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Riemannian Geometry and Geometric Analysis

  • Textbook
  • © 1995

Overview

Authors:
  1. Jürgen Jost

    1. Institut für Mathematik, Universität Bochum, Bochum, Germany

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Part of the book series: Universitext (UTX)

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Table of contents (9 chapters)

  1. Front Matter

    Pages I-XI
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  2. Foundational Material

    • Jürgen Jost
    Pages 1-54

  3. De Rham Cohomology and Harmonic Differential Forms

    • Jürgen Jost
    Pages 55-75

  4. Parallel Transport, Connections, and Covariant Derivatives

    • Jürgen Jost
    Pages 77-123

  5. Geodesics and Jacobi Fields

    • Jürgen Jost
    Pages 125-163

  6. A Short Survey on Curvature and Topology

    • Jürgen Jost
    Pages 165-171

  7. Morse Theory and Closed Geodesics

    • Jürgen Jost
    Pages 173-210

  8. Symmetric Spaces and Kähler Manifolds

    • Jürgen Jost
    Pages 211-261

  9. The Palais-Smale Condition and Closed Geodesics

    • Jürgen Jost
    Pages 263-276

  10. Harmonic Maps

    • Jürgen Jost
    Pages 277-384

  11. Back Matter

    Pages 385-404
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Keywords

About this book

The present textbook is a somewhat expanded version of the material of a three-semester course I gave in Bochum. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. In the first chapter, we introduce the basic geometric concepts, like dif­ ferentiable manifolds, tangent spaces, vector bundles, vector fields and one­ parameter groups of diffeomorphisms, Lie algebras and groups and in par­ ticular Riemannian metrics. We also derive some elementary results about geodesics. The second chapter introduces de Rham cohomology groups and the es­ sential tools from elliptic PDE for treating these groups. In later chapters, we shall encounter nonlinear versions of the methods presented here. The third chapter treats the general theory of connections and curvature. In the fourth chapter, we introduce Jacobi fields, prove the Rauch com­ parison theorems for Jacobi fields and apply these results to geodesics. These first four chapters treat the more elementary and basic aspects of the subject. Their results will be used in the remaining, more advanced chapters that are essentially independent of each other. In the fifth chapter, we develop Morse theory and apply it to the study of geodesics. The sixth chapter treats symmetric spaces as important examples of Rie­ mannian manifolds in detail.

Authors and Affiliations

  • Institut für Mathematik, Universität Bochum, Bochum, Germany

    Jürgen Jost

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