# Geometric invariants of a Riemannian manifold encoded in certain moment map

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.