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I am not sure what class of surfaces or topological spaces this is a theorem for but it should at least include the plane, circle, sphere and torus (hopefully also the Klein bottle) - so part of the question is what class of surfaces to consider:

Given a fixed base point consider every path in some homotopy class. How can one prove that a shortest path exists and that it is a geodesic?

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Geodesics or even distances are not defined for general topological spaces. –  lhf Apr 28 '11 at 1:57
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The objects for which the word "geodesic" makes sense are metric spaces, not topological spaces. Most people would specialize even further to Riemannian manifolds. –  Qiaochu Yuan Apr 28 '11 at 2:23
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up vote 2 down vote accepted

On a complete Riemannian manifold, there is always a shortest geodesic in a given homotopy class of loops based at a point. See e.g. here which treats the (more general) case of Finsler manifolds. (I found this particular reference just by googling, and you should be able to find many other references in the same way.)

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Matt E, thanks for the reference but that is not true that I would be able to find this by googling. I have tried for a long time without success before asking this here. –  quanta Apr 28 '11 at 7:12
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