Assume we have a vector space in which some equations of motion are governed by a Hamiltonian (kinetic + potential energy) as per $H(q,p)=T(p)+V(q)$, mostly with $T(p)=p^2/(2m)$.
In Physics, it is said that one can put the potential $V$ somehow in the curvature of the space. So we are in the setting of a Riemannian manifold, say $M$, equipped with a "metric" (inner product in $TM$, the tangent bundle) $g_x:T_xM\times T_xM\to\mathbb{R}$ defined at every $x\in M$. Then one can define the geodesics...
What is actually the notion of this equivalence roughly spoken "curvature without forces is equal to flat space with forces" ?
What is the mathematicaly precise way to "transform" the potential energy (the "forces") into the curvature defined by the Riemannian metric, which in Physics is normally given in coordinate reference system in matrix representation as $g_{ab}(x) \overset{?}{=} \text{some function of }V(x)$?
Edit: Is it somehow related to Maupertuis's principle, or Geodesics as Hamiltonian flows?