Geometry VI: Riemannian Geometry

M. M. Postnikov (auth.)3540411089, 9783540411086

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.

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Table of contents :
Front Matter….Pages I-XVIII
Affine Connections….Pages 1-13
Covariant Differentiation. Curvature….Pages 14-28
Affine Mappings. Submanifolds….Pages 29-43
Structural Equations. Local Symmetries….Pages 44-54
Symmetric Spaces….Pages 55-66
Connections on Lie Groups….Pages 67-76
Lie Functor….Pages 77-86
Affine Fields and Related Topics….Pages 87-100
Cartan Theorem….Pages 101-113
Palais and Kobayashi Theorems….Pages 114-126
Lagrangians in Riemannian Spaces….Pages 127-140
Metric Properties of Geodesics….Pages 141-158
Harmonic Functionals and Related Topics….Pages 159-175
Minimal Surfaces….Pages 176-192
Curvature in Riemannian Space….Pages 193-206
Gaussian Curvature….Pages 207-222
Some Special Tensors….Pages 223-237
Surfaces with Conformal Structure….Pages 238-247
Mappings and Submanifolds I….Pages 248-261
Submanifolds II….Pages 262-275
Fundamental Forms of a Hypersurface….Pages 276-287
Spaces of Constant Curvature….Pages 288-297
Space Forms….Pages 298-307
Four-Dimensional Manifolds….Pages 308-323
Metrics on a Lie Group I….Pages 324-332
Metrics on a Lie Group II….Pages 333-343
Jacobi Theory….Pages 344-359
Some Additional Theorems I….Pages 360-370
Some Additional Theorems II….Pages 371-380
Addendum….Pages 381-381
Smooth Manifolds….Pages 381-393
Tangent Vectors….Pages 394-403
Submanifolds of a Smooth Manifold….Pages 404-412
Vector and Tensor Fields. Differential Forms….Pages 413-435
Vector Bundles….Pages 436-453
Connections on Vector Bundles….Pages 454-474
Curvature Tensor….Pages 475-493
Back Matter….Pages 494-503