微分几何在现代理论物理和应用数学中扮演着越来越重要的角色。本书给出了在理论物理和应用数学中很重要的几何知识的引入，包括，流形、张量场、微分形式、联络、辛几何、李群作用、族以及自旋。

本书以一种非正式的形式写作，作者给出了1000多例子重在强调对一般理论的深刻理解。本书将要为读者很好的学习拉格郎日现代处理方法、哈密顿力学、电磁、规范场，相对论以及万有引力做充足的准备。

本书很适合作为物理、数学以及工程专业的高年级本科生以及研究生的教程，也是一本很难得自学教程。

Preface

Introduction

1 The concept of a manifold

2 Vector and tensor fields

3 Mappings of tensors induced by mappings of manifolds

4 Lie derivative

5 Exterior algebra

6 Differential calulus of forms

7 Integral calculus of forms

8 Particular cases and applications of Stokes'theorem

9 Poincare lemma and cohomologies

10 Lie groups:basic facts

11 Differential geometry on Lie groups

12 Representations of Lie groups and Lie algebras

13 Actions of Lie groups and Lie algebras on manifolds

14 Hamiltonian mechanics and symplectic manifolds

15 Parallel transport and linear connection on M

16 Field theory and the language of forms

17 Differential geometry on T M and T*M

18 Hamiltonian and Lagrangian equations

19 Linear connection and the frame bundle

20 Connection on a principal G-bundle

21 Gauge theories and connections

22 Spinor fields and the Dirac operator

Appendix A Some relevant algebrai structures

Appendix B Starring