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Let $(M,g)$ be a Riemannian manifold with isometric group $G=Iso(M,G)$. The metric gives an isomorphism between tangent and cotangent bundle of $M$. So $g$ induce a natural symplectic structure on $TM$,

The action of $G$ on $TM$ induce a moment map $\mu:TM\to \mathfrak{g}^*$. Here $\mathfrak{g}^*$ is the dual of the Lie algebra $\frak{g}$ of $G$.

To what extend this moment map encode the geometric invariants of the Riemannian manifold $(M,g)$? Can we extract geometric quantities of $(M,g)$ from this moment map? Are there some relations between the "Curvature" of the Riemannian manifold and certain properties of corresponding moment map?

Are there some research devoted to this question?

I found this related MO post.

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Yes, all has been solved by Jean-Marie Souriau: see Barbaresco, F.; Gay-Balmaz, F. Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics. Entropy 2020, 22, 498.

F. Barbaresco

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    $\begingroup$ A reference is a start, but could you please point to specific results in the reference that address the question precisely? We want you to show us your insights! (Sorry, what do 22 and 498 signify?) $\endgroup$
    – Todd Trimble
    Commented Nov 27, 2020 at 4:42
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    $\begingroup$ @ToddTrimble ,22 and 498 are just bibliographical information 22 is volume number and 498 is article number in the journal. $\endgroup$
    – i9Fn
    Commented Nov 27, 2020 at 9:56
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    $\begingroup$ While the paper seems extremely interesting and, in a sense, sensational, its content far exceeds the topic of the question: it is about connections of the moment maps in question with information geometry and machine learning. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 27, 2020 at 10:41
  • $\begingroup$ Thank you for your answer which leads me to be familiar with Jean Marie Souriau. BTW do you know any source for dowloading this video of J.M Souriau? archivesaudiovisuelles.fr/FR/… $\endgroup$
    – Ali Taghavi
    Commented Nov 29, 2020 at 12:24

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