I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a unique closed geodesic. I found a proof for surfaces of constant negative curvature in Stillwell's "Geometry of Surfaces" but I haven't been able to find a source which proves this in greater generality.
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4$\begingroup$ This result is a special case of the following more general result of Eells and Sampson (see Theorem B in the last section of "Harmonic mappings of Riemannian manifolds") that any continuous map between closed Riemannian manifolds $M$ and $N$ where $M$ has nonnegative Ricci curvature and $N$ has nonpositive sectional curvature is homotopic to a totally geodesic map; if, moreover, $N$ has somewhere strictly negative curvature, then the map is either null-homotopic or homotopic to a closed geodesic. Any survey on harmonic mappings should contain some version of this. $\endgroup$– Dan FoxCommented Dec 17, 2012 at 8:04
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4$\begingroup$ This is an easy corollary of the "flat strip" theorem: If $X$ is a CAT(0) space and $g_1, g_2$ are two geodesics in $X$ which are Hausdorff-close, then $g_1, g_2$ cobound a flat strip in $X$. You can also take a look at do Carmo's book "Riemannian geometry", I think there is a proof there. $\endgroup$– MishaCommented Dec 17, 2012 at 8:04
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3$\begingroup$ This question is probably more appropriate for Math.stackexchange - you can find the answer in many Riemannian geometry textbooks. $\endgroup$– Ian AgolCommented Dec 17, 2012 at 21:51
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$\begingroup$ Thank you, I had bad luck finding this with a few Riemannian geometry texts and I flipped through do Carmo's book and somehow missed this result, which is proven in the text. $\endgroup$– ClarkCommented Dec 18, 2012 at 3:00
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