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 encod 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 find this  related MO post. I share it here:

https://mathoverflow.net/questions/336519/moment-map-interpretation-of-einstein-equation?rq=1