Riemannian Manifolds

1. Front Matter

   Pages i-xv

   PDF

2. What is Curvature?

   • John M. Lee

   Pages 1-10

3. Review of Tensors, Manifolds, and Vector Bundles

   • John M. Lee

   Pages 11-21

4. Definitions and Examples of Riemannian Metrics

   • John M. Lee

   Pages 23-46

5. Connections

   • John M. Lee

   Pages 47-64

6. Riemannian Geodesics

   • John M. Lee

   Pages 65-89

7. Geodesics and Distance

   • John M. Lee

   Pages 91-113

8. Curvature

   • John M. Lee

   Pages 115-129

9. Riemannian Submanifolds

   • John M. Lee

   Pages 131-153

10. The Gauss-Bonnet Theorem

    • John M. Lee

    Pages 155-172

11. Jacobi Fields

    • John M. Lee

    Pages 173-191

12. Curvature and Topology

    • John M. Lee

    Pages 193-208

13. Back Matter

    Pages 209-226

    PDF

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.

• Riemannian geometry
• Tensor
• Volume
• curvature
• manifold

"This book is very well writen, pleasant to read, with many good illustrations. It deals with the core of the subject, nothing more and nothing less. Simply a recommendation for anyone who wants to teach or learn about the Riemannian geometry."
Nieuw Archief voor Wiskunde, September 2000

• Department of Mathematics, University of Washington, Seattle, USA

  John M. Lee