沃尔斯齐普所著的《微分几何中的度量结构(英文版)》是一部学习微分流形和纤维丛的入门书籍，从矩阵微分几何的观点出发研究纤维丛，讨论了欧几里得丛；黎曼连通；曲率和Chern-Weil理论；也包括Pontrjagin, Euler, 和Chern 的向量丛特征类，并通过球上的丛详细阐释了这些概念。适用于对微分几何、流形以及丛感兴趣的读者。

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