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Geometry VI : Riemannian Geometry
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| Abstract | 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.show more |
| Table of Contents | 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.show more |
| Print Version | https://hdl.handle.net/2324/1000691511 |
| View fulltext | Springer Book Archive - Mathematics: 2001 |
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| Created Date | 2023.09.29 |
| Modified Date | 2024.01.30 |