Let $M$ be a closed Riemannian manifold with non-positive sectional curvature, then it is well-known that there are no contractible closed geodesics in $M$. More generally, let $M$ be a closed manifold which is topologically a $K(\pi,1)$ space, is there a Riemannian metric on $M$ so that every closed geodesic in $M$ is non-contractible?
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$\begingroup$ related question: mathoverflow.net/questions/358943 $\endgroup$– Anton PetruninNov 22, 2022 at 18:39
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$\begingroup$ Did you think about connected sum of torus with exotic sphere? $\endgroup$– Anton PetruninNov 22, 2022 at 22:34
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