My research is mainly in Harmonic analysis, so I'm not sure if I'm missing some subtleties with the following Differential Geometry argument:
The unit sphere has periodic geodesic flow, of period $2 \pi$. By the Killing Hopf Theorem, every manifold with constant positive sectional curvature must be a quotient of the sphere by a group acting freely and properly discontinuously. Does it follow from this that all manifolds with constant positive sectional curvature equal to one must have a periodic geodesic flow, with period at most $2 \pi$?