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

Please help.


  • 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 $\pi$.

  • $\begingroup$ How about Section E 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

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

Their result is in the following lecture note : http://people.math.sc.edu/howard/Notes/hopf_note.pdf

  • $\begingroup$ By the way, do you know a reference to the second statement in my question? $\endgroup$ – Anton Petrunin Sep 15 '16 at 17:26
  • $\begingroup$ I do not know a reference to the second statement $\endgroup$ – Hee Kwon Lee Sep 15 '16 at 17:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.