附註:Includes bibliographical references (pages 209-211) and index.
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.
摘要:This text is designed for a one-quarter or one-semester graduate course on Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced study of Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the curvature tensor as a way of measuring whether a Riemannian manifold is locally equivalent to Euclidean space. Submanifold theory is developed next in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and the characterization of manifolds of constant curvature.
