Introduction to Smooth Manifolds

© 2018: This is the second edition of the book originally published as Riemannian Manifolds: An Introduction to Curvature.

From the back cover:

This textbook is designed for a one or two semester graduate course on Riemannian geometry for students who are familiar with topological and differentiable manifolds. The second edition has been adapted, expanded, and aptly retitled from Lee’s earlier book, Riemannian Manifolds: An Introduction to Curvature . Numerous exercises and problem sets provide the student with opportunities to practice and develop skills; appendices contain a brief review of essential background material.

While demonstrating the uses of most of the main technical tools needed for a careful study of Riemannian manifolds, this text focuses on ensuring that the student develops an intimate acquaintance with the geometric meaning of curvature. The reasonably broad coverage begins with a treatment of indispensable tools for working with Riemannian metrics such as connections and geodesics. Several topics have been added, including an expanded treatment of pseudo-Riemannian metrics, a more detailed treatment of homogeneous spaces and invariant metrics, a completely revamped treatment of comparison theory based on Riccati equations, and a handful of new local-to-global theorems, to name just a few highlights.

About problems with print quality:

Many people have reported receiving copies of Springer books, especially from Amazon, that suffer from extremely poor print quality (bindings that quickly break, thin paper, and low-resolution printing, for example). This seems to be less likely to happen if you purchase directly from Springer, but even then it's not unheard of. Springer has told me they will replace any book with substandard print quality regardless of where you purchased it. Contact sales-ny@springer.com for information.


About the first edition:
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