Differential Calculus3

1．Categories4

2．Topological Vector Spaces5

3．Derivatives and Composition of Maps8

4．Integration and Taylor's Formula12

5．The Inverse Mapping Theorem15

CHAPTER Ⅱ Manifolds22

1．Atlases,Charts,Morphisms22

2．Submanifolds,Immersions,Submersions25

3．Partitions of Unity33

4．Manifolds with Boundary39

CHAPTER Ⅲ Vector Bundles43

1．Definition,Pull Backs43

2．The Tangent Bundle51

3．Exact Sequences of Bundles52

4．Operations on Vector Bundles58

5．Splitting of Vector Bundles63

CHAPTER Ⅳ Vector Fields and Differential Equations66

1．Existence Theorem for Differential Equations67

2．Vector Fields,Curves,and Flows88

3．Sprays96

4．The Flow of a Spray and the Exponential Map105

5．Existence of Tubular Neighborhoods110

6．Uniqueness of Tubular Neighborhoods112

CHAPTER Ⅴ Operations on Vector Fields and Differential Forms116

1．Vector Fields,Differential Operators,Brackets116

2．Lie Derivative122

3．Exterior Derivative124

4．The Poincaré Lemma137

5．Contractions and Lie Derivative139

6．Vector Fields and 1-Forms Under Self Duality143

7．TheCanonical 2-Form149

8．Darboux's Theorem151

CHAPTER Ⅵ The Theorem of Frobenius155

1．Statement of the Theorem155

2．Differential Equations Depending on a Parameter160

3．Proof of the Theorem161

4．The Global Formulation162

5．Lie Groups and Subgroups165

PART Ⅱ Metrics,Covariant Derivatives,and Riemannian Geometry171

CHAPTER Ⅶ Metrics173

1．Definition and Functoriality173

2．The Hilbert Group177

3．Reduction to the Hilbert Group180

4．Hilbertian Tubular Neighborhoods184

5．The Morse-Palais Lemma186

6．The Riemannian Distance189

7．The Canonical Spray192

CHAPTER Ⅷ Covariant Derivatives and Geodesics196

1．Basic Properties196

2．Sprays and Covariant Derivatives199

3．Derivative Along a Curve and Parallelism204

4．The Metric Derivative209

5．More Local Results on the Exponential Map215

6．Riemannian Geodesic Length and Completeness221

CHAPTER Ⅸ Curvature231

1．The Riemann Tensor231

2．Jacobi Lifts239

3．Application of Jacobi Lifts to Texpx246

4．Convexity Theorems255

5．Taylor Expansions263

CHAPTER Ⅹ Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle267

1．Convexity of Jacobi Lifts267

2．Global Tubular Neighborhood of a Totally Geodesic Submanifold271

3．More Convexity and Comparison Results276

4．Splitting of the Double Tangent Bundle279

5．Tensorial Derivative of a Curve in TX and of the Exponential Map286

6．The Flow and the Tensorial Derivative291

CHAPTER Ⅺ Curvature and the Variation Formula294

1．The Index Form,Variations,and the Second Variation Formula294

2．Growth of a Jacobi Lift304

3．The Semi Parallelogram Law and Negative Curvature309

4．Totally Geodesic Submanifolds315

5．Rauch Comparison Theorem318

CHAPTER Ⅻ An Example of Seminegative Curvature322

1．POSn(R) as a Riemannian Manifold322

2．The Metric Increasing Property of the Exponential Map327

3．Totally Geodesic and Symmetric Submanifolds332

CHAPTER ⅩⅢ Automorphisms and Symmetries339

1．The Tensorial Second Derivative342

2．Alternative Definitions of Killing Fields347

3．Metric Killing Fields351

4．Lie Algebra Properties of Killing Fields354

5．Symmetric Spaces358

6．Parallelism and the Riemann Tensor365

CHAPTER ⅩⅣ Immersions and Submersions369

1．The Covariant Derivative on a Submanifold369

2．The Hessian and Laplacian on a Submanifold376

3．The Covariant Derivative on a Riemannian Submersion383

4．The Hessian and Laplacian on a Riemannian Submersion387

5．The Riemann Tensor on Submanifolds390

6．The Riemann Tensor on a Riemannian Submersion393

PART Ⅲ Volume Forms and Integration395

CHAPTER ⅩⅤ Volume Forms397

1．Volume Forms and the Divergence397

2．Covariant Derivatives407

3．The Jacobian Determinant of the Exponential Map412

4．The Hodge Star on Forms418

5．Hodge Decomposition of Differential Forms424

6．Volume Forms in a Submersion428

7．Volume Forms on Lie Groups and Homogeneous Spaces435

8．Homogeneously Fibered Submersions440

CHAPTER ⅩⅥ Integration of Differential Forms448

1．Sets of Measure 0448

2．Change of Variables Formula453

3．Orientation461

4．The Measure Associated with a Differential Form463

5．Homogeneous Spaces471

CHAPTER ⅩⅦ Stokes'Theorem475

1．Stokes'Theorem for a Rectangular Simplex475

2．Stokes'Theorem on a Manifold478

3．Stokes'Theorem with Singularities482

CHAPTER ⅩⅧ Applications of Stokes'Theorem489

1．The Maximal de Rham Cohomology489

2．Moser's Theorem496

3．The Divergence Theorem497

4．The Adjoint of d for Higher Degree Forms501

5．Cauchy's Theorem503

6．The Residue Theorem507

APPENDIX The Spectral Theorem511

1．Hilbert Space511

2．Functionals and Operators512

3．Hermitian Operators515

Bibliography523

Index531