This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.

本书由菲尔兹奖和沃尔夫奖得主 J.W. Milnor 所著，是一本蜚声国际数学界的经典之作。内容涉及光滑流形和光滑映射，Sard 定理和 Brown 定理，映射的模2度，定向流形，向量场与 Euler 数，标架式协边，Pontryagin 构造等。全书内容简要，短小精悍。
本书为双语版，可用于双语教学。既适合高等院校数学专业高年级本科生和研究生阅读，也可供对微分拓扑有兴趣的专业人士参考。