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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 $\pi$.

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  • $\begingroup$ 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
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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

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  • $\begingroup$ 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

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