Periodic geodesics in singular spaces

We extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if $X$ is a compact geodesic metric space satisfying the CAT($κ$) condition for some fixed $κ>0$ and $π_n(X)\ne 0$ for some $n>0$ then $X$ has a periodic geodesic. This condition is satisfied for example by locally CAT($κ$) manifolds. Our result applies more generally to compact locally uniquely geodesic spaces.

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