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.
目录
Preface 1 What Is Curvature? The Euclidean Plane Surfaces in Space Curvature in Higher Dimensions 2 Review of Tensors， Manifolds， and Vector Bundles Tensors on a Vector Space Manifolds Vector Bundles Tensor Bundles and Tensor Fields 3 Definitions and Examples of Riemannian Metrics Riemannian Metrics Elementary Constructions Associated with Riemannian Metrics Generalizations of Riemannian Metrics The Model Spaces of Riemannian Geometry Problems 4 Connections The Problem of Differentiating Vector Fields Connections Vector Fields Along Curves …… 5 Riemannian Geodeics 6 Geodesis and Distance 7 Curvature 8 Riemannian Submanifolds 9 The Gauss-Bonnet Theorem 10 Jacobi Fields 11 Cuvature and Topology References Index
