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Riemannian geometry. 3rd edition. (English) Zbl 1417.53001
Graduate Texts in Mathematics 171. Cham: Springer (ISBN 978-3-319-26652-7/hbk; 978-3-319-26654-1/ebook). xviii, 499 p. (2016).
Publisher’s description: Intended for a one year course, this text serves as a single source, introducing readers 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. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups.
Important revisions to the third edition include:
– a substantial addition of unique and enriching exercises scattered throughout the text;
– inclusion of an increased number of coordinate calculations of connection and curvature;
– addition of general formulas for curvature on Lie groups and submersions;
– integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger;
– incorporation of several recent results about manifolds with positive curvature;
– presentation of a new simplifying approach to the Bochner technique for tensors with application to bound topological quantities with general lower curvature bounds.
For the first two editions see Zbl 0914.53001 and Zbl 1220.53002.

MSC:
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53Cxx Global differential geometry
53C20 Global Riemannian geometry, including pinching
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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