Geometry VI : Riemannian Geometry

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 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 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 references book for professional mathematicians and physicists.

• Preface
• Ch. 1 Affine Connections 1
• Ch. 2 Covariant Differentiation. Curvature 14
• Ch. 3 Affine Mappings. Submanifolds 29
• Ch. 4 Structural Equations. Local Symmetries 44
• Ch. 5 Symmetric Spaces 55
• Ch. 6 Connections on Lie Groups 67
• Ch. 7 Lie Functor 77
• Ch. 8 Affine Fields and Related Topics 87
• Ch. 9 Cartan Theorem 101
• Ch. 10 Palais and Kobayashi Theorems 114
• Ch. 11 Lagrangians in Riemannian Spaces 127
• Ch. 12 Metric Properties of Geodesics 141
• Ch. 13 Harmonic Functionals and Related Topics 159
• Ch. 14 Minimal Surfaces 176
• Ch. 15 Curvature in Riemannian Space 193
• Ch. 16 Gaussian Curvature 207
• Ch. 17 Some Special Tensors 223
• Ch. 18 Surfaces with Conformal Structure 238
• Ch. 19 Mappings and Submanifolds I 248
• Ch. 20 Submanifolds II 262
• Ch. 21 Fundamental Forms of a Hypersurface 276 vCh. 22 Spaces of Constant Curvature 288
• Ch. 23 Space Forms 298
• Ch. 24 Four-Dimensional Manifolds 308
• Ch. 25 Metrics on a Lie Group I 324
• Ch. 26 Metrics on a Lie Group II 333
• Ch. 27 Jacobi Theory 344
• Ch. 28 Some Additional Theorems I 360
• Ch. 29 Some Additional Theorems II 371 Addendum 381
• Ch. 30 Smooth Manifolds 381
• Ch. 31 Tangent Vectors 394
• Ch. 32 Submanifolds of a Smooth Manifold 404
• Ch. 33 Vector and Tensor Fields. Differential Forms 413
• Ch. 34 Vector Bundles 436
• Ch. 35 Connections on Vector Bundles 454
• Ch. 36 Curvature Tensor 475
• Suggested Reading 494
• Index 495