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The length of the shortest closed geodesic on a 2-dimensional sphere. (English) Zbl 1003.53030
Let \(\left(M,g\right)\) be a compact Riemannian manifold of dimension \(n\). The study of the length of the shortest closed geodesic, the diameter \(d\) or the volume of \(M\) has been an important problem.
We denote by \(\ell\left(M\right)\) the shortest closed geodesic, \(d\) the diameter and \(A\) the volume of \(M\). The aim of the present paper is to study this problem for a compact Riemannian manifold of dimension two. The main results of this work are the following: Let \(M\) be a manifold diffeomorphic to the 2-dimensional sphere. Then \(\ell(M)\leq 4\sqrt d\). Let \(M\) be a manifold diffeomorphic to the 2-dimensional sphere. Then \(\ell(M)\leq 8\sqrt A\).

MSC:
53C22 Geodesics in global differential geometry
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