微分几何中的度量结构

Preface

Chapter 1.Differentiable Manifolds

1.Basic Definitions

2.Differentiable Maps

3.Tangent Vectors

4.The Derivative

5.The Inverse and Implicit Function Theorems

6.Submanifolds

7.Vector Fields

8.The Lie Bracket

9.Distributions and Frobenius Theorem

10.Multilinear Algebra and Tensors

11.Tensor Fields and Differential Forms

12.Integration on Chains

13.The Local Version of Stokes' Theorem

14.Orientation and the Global Version of Stokes' Theorem

15.Some Applications of Stokes' Theorem

Chapter 2.Fiber Bundles

1.Basic Definitions and Examples

2.Principal and Associated Bundles

3.The Tangent Bundle of Sn

4.Cross—Sections of Bundles

5.Pullback and Normal Bundles

6.Fibrations and the Homotopy Lifting／Covering Properties

7.Grassmannians and Universal Bundles

Chapter 3.Homotopy Groups and Bundles Over Spheres

1.Differentiable Approximations

2.Homotopy Groups

3.The Homotopy Sequence of a Fibration

4.Bundles Over Spheres

5.The Vector Bundles Over Low—Dimensional Spheres

Chapter 4.Connections and Curvature

1.Connections on Vector Bundles

2.Covariant Derivatives

3.The Curvature Tensor of a Connection

4.Connections on Manifolds

5.Connections on Principal Bundles

Chapter 5.Metric Structures

1.Euclidean Bundles and Riemannian Manifolds

2.Riemannian Connections

3.Curvature Quantifiers

4.Isometric Immersions

5.Riemannian Submersions

6.The Gauss Lemma

7.Length—Minimizing Properties of Geodesics

8.First and Second Variation of Arc—Length

9.Curvature and Topology

10.Actions of Compact Lie Groups

Chapter 6.Characteristic Classes

1.The Weil Homomorphism

2.Pontrjagin Classes

3.The Euler Class

4.The Whitney Sum Formula for Pontrjagin and Euler Classes

5.Some Examples

6.The Unit Sphere Bundle and the Euler Class

7.The Generalized Gauss—Bonnet Theorem

8.Complex and Symplectic Vector Spaces

9.Chern Classes

Bibliography

Index

This text is an elementary introduction to differential geometry. Although it was written for a graduate-level audience, the only requisite is a solid back-ground in calculus, linear algebra, and basic point-set topology.
　　The first chapter covers the fundamentals of differentiable manifolds that are the bread and butter of differential geometry. All the usual topics are covered, culnunating in Stokes' theorem together with some applications. The stu dents' first contact with the subject can be overwhelming because of the wealth of abstract definitions involved, so examples have been stressed throughout. One concept, for instance, that students often find confusing is the definition of tangent vectors. They are first told that these are derivations on certain equiv-alence classes of functions, but later that the tangent space of Rl is "the same" as Rn. We have tried to keep these spaces separate and to carefully explain how a vector space E is canonically isomorphic to its tangent space at a point. This subtle distinction becomes essential when later discussing the vertical bundle of a given vector bundle.