Riemannian Geometry ( De Gruyter Studies in Mathematics )

Publication series ：De Gruyter Studies in Mathematics

Author： Wilhelm P.A. Klingenberg  

Publisher： De Gruyter‎

Publication year： 1995

E-ISBN: 9783110905120

P-ISBN(Paperback): 9783110145939

Subject： O186.12 Riemannian geometry

Language： ENG

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Description

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist.

The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level.

The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.
While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies

Chapter

Chapter 1: Foundations

pp.：  1 – 11

1.0 Review of Differential Calculus and Topology

pp.：  11 – 11

1.1 Differentiable Manifolds

pp.：  11 – 18

1.2 Tensor Bundles

pp.：  18 – 23

1.3 Immersions and Submersions

pp.：  23 – 33

1.4 Vector Fields and Tensor Fields

pp.：  33 – 41

1.5 Covariant Derivation

pp.：  41 – 49

1.6 The Exponential Mapping

pp.：  49 – 63

1.7 Lie Groups

pp.：  63 – 69

1.8 Riemannian Manifolds

pp.：  69 – 77

1.9 Geodesics and Convex Neighborhoods

pp.：  77 – 88

1.10 Isometric Immersions

pp.：  88 – 96

1.11 Riemannian Curvature

pp.：  96 – 107

1.12 Jacobi Fields

pp.：  107 – 119

Chapter 2: Curvature and Topology

pp.：  119 – 134

2.1 Appendix – Orientation

pp.：  134 – 146

2.1 Completeness and Cut Locus

pp.：  134 – 134

2.2 Symmetric Spaces

pp.：  146 – 151

2.3 The Hilbert Manifold of H1-curves

pp.：  151 – 168

2.4 The Loop Space and the Space of Closed Curves

pp.：  168 – 180

2.5 The Second Order Neighborhood of a Critical Point

pp.：  180 – 191

2.5 Appendix – The S1- and the Z2-action on AM

pp.：  191 – 206

2.6 Index and Curvature

pp.：  206 – 213

2.6 Appendix – The Injectivity Radius for 1/4-pinched Manifolds

pp.：  213 – 222

2.7 Comparison Theorems for Triangles

pp.：  222 – 225

2.8 The Sphere Theorem

pp.：  225 – 239

2.9 Non-compact Manifolds of Positive Curvature

pp.：  239 – 250

Chapter 3: Structure of the Geodesic Flow

pp.：  250 – 266

3.2 Properties of the Geodesic Flow

pp.：  266 – 275

3.1 Hamiltonian Systems

pp.：  266 – 266

3.3 Stable and Unstable Motions

pp.：  275 – 289

3.4 Geodesics on Surfaces

pp.：  289 – 298

3.5 Geodesics on the Ellipsoid

pp.：  298 – 313

3.6 Closed Geodesies on Spheres

pp.：  313 – 334

3.7 The Theorem of the Three Closed Geodesics

pp.：  334 – 347

3.8 Manifolds of Non-Positive Curvature

pp.：  347 – 360

3.9 The Geodesic Flow on Manifolds of Negative Curvature

pp.：  360 – 373

3.10 The Main Theorem for Surfaces of Genus 0

pp.：  373 – 390

References

pp.：  390 – 403

Index

pp.：  403 – 413

LastPages

pp.：  413 – 421

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