# Nice applications of Liouville's theorem

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

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 $\pi$.

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

Recall that Hopf and Green showed that integral of scalar curvature on a closed manifold without $conjugate$ points is non-positive. For the proof they used the fact that geodesic flow on unit tangent bundle preserves a volume