# Riemannian Manifolds

## An Introduction to Curvature

• John M. Lee
Book

Part of the Graduate Texts in Mathematics book series (GTM, volume 176)

1. Front Matter
Pages i-xv
2. John M. Lee
Pages 1-10
3. John M. Lee
Pages 11-21
4. John M. Lee
Pages 23-46
5. John M. Lee
Pages 47-64
6. John M. Lee
Pages 65-89
7. John M. Lee
Pages 91-113
8. John M. Lee
Pages 115-129
9. John M. Lee
Pages 131-153
10. John M. Lee
Pages 155-172
11. John M. Lee
Pages 173-191
12. John M. Lee
Pages 193-208
13. Back Matter
Pages 209-226

### Introduction

This book is designed as a textbook for a one-quarter or one-semester graduate course on Riemannian geometry, for students who are familiar with topological and differentiable manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. The author has selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics,without which one cannot claim to be doing Riemannian geometry. 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. From then on, all efforts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss–Bonnet theorem (expressing the total curvature of a surface in term so fits topological type), the Cartan–Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose–Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints.

### Keywords

Riemannian geometry Tensor Volume curvature manifold

#### Authors and affiliations

• John M. Lee
• 1
1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

### Bibliographic information

• DOI https://doi.org/10.1007/b98852