The Length of a Shortest Closed Geodesic on a Two-Dimensional Sphere and Coverings by Metric Balls

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.