微分流形与黎曼几何 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Ⅰ.Introduction to Manifolds1

1.Preliminary Comments on Rn1

2.Rn and Euclidean Space4

3.Topological Manifolds6

4.Further Examples of Manifolds Cutting and Pasting11

5.Abstract Manifolds Some Examples14

Ⅱ.Functions of Several Variables and Mappings20

1.Differentiability for Functions of Several Variables20

2.Differentiability of Mappings and Jacobians25

3.The Space of Tangent Vectors at a Point of Rn29

4.Another Definition of Ta （Rn）32

5.Vector Fields on Open Subsets of Rn36

6.The Inverse Function Theorem41

7.The Rank of a Mapping46

Ⅲ.Differentiable Manifolds and Submanifolds52

1.The Definition of a Differentiable Manifold52

2.Further Examples59

3.Differentiable Functions and Mappings65

4.Rank of a Mapping, Immersions68

5.Submanifolds74

6.Lie Groups80

7.The Action of a Lie Group on a Manifold Transformation Groups87

8.The Action of a Discrete Group on a Manifold93

9.Covering Manifolds98

Ⅳ Vector Fields on a Manifold104

1.The Tangent Space at a Point of a Manifold104

2.Vector Fields113

3.One-Parameter and Local One-Parameter Groups Acting on a Manifold119

4.The Existence Theorem for Ordinary Differential Equations127

5.Some Examples of One-Parameter Groups Acting on a Manifold135

6.One-Parameter Subgroups of Lie Groups142

7.The Lie Algebra of Vector Fields on a Manifold146

8.Frobenius’s Theorem153

9.Homogeneous Spaces160

Ⅴ Tensors and Tensor Fields on Manifolds171

1.Tangent Covectors171

Covectors on Manifolds172

Covector Fields and Mappings174

2.Bilinear Forms.The Riemannian Metric177

3.Riemannian Manifolds as Metric Spaces181

4.Partitions of Unity186

Some Applications of the Partition of Unity188

5.Tensor Fields192

Tensors on a Vector Space192

Tensor Fields194

Mappings and Covariant Tensors195

The Symmetrizing and Alternating Transformations196

6.Multiplication of Tensors199

Multiplication of Tensors on a Vector Space199

Multiplication of Tensor Fields201

Exterior Multiplication of Alternating Tensors202

The Exterior Algebra on Manifolds206

7.Orientation of Manifolds and the Volume Element207

8.Exterior Differentiation212

An Application to Frobenius’s Theorem216

Ⅵ.Integration on Manifolds223

1.Integration in Rn Domains of Integration223

Basic Properties of the Riemann Integral224

2.A Generalization to Manifolds229

Integration on Riemannian Manifolds232

3.Integration on Lie Groups237

4.Manifolds with Boundary243

5.Stokes’s Theorem for Manifolds251

6.Homotopy of Mappings.The Fundamental Group258

Homotopy of Paths and Loops.The Fundamental Group259

7.Some Applications of Differential Forms.The de Rham Groups265

The Homotopy Operator268

8.Some Further Applications of de Rham Groups272

The de Rham Groups of Lie Groups276

9.Covering Spaces and Fundamental Group280

Ⅶ.Differentiation on Riemannian Manifolds289

1.Differentiation of Vector Fields along Curves in Rn289

The Geometry of Space Curves292

Curvature of Plane Curves296

2.Differentiation of Vector Fields on Submanifolds of Rn298

Formulas for Covariant Derivatives303

？xp Y and Differentiation of Vector Fields305

3.Differentiation on Riemannian Manifolds308

Constant Vector Fields and Parallel Displacement314

4.Addenda to the Theory of Differentiation on a Manifold316

The Curvature Tensor316

The Riemannian Connection and Exterior Differential Forms319

5.Geodesic Curves on Riemannian Manifolds321

6.The Tangent Bundle and Exponential Mapping.Normal Coordinates326

7.Some Further Properties of Geodesics332

8.Symmetric Riemannian Manifolds340

9.Some Examples346

Ⅷ.Curvature355

1.The Geometry of Surfaces in E3355

The Principal Curvatures at a Point of a Surface359

2.The Gaussian and Mean Curvatures of a Surface363

The Theorema Egregium of Gauss366

3.Basic Properties of the Riemann Curvature Tensor371

4.Curvature Forms and the Equations of Structure378

5.Differentiation of Covariant Tensor Fields384

6.Manifolds of Constant Curvature391

Spaces of Positive Curvature394

Spaces of Zero Curvature396

Spaces of Constant Negative Curvature397

REFERENCES403

INDEX411