John M. Lee - Introduction to Riemannian Manifolds (Graduate Texts in Mathematics)

Advisory Board:

Alejandro Adem, University of British Columbia

David Eisenbud, University of California, Berkeley & MSRI

Brian C. Hall, University of Notre Dame

J.F. Jardine, University of Western Ontario

Jeffrey C. Lagarias, University of Michigan

Ken Ono, Emory University

Jeremy Quastel, University of Toronto

Fadil Santosa, University of Minnesota

Barry Simon, California Institute of Technology

Ravi Vakil, Stanford University

Steven H. Weintraub, Lehigh University

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

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Riemannian geometry is the study of manifolds endowed with Riemannian metrics , which are, roughly speaking, rules for measuring lengths of tangent vectors and angles between them. It is the most geometric branch of differential geometry. Riemannian metrics are named for the great German mathematician Bernhard Riemann (18261866).

This book is designed as a textbook for a graduate course on Riemannian geometry for students who are familiar with the basic theory of smooth manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature, and in particular introducing many of the fundamental results that relate the local geometry of a Riemannian manifold to its global topology (the kind of results I like to call local-to-global theorems, as explained in Chapter ). In so doing, it introduces and demonstrates the uses of most of the main technical tools needed for a careful study of Riemannian manifolds.

The book is meant to be introductory, not encyclopedic. Its coverage is reasonably broad, but not exhaustive. It begins with a careful treatment of the machinery of metrics, connections, and geodesics, which are the indispensable tools in the subject. Next comes a discussion of Riemannian manifolds as metric spaces, and the interactions between geodesics and metric properties such as completeness. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation.

The first local-to-global theorem I discuss is the GaussBonnet theorem for compact surfaces. Many students will have seen a treatment of this in undergraduate courses on curves and surfaces, but because I do not want to assume such a course as a prerequisite, I include a complete proof.

From then on, all efforts are bent toward proving a number of fundamental local-to-global theorems for higher-dimensional manifolds, most notably the KillingHopf theorem about constant-curvature manifolds, the CartanHadamard theorem about nonpositively curved manifolds, and Myerss theorem about positively curved ones. The last chapter also contains a selection of other important local-to-global theorems.

Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry book, but they would not fit in this book without drastically broadening its scope. In particular, I do not treat the Morse index theorem, Toponogovs theorem, or their important applications such as the sphere theorem; Hodge theory, gauge theory, minimal surface theory, or other applications of elliptic partial differential equations to Riemannian geometry; or evolution equations such as the Ricci flow or the mean curvature flow. These important topics are for other, more advanced, books.

When I wrote the first edition of this book twenty years ago, a number of superb reference books on Riemannian geometry were already available; in the intervening years, many more have appeared. I invite the interested reader, after reading this book, to consult some of those for a deeper treatment of some of the topics introduced here, or to explore the more esoteric aspects of the subject. Some of my favorites are Peter Petersens admirably comprehensive introductory text [Pet16]; the elegant introduction to comparison theory by Jeff Cheeger and David Ebin [CE08] (which was out of print for a number of years, but happily has been reprinted by the American Mathematical Society); Manfredo do Carmos much more leisurely treatment of the same material and more [dC92]; Barrett ONeills beautifully integrated introduction to pseudo-Riemannian and Riemannian geometry [ON83]; Michael Spivaks classic multivolume tome [Spi79], which can be used as a textbook if plenty of time is available, or can provide enjoyable bedtime reading; the breathtaking survey by Marcel Berger [Ber03], which richly earns the word panoramic in its title; and the Encyclopaedia Britannica of differential geometry books,