Riemannian Manifolds: An Introduction to Curvature

by John M. Lee

The preface and Chapter 1 are available for downloading in PDF format.  You need the Adobe Acrobat Reader to view them.
Front Matter and Preface
  1. What Is Curvature?
  2. Review of Tensors, Manifolds, and Vector Bundles
  3. Definitions and Examples of Riemannian Metrics
  4. Connections
  5. Riemannian Geodesics
  6. Geodesics and Distance
  7. Curvature
  8. Riemannian Submanifolds
  9. The Gauss-Bonnet Theorem
  10. Jacobi Fields
  11. Curvature and Topology
References

Index

Riemannian Manifolds: An Introduction to Curvature