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Ⅰ.Introduction to Manifolds 1

1.Preliminary Comments on Rn 1

2.Rn and Euclidean Space 4

3.Topological Manifolds 6

4.Further Examples of Manifolds Cutting and Pasting 11

5.Abstract Manifolds Some Examples 14

Ⅱ.Functions of Several Variables and Mappings 20

1.Differentiability for Functions of Several Variables 20

2.Differentiability of Mappings and Jacobians 25

3.The Space of Tangent Vectors at a Point of Rn 29

4.Another Definition of Ta （Rn） 32

5.Vector Fields on Open Subsets of Rn 36

6.The Inverse Function Theorem 41

7.The Rank of a Mapping 46

Ⅲ.Differentiable Manifolds and Submanifolds 52

1.The Definition of a Differentiable Manifold 52

2.Further Examples 59

3.Differentiable Functions and Mappings 65

4.Rank of a Mapping, Immersions 68

5.Submanifolds 74

6.Lie Groups 80

7.The Action of a Lie Group on a Manifold Transformation Groups 87

8.The Action of a Discrete Group on a Manifold 93

9.Covering Manifolds 98

Ⅳ Vector Fields on a Manifold 104

1.The Tangent Space at a Point of a Manifold 104

2.Vector Fields 113

3.One-Parameter and Local One-Parameter Groups Acting on a Manifold 119

4.The Existence Theorem for Ordinary Differential Equations 127

5.Some Examples of One-Parameter Groups Acting on a Manifold 135

6.One-Parameter Subgroups of Lie Groups 142

7.The Lie Algebra of Vector Fields on a Manifold 146

8.Frobenius’s Theorem 153

9.Homogeneous Spaces 160

Ⅴ Tensors and Tensor Fields on Manifolds 171

1.Tangent Covectors 171

Covectors on Manifolds 172

Covector Fields and Mappings 174

2.Bilinear Forms.The Riemannian Metric 177

3.Riemannian Manifolds as Metric Spaces 181

4.Partitions of Unity 186

Some Applications of the Partition of Unity 188

5.Tensor Fields 192

Tensors on a Vector Space 192

Tensor Fields 194

Mappings and Covariant Tensors 195

The Symmetrizing and Alternating Transformations 196

6.Multiplication of Tensors 199

Multiplication of Tensors on a Vector Space 199

Multiplication of Tensor Fields 201

Exterior Multiplication of Alternating Tensors 202

The Exterior Algebra on Manifolds 206

7.Orientation of Manifolds and the Volume Element 207

8.Exterior Differentiation 212

An Application to Frobenius’s Theorem 216

Ⅵ.Integration on Manifolds 223

1.Integration in Rn Domains of Integration 223

Basic Properties of the Riemann Integral 224

2.A Generalization to Manifolds 229

Integration on Riemannian Manifolds 232

3.Integration on Lie Groups 237

4.Manifolds with Boundary 243

5.Stokes’s Theorem for Manifolds 251

6.Homotopy of Mappings.The Fundamental Group 258

Homotopy of Paths and Loops.The Fundamental Group 259

7.Some Applications of Differential Forms.The de Rham Groups 265

The Homotopy Operator 268

8.Some Further Applications of de Rham Groups 272

The de Rham Groups of Lie Groups 276

9.Covering Spaces and Fundamental Group 280

Ⅶ.Differentiation on Riemannian Manifolds 289

1.Differentiation of Vector Fields along Curves in Rn 289

The Geometry of Space Curves 292

Curvature of Plane Curves 296

2.Differentiation of Vector Fields on Submanifolds of Rn 298

Formulas for Covariant Derivatives 303

？xp Y and Differentiation of Vector Fields 305

3.Differentiation on Riemannian Manifolds 308

Constant Vector Fields and Parallel Displacement 314

4.Addenda to the Theory of Differentiation on a Manifold 316

The Curvature Tensor 316

The Riemannian Connection and Exterior Differential Forms 319

5.Geodesic Curves on Riemannian Manifolds 321

6.The Tangent Bundle and Exponential Mapping.Normal Coordinates 326

7.Some Further Properties of Geodesics 332

8.Symmetric Riemannian Manifolds 340

9.Some Examples 346

Ⅷ.Curvature 355

1.The Geometry of Surfaces in E3 355

The Principal Curvatures at a Point of a Surface 359

2.The Gaussian and Mean Curvatures of a Surface 363

The Theorema Egregium of Gauss 366

3.Basic Properties of the Riemann Curvature Tensor 371

4.Curvature Forms and the Equations of Structure 378

5.Differentiation of Covariant Tensor Fields 384

6.Manifolds of Constant Curvature 391

Spaces of Positive Curvature 394

Spaces of Zero Curvature 396

Spaces of Constant Negative Curvature 397

REFERENCES 403

INDEX 411