Let $(X,g_1)$ and $(X,g_2)$ be two Riemannian manifolds over the same space $X$.
My (vague) question is the following : If I know that the two induced Riemannian volume form coincide, what can I say on $g_1$ and $g_2$ ?
A variant of the question : what kind of extra assumption(s) (like curvature, etc) can I add on $X$ and/or $g_1$ and $g_2$ so that the equality of the volume forms implies the equality of the Riemannian metrics (maybe up to identifiable "easy" transformations).
For a concrete example, we can consider for $X$ the $n$-th space of (unordered) configurations on $\mathbb R$, that is $$ X=\Big\{(x_1,\ldots,x_n)\in\mathbb R^n : \; x_i\neq x_j \mbox{ for } i\neq j\Big\}/S_n, $$ where $S_n$ stands for the symmetric group over $n$ elements.