Overview

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.

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9783540411086

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Reviews

From the reviews of the first edition: <p>.,. 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... <p>H.Geiges, Nieuw Archief voor Wiskunde 2002, Vol. 5/3, Issue 4 <p>.,. 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. <p>Robert J. Low, Mathematical Reviews, Issue 2002g <p>.,. 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 fA1/4r Studierende mathematischer Disziplinen als auch fA1/4r Dozenten als Grundlage einschlAgiger Lehrveranstaltungen. <p>P. Paukowitsch, Wien (IMN - Internationale Mathematische Nachrichten 190, S. 64-65, 2002) <p> 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 acourse on abstract Riemannian geometry (after an introduction to differentiable manifolds) but also as a reference work. (Eric Boeckx, Zentralblatt MATH, Vol. 993 (18), 2002)

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