Estimation of the Length of a Simple Geodesic on a Convex Surface
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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.
KeywordsGaussian Curvature Convex Surface Positive Gaussian Curvature Simple Geodesic
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- 1.Libermann I. M., “Geodesic lines on convex surfaces,” Dokl. Akad. Nauk SSSR, 32, 310-312 (1941).Google Scholar
- 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.Gromoll D., Klingenberg W., and Meyer W., Riemannian Geometry in the Large [Russian translation], Mir, Moscow (1971).Google Scholar
- 4.Alexandrov A. D., Intrinstic Geometry of Convex Surfaces [in Russian], Gostekhizdat, Moscow and Leningrad (1948).Google Scholar
- 5.Rozenfel'd B. A., Non-Euclidean Geometries [in Russian], Gostekhizdat, Moscow (1955).Google Scholar