《An Introduction to Differentiable Manifolds and Riemannian Geometry》

Ⅳ. Vector Fields on a Manifold106

1.The Tangent Space at a Point of a Manilold106

2. Vector Fields115

3. One-Parameter and Local One-Parameter Groups Acting on a Manifold122

4. The Existence Theorem for Ordinary Differential Equations130

5. Some Examples of One-Parameter Groups Acting on a Manifold138

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

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

8. Frobenius Theorem156

9.Homogeneous Spaces164

Notes171

Appendix Partial Proof of Theorem 4.1172

Ⅴ. Tensors and Tensor Fields on Manifolds175

1. Tangent Covectors175

Covectors on Manifolds176

Covector Fields and Mappings178

2.Bilinear Forms. The Riemannian Metric181

3. Riemannian Manifolds as Metric Spaccs185

4.Partitions of Unity191

Some Applications of the Partition of Unity193

5. Tensor Fields197

Tensors on a Vector Space197

Tensor Fields199

Mappings and Covariant Tensors200

The Symmetrizing and Alternating Transformations201

6.Multiplication of Tensors204

Multiplication of Tensors on a Vector Space205

Multiplication of Tensor Fields206

Exterior Multiplication of A207

ernating Tensors207

The Exterior Algebra on Manifolds211

7. Orientation of Manifolds and the Volume Element213

8.Exterior Differentiation217

An Application to Frobenius' Theorem221

Notes225

Ⅵ. Integration on Manifolds227

1.Integration in Rn. Domains of Integration227

Basic Properties of the Riemann Integral228

2.A Generalization to Manifolds233

Integration on Riemannian Manifolds237

3. Integration on Lie Groups241

4. Manifolds with Boundary248

5. Stokes 's Theorem for Manifolds with Boundary256

6.Homotopy of Mappings. The Fundnmental Group263

Homotopy of Paths and Loops. The Fundamental Group265

7. Some Applications of Differential Forms. The de Rham Groups271

The Homotopy Operator274

Ⅳ. Vector Fields on a Manifold106

1.The Tangent Space at a Point of a Manifold106

2. Vector Fields115

3. One-Parameter and Local One-Parameter Groups Acting on a Manifold122

4. The Existence Theorem for Ordinar Differemial Equations130

5. Some Examples of One-Parameter Groups Actmg on a Manifold13

6. One-Parameter Subgroups of Lic Groups145

7. The Lie Algebra of Vector Fields on a Mamfold149

8. Frobenius' Theorem156

9.Homogeneous Spaces164

Notes171

Appendix Partial Proof of Theorem 41172

Ⅴ.Tensors and Tensor Fields on Manifolds175

1.Tangent Covectors175

Covectors on Manifolds176

Covector Fields and Mappings178

2. Bilinear Forms. The Riemannian Metrie181

3. Riemannian Manifolds as Metric Spaces185

4.Partitions of Unity191

Some Applications of the Partition of Unity193

5.Tensor Fields197

Tensors on a Vector Space197

Tensor Fields199

Mappings and Covariant Tensors200

The Symmetrizing and Alternating Transformations201

6.Multiplication of Tensors204

Multiplication of Tensors on a Vector Space205

Multiplication of Tensor Fields206

Exterior Multiplication of Alternating Tensors207

The Exterior Algebra on Manifolds211

7. Orientation of Manifolds and the Volume Element213

8.Exterior Differentiation217

An Application to Frobenius' Theorem221

Notes225

Ⅵ. Integration on Manifolds227

1.Integration in Rn. Domains of Integration227

Basic Properties of the Riemann Integral228

2.A Generalization to Manifolds233

Integration on Riemannian Manifolds237

3. Integration on Lie Groups241

4. Manifolds with Boundary248

5. Stokes 's Theorem for Manifolds with Boundary256

6.Homnotopy of Mappings. The Fundamental Group263

Homootopy of Paths and Loops. The Fundamental Group265

7.Some Applications of Differential Forms. The de Rham Groups271

The Homotopy Operator274

8.Some Further Applications of de R ham Groups278

The de Rham Groups of Lie Groups282

9. Covering Spaces and the Fundamental Group286

Notes292

Ⅶ.Differentiation on Riemannian Manifolds294

1.Differentiation of Vector Fields along Curves in Rn294

The Geometry of Space Curves297

Curvature of Plane Curves301

2.Differentiation of Vector Fields on Submanifolds of Rn303

Formulas for Covariant Derivatives308

▽ x Y and Differentiation of Vector Fields310

3.Differentiation on Riemannian Manifolds313

Constant Vector.Fields and Parallel Displacement319

4.Addenda to the Theory of Differentiation on a Manifold321

The Curvature Tensor321

The Riemannian Connection and Exterior Differential Forms324

5. Geodesic Curves on Riemannian Manifolds326

6. The Tangent Bundle and Exponential Mapping. Normal Coordinates331

7. Some Further Properties of Geodesics338

8. Symmetric Riemannian Manifolds347

9.Some Examples353

Notes360

Ⅷ.Curvature362

1. The Geometry of Surfaces in E3362

The Principal Curvatures at a Point of a Surface366

2.The Gaussian and Mean Curvatures of a Surface370

The Theorema Egregium of Gauss373

3. Basic Properties of the Riemann Curvature Tensor378

4. The Curvature Forms and the Equations of Structure385

5. Differentiation of Covariant Tensor Fields391

6. Manifolds of Constant Curvature399

Spaces of Positive Curvature402

Spaces of Zero Curvature404

Spaces of Constant Negative Curvature405

Notes410

REFERENCES413

INDEX417