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.
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
