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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
##### Keywords:
closed geodesic; diameter; volume; two-dimensional sphere
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