本书从黎曼度量及联络出发，介绍了黎曼流形研究中的各种基本概念和技巧。以测地线的研究为重点讨论了各种形式的比较定理和Morse指数定理，同时还介绍了子流形几何学。书中也勾画了近代微分几何中的一些重大成果，如球面定理、正质量猜想以及几乎平坦流形等，最后还列举了当今微分几何研究中一些尚待解决的问题。

Starting from Riemannian metrics and connections, this book introduces various basic concepts and techniques in the study of Riemannian manifolds. Focusing on the study of geodesics, various forms of comparison theorems and Morse index theorems were discussed, and submanifold geometry was also introduced. The book also outlines some major achievements in modern differential geometry, such as the spherical theorem, the conjecture of positive mass, and almost flat manifolds. Finally, it lists some unresolved problems in current differential geometry research.