Periodic Geodesics in Singular Spaces

In this paper, we extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if [Formula: see text] is a compact geodesic metric space satisfying the CAT([Formula: see text]) condition for some fixed [Formula: see text] and [Formula: see text] for some [Formula: see text] then [Formula: see text] has a periodic geodesic. This condition is satisfied for example by locally CAT([Formula: see text]) manifolds. Our result applies more generally to compact locally uniquely geodesic spaces.