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The length of a shortest closed geodesic and the area of a $$2$$-dimensional sphere. (English) Zbl 1098.53035
The author proves that if $$M$$ is a Riemannian manifold diffeomorphic to the 2-dimensional sphere then $$l(M)\leq 4\sqrt{2}\sqrt{A}$$, where $$l(M)$$ and $$A$$ denote the length of a shortest closed non-trivial geodesic on $$M$$ and the area of $$M$$, respectively. This inequality improves previous results.

##### MSC:
 53C22 Geodesics in global differential geometry 58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
##### Keywords:
closed geodesic; area; Riemannian manifold; geometric inequality
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