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this is true for all spherical spaceforms. given two non-antipodal points on the sphere, the geodesic circle passing through them is unique. therefore given any point $p\in\mathbb S^n$ with $n>1$ at most finitely many geodesic circles through $p$ can project to closed geodesics of length less than $2\pi$. on the other hand there will obviously be some shorter closed geodesics in the quotient too.