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Ⅳ. Vector Fields on a Manifold 106

1. The Tangent Space at a Point of a Manilold 106

2. Vector Fields 115

3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 122

4. The Existence Theorem for Ordinary Differential Equations 130

5. Some Examples of One-Parameter Groups Acting on a Manifold 138

6. One-Parameter Subgroups of Lie Groups 1- 45

7. The Lie Algebra of Vector Fields on a Manifold 1- 49

8. Frobenius Theorem 156

9. Homogeneous Spaces 164

Notes 171

Appendix Partial Proof of Theorem 4.1 172

Ⅴ. Tensors and Tensor Fields on Manifolds 175

1. Tangent Covectors 175

Covectors on Manifolds 176

Covector Fields and Mappings 178

2. Bilinear Forms. The Riemannian Metric 181

3. Riemannian Manifolds as Metric Spaccs 185

4. Partitions of Unity 191

Some Applications of the Partition of Unity 193

5. Tensor Fields 197

Tensors on a Vector Space 197

Tensor Fields 199

Mappings and Covariant Tensors 200

The Symmetrizing and Alternating Transformations 201

6. Multiplication of Tensors 204

Multiplication of Tensors on a Vector Space 205

Multiplication of Tensor Fields 206

Exterior Multiplication of A 207

ernating Tensors 207

The Exterior Algebra on Manifolds 211

7. Orientation of Manifolds and the Volume Element 213

8. Exterior Differentiation 217

An Application to Frobenius' Theorem 221

Notes 225

Ⅵ. Integration on Manifolds 227

1. Integration in Rn. Domains of Integration 227

Basic Properties of the Riemann Integral 228

2. A Generalization to Manifolds 233

Integration on Riemannian Manifolds 237

3. Integration on Lie Groups 241

4. Manifolds with Boundary 248

5. Stokes 's Theorem for Manifolds with Boundary 256

6. Homotopy of Mappings. The Fundnmental Group 263

Homotopy of Paths and Loops. The Fundamental Group 265

7. Some Applications of Differential Forms. The de Rham Groups 271

The Homotopy Operator 274

Ⅳ. Vector Fields on a Manifold 106

1. The Tangent Space at a Point of a Manifold 106

2. Vector Fields 115

3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 122

4. The Existence Theorem for Ordinar Differemial Equations 130

5. Some Examples of One-Parameter Groups Actmg on a Manifold 13

6. One-Parameter Subgroups of Lic Groups 145

7. The Lie Algebra of Vector Fields on a Mamfold 149

8. Frobenius' Theorem 156

9. Homogeneous Spaces 164

Notes 171

Appendix Partial Proof of Theorem 41 172

Ⅴ. Tensors and Tensor Fields on Manifolds 175

1. Tangent Covectors 175

Covectors on Manifolds 176

Covector Fields and Mappings 178

2. Bilinear Forms. The Riemannian Metrie 181

3. Riemannian Manifolds as Metric Spaces 185

4. Partitions of Unity 191

Some Applications of the Partition of Unity 193

5. Tensor Fields 197

Tensors on a Vector Space 197

Tensor Fields 199

Mappings and Covariant Tensors 200

The Symmetrizing and Alternating Transformations 201

6. Multiplication of Tensors 204

Multiplication of Tensors on a Vector Space 205

Multiplication of Tensor Fields 206

Exterior Multiplication of Alternating Tensors 207

The Exterior Algebra on Manifolds 211

7. Orientation of Manifolds and the Volume Element 213

8. Exterior Differentiation 217

An Application to Frobenius' Theorem 221

Notes 225

Ⅵ. Integration on Manifolds 227

1. Integration in Rn. Domains of Integration 227

Basic Properties of the Riemann Integral 228

2. A Generalization to Manifolds 233

Integration on Riemannian Manifolds 237

3. Integration on Lie Groups 241

4. Manifolds with Boundary 248

5. Stokes 's Theorem for Manifolds with Boundary 256

6. Homnotopy of Mappings. The Fundamental Group 263

Homootopy of Paths and Loops. The Fundamental Group 265

7. Some Applications of Differential Forms. The de Rham Groups 271

The Homotopy Operator 274

8. Some Further Applications of de R ham Groups 278

The de Rham Groups of Lie Groups 282

9. Covering Spaces and the Fundamental Group 286

Notes 292

Ⅶ. Differentiation on Riemannian Manifolds 294

1. Differentiation of Vector Fields along Curves in Rn 294

The Geometry of Space Curves 297

Curvature of Plane Curves 301

2. Differentiation of Vector Fields on Submanifolds of Rn 303

Formulas for Covariant Derivatives 308

▽ x Y and Differentiation of Vector Fields 310

3. Differentiation on Riemannian Manifolds 313

Constant Vector.Fields and Parallel Displacement 319

4. Addenda to the Theory of Differentiation on a Manifold 321

The Curvature Tensor 321

The Riemannian Connection and Exterior Differential Forms 324

5. Geodesic Curves on Riemannian Manifolds 326

6. The Tangent Bundle and Exponential Mapping. Normal Coordinates 331

7. Some Further Properties of Geodesics 338

8. Symmetric Riemannian Manifolds 347

9. Some Examples 353

Notes 360

Ⅷ. Curvature 362

1. The Geometry of Surfaces in E3 362

The Principal Curvatures at a Point of a Surface 366

2. The Gaussian and Mean Curvatures of a Surface 370

The Theorema Egregium of Gauss 373

3. Basic Properties of the Riemann Curvature Tensor 378

4. The Curvature Forms and the Equations of Structure 385

5. Differentiation of Covariant Tensor Fields 391

6. Manifolds of Constant Curvature 399

Spaces of Positive Curvature 402

Spaces of Zero Curvature 404

Spaces of Constant Negative Curvature 405

Notes 410

REFERENCES 413

INDEX 417