About the book

This book of problems is the result of a course in differential geometry and topology, given at the mechanics-and-mathematics department of Moscow State University.

It contains problems practically for all sections of the seminar course. Although certain textbooks and books of problems indicated in the bibliography list were used in preparation of this volume, a considerable number of the problems were prepared for this book expressly.

The material is distributed over the sections as in text book [3]. Some problems, however, touch upon topics outside the lectures. In these cases, the corresponding sections are supplied with additional definitions and explanations.

In conclusion, the authors express their sincere gratitude to all those who helped to publish this work.

Contents

Preface 5
1. Application of Linear Algebra to Geometry 7
2. Systems of Coordinates 9
3. Riemannian Metric 14
4. Theory of Curves 16
5. Surfaces 34
6. Manifolds 53
7. Transformation Groups 60
8. Vector Fields 64
9. Tensor Analysis 70
10. Differential Forms, Integral Formulae, De Rham
Cohomology 75
11. General Topology 81
12. Homotopy Theory 87
13. Covering Maps, Fibre Spaces, Riemann Surfaces 97
14. Degree of Mapping 105
15. Simplest Variational Problems 108
Answers and Hints 113
Bibliography 208