I need a big list of nice-looking and simple applications of Liouville's theorem on geodesic flow in Riemannian geometry.

*Please help.*

**Examples:**

A Riemannian manifold with finite volume does not admit a strictly convex function.

If $M$ is a closed $m$-dimensional Riemannian manifold and $\mathrm{Sc}_M\ge \mathrm{Sc}_{\mathbb S^m}$ then injectivity radius of $M$ is at most $\pi$.

here? (Citation: Joyce, W. B. (1974). Classical-particle description of photons and phonons.Physical Review D, 9(12), 3234.) $\endgroup$ – Benjamin Dickman Jun 1 '15 at 10:45