リーマン多様体<br>Riemannian Manifolds : An Introduction to Curvature (Graduate Texts in Mathematics Vol.176)

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リーマン多様体
Riemannian Manifolds : An Introduction to Curvature (Graduate Texts in Mathematics Vol.176)

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  • 製本 Hardcover:ハードカバー版/ページ数 240 p.
  • 商品コード 9780387982717

基本説明

Designed for a graduate course on Riemannian geometry. Contents: What is curvature; Review of Tensors, Manifolds, and Vector bundles; Definitions and Examples of Riemannian Metrics; Connections; Riemann Geodexics; Geodesics and Distance; and more.

Full Description


This text 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 course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

Table of Contents

Preface                                            vii
1 What Is Curvature? 1 (10)
The Euclidean Plane 2 (2)
Surfaces in Space 4 (4)
Curvature in Higher Dimensions 8 (3)
2 Review of Tensors, Manifolds, and Vector 11 (12)
Bundles
Tensors on a Vector Space 11 (3)
Manifolds 14 (2)
Vector Bundles 16 (3)
Tensor Bundles and Tensor Fields 19 (4)
3 Definitions and Examples of Riemannian 23 (24)
Metrics
Riemannian Metrics 23 (4)
Elementary Constructions Associated with 27 (3)
Riemannian Metrics
Generalizations of Riemannian Metrics 30 (3)
The Model Spaces of Riemannian Geometry 33 (10)
Problems 43 (4)
4 Connections 47 (18)
The Problem of Differentiating Vector Fields 48 (1)
Connections 49 (6)
Vector Fields Along Curves 55 (3)
Geodesics 58 (5)
Problems 63 (2)
5 Riemannian Geodesics 65 (26)
The Riemannian Connection 65 (7)
The Exponential Map 72 (4)
Normal Neighborhoods and Normal Coordinates 76 (5)
Geodesics of the Model Spaces 81 (6)
Problems 87 (4)
6 Geodesics and Distance 91 (24)
Lengths and Distances on Riemannian 91 (5)
Manifolds
Geodesics and Minimizing Curves 96 (12)
Completeness 108(4)
Problems 112(3)
7 Curvature 115(16)
Local Invariants 115(4)
Flat Manifolds 119(2)
Symmetries of the Curvature Tensor 121(3)
Ricci and Scalar Curvatures 124(4)
Problems 128(3)
8 Riemannian Submanifolds 131(24)
Riemannian Submanifolds and the Second 132(7)
Fundamental Form
Hypersurfaces in Euclidean Space 139(6)
Geometric Interpretation of Curvature in 145(5)
Higher Dimensions
Problems 150(5)
9 The Gauss-Bonnet Theorem 155(18)
Some Plane Geometry 156(6)
The Gauss-Bonnet Formula 162(4)
The Gauss-Bonnet Theorem 166(5)
Problems 171(2)
10 Jacobi Fields 173(20)
The Jacobi Equation 174(4)
Computations of Jacobi Fields 178(3)
Conjugate Points 181(4)
The Second Variation Formula 185(2)
Geodesics Do Not Minimize Past Conjugate 187(4)
Points
Problems 191(2)
11 Curvature and Topology 193(16)
Some Comparison Theorems 194(2)
Manifolds of Negative Curvature 196(3)
Manifolds of Positive Curvature 199(5)
Manifolds of Constant Curvature 204(4)
Problems 208(1)
References 209(4)
Index 213