本书是公认的微分流形最佳入门教材之一，自初版出版30多年以来，一直畅销不衰，被国外各大高校广泛采用为教材。本书对国内的教学也产生了深远的影响，北京大学、复旦大学和四川大学等名校长期将其作为主要参考书。

本书的特点是对基础知识要求较少，只需有微积分、线性代数和少量拓扑学背景即可研读；而且在内容安排上力求循序渐进，透彻讲述基本概念，避免不必要的抽象和推广，并配合丰富且有详细解答的例子。非常好地降低了这门课程的学习难度。修订版在第2版色经做出较大调整的基础上，又做了一些小的但很重要的订正和更新。使本书更趋完美。

这是一本非常好的微分流形入门书。全书从一些基本的微积分知识入手，然后一点点深入介绍，主要内容有：流形介绍、多变量函数和映射、微分流形和子流形、流形上的向量场、张量和流形上的张量场、流形上的积分法、黎曼流形上的微分法以及曲率。书后有难度适中的习题，全书配有很多精美的插图。

本书非常适合初学者阅读，可作为数学系、物理系、机械系等理工科高年级本科生和研究生的教材。

I. Introduction to Manifolds

1. Preliminary Comments on Rn  1

2. Rn and Euclidean Space  4

3. Topological Manifolds  6

4. Further Examples of Manifolds. Cutting and Pasting  I l

5. Abstract Manifolds. Some Examples  14

II. Functions of Several Variables and Mappings

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 R"  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. Oifferentiable Manifolds and Submanifolds

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

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

I. 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 ofTensor 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

1. Integration in R" 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

1. Differentiation of Vector Fields along Curves in R"  289

   The Geometry of Space Curves  292

   Curvature of Plane Curves  296

2. Differentiation of Vector Fields on Submanifolds of R"  298

   Formulas for Covariant Derivatives  303

   Vxp, 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

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