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Siberian Mathematical Journal

, Volume 42, Issue 5, pp 833–845 | Cite as

Estimation of the Length of a Simple Geodesic on a Convex Surface

  • V. A. Vaigant
  • O. Yu. Matukevich
Article
  • 32 Downloads

Abstract

It was proved by I. M. Liberman that for a C2-smooth closed surface M of positive Gaussian curvature there exists a number l such that any geodesic arc on M of length at least l is not simple. In this article we indicate a lower bound for l. We exhibit an example showing that our estimate is unimprovable.

Keywords

Gaussian Curvature Convex Surface Positive Gaussian Curvature Simple Geodesic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Libermann I. M., “Geodesic lines on convex surfaces,” Dokl. Akad. Nauk SSSR, 32, 310-312 (1941).Google Scholar
  2. 2.
    Toponogov V. A., “Estimation of the length of a convex curve on a two-dimensional surface,” Sibirsk. Mat. Zh., 4, No. 5, 1189-1183 (1963).Google Scholar
  3. 3.
    Gromoll D., Klingenberg W., and Meyer W., Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1971).Google Scholar
  4. 4.
    Alexandrov A. D., Intrinstic Geometry of Convex Surfaces [in Russian], Gostekhizdat, Moscow and Leningrad (1948).Google Scholar
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    Rozenfel'd B. A., Non-Euclidean Geometries [in Russian], Gostekhizdat, Moscow (1955).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • V. A. Vaigant
    • 1
  • O. Yu. Matukevich
    • 2
  1. 1.Munster UniversityMunster
  2. 2.The Altai State UniversityBarnaul

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