This is the translation from the Russian of a book on differential geometry and topology, written by two world famous mathematicians, published by Nauka in 1987. It reflects the authors’ experience in teaching a modern and coherent course on these topics. Concerning the authors’ intention and the principles of the presentation we quote from their preface: “The book is intended for the mathematics, physics and mechanics education of second and third year university students. A minimal degree of abstractness of the language, and the style of presentation of the material, consistency with the language of mechanics and physics, and the preference for the material important for natural sciences were the basic principles of the presentation.
The book contains three parts: Part I. Basic concepts of differential geometry (including the differential geometry of curves and surfaces, the concept of a manifold and the simplest examples); Part II. Tensors. Riemannian geometry (including the fundamental principles of tensor analysis and Riemannian geometry, the general Stokes formula and examples); Part III. Basic elements of topology (including examples of differential forms, the degree of a mapping and applications, homotopy, vector fields, singular points of vector fields, the fundamental group and coverings. Seven appendices are added (Appendices 2 and 3 give the simplest geometric ideas fundamental to physics; Appendix 7 includes selected problems and exercises). A list of references with 36 items and an index is presented at the end.
The book is written in an elegant style with a large number of very instructive examples. Beginners can benefit from the authors’ ability to explain details accurately while preserving a clear emphasis on the main ideas. Despite a few typographical errors, the book can be highly recommended to students as a supplementary final undergraduate or beginning graduate textbook.