I am seeking the answer to the following question: Given a simple closed curve on a compact Riemannian surface(a compact surface with a Riemannian metric), whether there exists, in the homology class of this simple closed curve, a (single) closed curve which has the shortest length measured with respect to the given Riemannian metric? In other words, whether there exists a shortest path which is composed by a single closed curve in the homology class of a given simple closed curve? I think this might be interesting, can you help me?(adsbygoogle = window.adsbygoogle || []).push({});

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# Is there exist a shortest closed curve in the homology class of a simple closed curve

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