微分拓扑是每个人都应该了解的理论。
《微分拓扑短期课程（英文）》主要介绍了微分拓扑学的相关理论，通过对机器人手臂的介绍引入课程。
《微分拓扑短期课程（英文）》共八章，包括微分拓扑简介、光滑映射、切线空间、常规值、向量丛、向量丛的结构、可积性和走向全球的局部现象。
《微分拓扑短期课程（英文）》首先讨论了流形、切线空间、余切空间，其次讨论了丛的相关知识，最后自然地以切线和余切丛的讨论而告终。
《微分拓扑短期课程（英文）》是一本适合具有一定数学水平的学生使用的教科书，内容由浅入深，适合高等院校师生、研究生及数学爱好者参考阅读。
In his inaugural lecture in 18541， Riemann introduced the concept of an \"n-fach ausgedehnte Grosse\"-roughly something that has \"n degrees of freedom\" and which we now would call an n-dimensional manifold.
Examples of manifolds are all around us and arise in many applications， but formulating the ideas in a satisfying way proved to be a challenge inspiring the creation of beautiful mathematics. As a matter of fact， much of the mathematical language of the twentieth century was created with manifolds in mind.
Modern texts often leave readers with the feeling that they are getting the answer before they know there is a problem. Taking the historical approach to this didactic problem has several disadvantages. The pioneers were brilliant mathematicians， but still they struggled for decades getting the concepts right. We must accept that we are standing on the shoulders of giants.
Preface
1 Introduction
1．1 A Robot's Arm
1．2 The Configuration Space of Two Electrons
1．3 State Spaces and Fiber Bundles
1．4 Further Examples
1．5 Compact Surfaces
1．6 Higher Dimensions
2 Smooth Manifolds
2．1 Topological Manifolds
2．2 Smooth Structures
2．3 Maximal Atlases
2．4 Smooth Maps
2．5 Submanifolds
2．6 Products and Sums
3 The Tangent Space
3．1 Germs
3．2 Smooth Bump Functions
3．3 The Tangent Space
3．4 The Cotangent Space
3．5 Derivations
4 Regular Values
4．1 The Rank
4．2 The Inverse Function Theorem
4．3 The Rank Theorem
4．4 Regular Values
4．5 Transversality
4．6 Sard's Theorem
4．7 Immersions and Imbeddings
5 Vector Bundles
5．1 Topological Vector Bundles
5．2 Transition Functions
5．3 Smooth Vector Bundles
5．4 Pre-vector Bundles
5．5 The Tangent Bundle
5．6 The Cotangent Bundle
……
6 Constructions on Vector Bundles
7 Integrability
8 Local Phenomena that Go Global
Appendix A Point Set Topology
References
Index
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