黎曼流形

內(nèi)容概要

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.

書籍目錄

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 Geodeics6 Geodesis and Distance7 Curvature8 Riemannian Submanifolds9 The Gauss-Bonnet Theorem10 Jacobi Fields11 Cuvature and TopologyReferencesIndex

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