Recover Riemannian metric from volume form

Question

If I know a Riemannian volume form on a Riemannian manifold, then is it always possible to recover the Riemannian metric which induces that volume?

Answer 1

The intrinsic volume form is given by $\left|\det G\right|^{1/2}$, where $G$ is the matrix representation of the metric $g$, i.e. $G:=\left(g_{ij}\right)$ where $g_{ij} := g\left(X_i,X_j\right)$, and $X_k$ are coordinate vector fields.

Metrics $g$ and $g'$ with matrices $G$ and $G'$ with $\left|\det G\right| = \left| \det G' \right|$ will have the same volume form.