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.
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.
Review :
From the reviews of the first edition:
"... The book is .. comprehensive and original enough to be of interest to any professional geometer, but I particularly recommend it to the advanced student, who will find a host of instructive examples, exercises and vistas that few comparable texts offer... "
H.Geiges, Nieuw Archief voor Wiskunde 2002, Vol. 5/3, Issue 4
"... I found the presentation insightful and stimulating. A useful paedagogical device of the text is to make much use of both the index and coordinate-free notations, encouraging flexibility (and pragmatism) in the reader. ... the book should be of use to a wide variaty of readers: the relative beginner, with perhaps an introductory course in differential geometry, will find his horizons greatly expanded in the material for which this prepares him; while the more experienced reader will surely find the more specialised sections informative."
Robert J. Low, Mathematical Reviews, Issue 2002g
"... Insgesamt liegt ein sehr empfehlenswertes Lehrbuch einerseits zur Riemannschen Geometrie und andererseits zur Theorie differenzierbarer Mannigfaltigkeiten vor, wegen der strukturierten Breite der Darstellung sehr gut geeignet sowohl zum Selbststudium für Studierende mathematischer Disziplinen als auch für Dozenten als Grundlage einschlägiger Lehrveranstaltungen."
P. Paukowitsch, Wien (IMN - Internationale Mathematische Nachrichten 190, S. 64-65, 2002)
"M.M. Postnikov has written a well-structured and readable book with a satisfying sense of completeness to it. The reviewer believes this book deserves a place next to the already existing literature on Riemannian geometry, principally as a basis for teaching a course on abstract Riemannian geometry (after an introduction to differentiable manifolds) but also as a reference work." (Eric Boeckx, Zentralblatt MATH, Vol. 993 (18), 2002)
