本书为《国外数学名著系列》丛书之一。该丛书是科学出版社组织学术界多位知名院士、专家精心筛选出来的一批基础理论类数学著作，读者对象面向数学系高年级本科生、研究生及从事数学专业理论研究的科研工作者。

本册为《几何(Ⅵ黎曼几何影印版)60》。

This book treats that part of Riemannian geometry related to more classical topics in a very original，clear and solid style.Before going to Riemannian geometry，the author presents a more general theory of manifolds with a linear connection.Having in mind different generalizations of Riemannian manifolds，it is clearly stressed which notions and theorems belong to Riemannian geometry and which of them are of a more general nature.Much attention is paid to transformation groups of smooth manifolds.

Throughout the book，different aspects of symmetric spaces are treated.The author successfully combines the co-ordinate and invariant approaches to differential geometry，which give the reader tools for practical calculations as well as a theoretical understanding of the subject.The book contains a very useful large appendix on foundations of differentiable manifolds and basic structures on them which makes it self-contained and practically independent from other sources.

The results are well presented and useful for students in mathematics and theoretical physics，and for experts in these fields.The book can serve as a textbook for students doing geometry，as well as a reference book for professional mathematicians and physicists.

Preface

Chapter 1 .Affine Connections

 1.Connection on a Manifold

 2.Covariant Differentiation and Parallel Translation Along a Curve

 3.Geodesics

 4.Exponential Mapping and Normal Neighborhoods

 5.Whitehead Theorem

 6.Normal Convex Neighborhoods

 7.Existence of Leray Coverings

Chapter 2. Covariant Differentiation.Curvature

Chapter 3. Affine Mappings.Submanifolds

Chapter 4. Structural Equations.Local Symmetries

Chapter 5. Symmetric Spaces

Chapter 6. Connections on Lie Groups

Chapter 7. Lie Functor

Chapter 8. Affine Fields and Related Topics

Chapter 9. Cartan Theorem

Chapter 10. Palais and Kobayashi Theorems

Chapter 11. Lagrangians in Riemannian Spaces

Chapter 12. Metric Properties of Geodesics

Chapter 13. Harmonic Functionals and Related Topics

Chapter 14. Minimal Surfaces

Chapter 15. Curvature in Riemannian Space

Chapter 16. Gaussian Curvature

Chapter 17. Some Special Tensors

Chapter 18. Surfaces with Conformal Structure

Chapter 19. Mappings and Submanifolds Ⅰ

Chapter 20. Submanifolds Ⅱ

Chapter 21. Fundamental Forms of a Hypersurface

Chapter 22. Spaces of Constant Curvature

Chapter 23. Space Forms

Chapter 24. Four-Dimensional Manifolds

Chapter 25. Metrics on a Lie Group Ⅰ

Chapter 26. Metrics on a Lie Group Ⅱ

Chapter 27. Jacobi Theory

Chapter 28. Some Additional Theorems Ⅰ

Chapter 29. Some Additional Theorems Ⅱ

Chapter 30. Smooth Manifolds

Chapter 31. Tangent Vectors

Chapter 32. Submanifolds of a Smooth Manifold

Chapter 33. Vector and Tensor Fields Differential Forms

Chapter 34. Vector Bundles

Chapter 35. Connections on Vector Bundles

Chapter 36. Curvature Tensor

Suggested Reading

Index