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I want to apply a theorem that describes the eigenfunctions of the Laplacian on a Riemannian manifold:

  • This theorem assumes that the geodesic flow of $M$ is ergodic with respect to the Liouville measure.
  • What is exactly the definition of the Liouville measure ?: Is it defined on $M$ or $TM$ ?.
  • In another reference, I saw that this measure is expressed as $$ \frac{{\rm d}x \wedge {\rm d}\xi}{{\rm d}\left\vert\,\xi\,\right\vert} $$
    • Could someone please explain this definition ?. I could not find it in any textbook.
Felix Marin
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rafael
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    Related: https://math.stackexchange.com/questions/1356069/two-different-definitions-of-a-liouville-measure?rq=1 – Lee Mosher Nov 29 '24 at 13:44
  • The measure is defined on the unit tangent bundle. You can find it's definition in many differential geometry textbooks. See also this MO post. – Moishe Kohan Nov 29 '24 at 17:36

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