Abstract
It was proved by I. M. Liberman that for a C 2-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.
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References
Libermann I. M., “Geodesic lines on convex surfaces,” Dokl. Akad. Nauk SSSR, 32, 310-312 (1941).
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).
Gromoll D., Klingenberg W., and Meyer W., Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1971).
Alexandrov A. D., Intrinstic Geometry of Convex Surfaces [in Russian], Gostekhizdat, Moscow and Leningrad (1948).
Rozenfel'd B. A., Non-Euclidean Geometries [in Russian], Gostekhizdat, Moscow (1955).
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Vaigant, V.A., Matukevich, O.Y. Estimation of the Length of a Simple Geodesic on a Convex Surface. Siberian Mathematical Journal 42, 833–845 (2001). https://doi.org/10.1023/A:1011951207751
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DOI: https://doi.org/10.1023/A:1011951207751