Abstract
In this paper we will present upper bounds for the length of a shortest closed geodesic on a manifold M diffeomorphic to the standard two-dimensional sphere. The first result is that the length of a shortest closed geodesic l(M) is bounded from above by 4r , where r is the radius of M . (In particular that means that l(M) is bounded from above by 2d, when M can be covered by a ball of radius d/2, where d is the diameter of M.) The second result is that l(M) is bounded from above by 2( max{r 1,r 2}+r 1+r 2), when M can be covered by two closed metric balls of radii r 1,r 2 respectively. For example, if r 1 = r 2= d/2 , thenl(M)≤ 3d. The third result is that l(M)≤ 2(max{r 1,r 2 r 3}+r 1+r 2+r 3), when M can be covered by three closed metric balls of radii r 1,r 2,r 3. Finally, we present an estimate for l(M) in terms of radii of k metric balls covering M, where k ≥ 3, when these balls have a special configuration.
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Rotman, R. The Length of a Shortest Closed Geodesic on a Two-Dimensional Sphere and Coverings by Metric Balls. Geom Dedicata 110, 143–157 (2005). https://doi.org/10.1007/s10711-004-3734-7
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DOI: https://doi.org/10.1007/s10711-004-3734-7