Riemannian Geometry

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.

From reviews of the first edition:

“The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type.”

―Bernd Wegner, ZbMATH

Table of Contents

1 Riemannian Metrics

2 Derivatives

3 Curvature

4 Examples

5 Geodesics and Distance

6 Sectional Curvature Comparison I

7 Ricci Curvature Comparison

8 Killing Fields

9 The Bochner Technique

10 Symmetric Spaces and Holonomy

11 Convergence

12 Sectional Curvature Comparison II

About the Author

Peter Petersen is a Professor of Mathematics at UCLA. His current research is on various aspects of Riemannian geometry. Professor Petersen has authored two important textbooks for Springer: Riemannian Geometry in the GTM series and Linear Algebra in the UTM series.