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 pre- sents a more general theory of manifolds with a linear con- nection. Having in mind different generalizations of Rieman- nian 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 trans- formation 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.

Inhalt

1. Affine Connections.- 2. Covariant Differentiation. Curvature.- 3. Affine Mappings. Submanifolds.- 4. Structural Equations. Local Symmetries.- 5. Symmetric Spaces.- 6. Connections on Lie Groups.- 7. Lie Functor.- 8. Affine Fields and Related Topics.- 9. Cartan Theorem.- 10. Palais and Kobayashi Theorems.- 11. Lagrangians in Riemannian Spaces.- 12. Metric Properties of Geodesics.- 13. Harmonic Functionals and Related Topics.- 14. Minimal Surfaces.- 15. Curvature in Riemannian Space.- 16. Gaussian Curvature.- 17. Some Special Tensors.- 18. Surfaces with Conformal Structure.- 19. Mappings and Submanifolds I.- 20. Submanifolds II.- 21. Fundamental Forms of a Hypersurface.- 22. Spaces of Constant Curvature.- 23. Space Forms.- 24. Four-Dimensional Manifolds.- 25. Metrics on a Lie Group I.- 26. Metrics on a Lie Group II.- 27. Jacobi Theory.- 28. Some Additional Theorems I.- 29. Some Additional Theorems II.- Addendum.- 30. Smooth Manifolds.- 31. Tangent Vectors.- 32. Submanifolds of a Smooth Manifold.- 33. Vector and Tensor Fields. Differential Forms.- 34. Vector Bundles.- 35. Connections on Vector Bundles.- 36. Curvature Tensor.- Bianchi Identity.- Suggested Reading.