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Lectures on Closed Geodesics

  • Book
  • © 1978

Overview

Authors:
  1. Wilhelm Klingenberg

    1. Mathematisches Institut der Universität Bonn, Bonn, Germany

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Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 230)

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Table of contents (5 chapters)

  1. Front Matter

    Pages I-XI
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  2. The Hilbert Manifold of Closed Curves

    • Wilhelm Klingenberg
    Pages 1-31

  3. The Morse-Lusternik-Schnirelmann Theory on the Manifold of Closed Curves

    • Wilhelm Klingenberg
    Pages 32-76

  4. The Geodesic Flow

    • Wilhelm Klingenberg
    Pages 77-121

  5. On the Existence of Many Closed Geodesics

    • Wilhelm Klingenberg
    Pages 122-166

  6. Miscellaneous Results

    • Wilhelm Klingenberg
    Pages 167-202

  7. Back Matter

    Pages 203-230
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Keywords

About this book

The question of existence of c10sed geodesics on a Riemannian manifold and the properties of the corresponding periodic orbits in the geodesic flow has been the object of intensive investigations since the beginning of global differential geo­ metry during the last century. The simplest case occurs for c10sed surfaces of negative curvature. Here, the fundamental group is very large and, as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed curve can be deformed into a c10sed curve having minimallength in its free homotopy c1ass. This minimal curve is, up to the parameterization, uniquely determined and represents a c10sed geodesic. The question of existence of a c10sed geodesic on a simply connected c10sed surface is much more difficult. As pointed out by Poincare [po 1] in 1905, this problem has much in common with the problem ofthe existence of periodic orbits in the restricted three body problem. Poincare [l.c.] outlined a proof that on an analytic convex surface which does not differ too much from the standard sphere there always exists at least one c10sed geodesic of elliptic type, i. e., the corres­ ponding periodic orbit in the geodesic flow is infinitesimally stable.

Authors and Affiliations

  • Mathematisches Institut der Universität Bonn, Bonn, Germany

    Wilhelm Klingenberg

Bibliographic Information

  • Book Title: Lectures on Closed Geodesics

  • Authors: Wilhelm Klingenberg

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-642-61881-9

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Sringer-Verlag Berlin Heidelberg 1978

  • Hardcover ISBN: 978-3-540-08393-1Published: 01 January 1978

  • Softcover ISBN: 978-3-642-61883-3Published: 13 October 2011

  • eBook ISBN: 978-3-642-61881-9Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: XI, 230

  • Topics: Differential Geometry

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