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Riemannian geometry. 2nd ed. (English) Zbl 1220.53002
Graduate Texts in Mathematics 171. New York, NY: Springer (ISBN 0-387-29246-2/hbk). xvi, 401 p. (2006).
Publisher’s description: Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.
Important additions to this new edition include:
– A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise;
– An increased number of coordinate calculations of connection and curvature;
– General fomulas for curvature on Lie Groups and submersions;
– Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgotten proof by Berger;
– Several recent results about manifolds with positive curvature.
See the review of the first edition [Graduate Texts in Mathematics. 171. New York, NY: Springer (1998; Zbl 0914.53001)].

MSC:
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching
53Cxx Global differential geometry
57R19 Algebraic topology on manifolds and differential topology
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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