Comparison Theorems in Riemannian Geometry

Comparison Theorems in Riemannian Geometry

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  • 製本 Hardcover:ハードカバー版/ページ数 161 p.
  • 言語 ENG
  • 商品コード 9780821844175
  • DDC分類 516.373

Full Description


The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry.

Table of Contents

Preface to the AMS Chelsea Printing                vii
Preface ix
Basic Concepts and Results 1 (34)
Notation and preliminaries 1 (2)
First variation of arc length 3 (3)
Exponetial map and normal coordinates 6 (3)
The Hopf-Rinow Theorem 9 (2)
The curvature tensor and Jacobi fields 11 (3)
Conjugate points 14 (2)
Second variation of arc length 16 (2)
Submanifolds and the second fundamental 18 (2)
form
Basic index lemmas 20 (3)
Ricci curvature and Myers' and Bonnet's 23 (1)
Theorems
Rauch Comparison Theorem 24 (6)
The Cartan-Hadamard Theorem 30 (1)
The Cartan-Ambrose-Hicks Theorem 31 (3)
Spaces of constant curvature 34 (1)
Toponogov's Theorem 35 (12)
Homogeneous Spaces 47 (22)
Morse Theory 69 (12)
Closed Geodesics and the Cut Locus 81 (12)
The Sphere Theorem and its Generalizations 93 (12)
The Differentiable Sphere Theorem 105(14)
Complete Manifolds of Nonnegative Curvature 119(18)
Compact Manifolds of Nonpositive Curvature 137(12)
Bibliography 149(6)
Additional Bibliography 155(2)
Index 157