Given a (smooth, orientable) n-dimensional manifold $M$ with two (pseudo-)Riemannian metrics $g\_{1}$ and $g\_{2}$ of the same signature that induce the same Levi-Civita connection and satisfy $\sqrt{|\det{g\_1}|}\mathrm{d}x^{1}\wedge...\wedge\mathrm{d}x^{n} = \sqrt{|\det{g\_2}|}\mathrm{d}x^{1}\wedge...\wedge\mathrm{d}x^{n}$, 


is $g\_{1}=g\_{2}$?



Are two metrics with the same Levi-Civita connection and the same volume form identical?
The standard Minkowski metric, transformed by scaling one variable by 1/2, and another by 2, has the usual volume form, and the same Levi--Civita connection (which depends only on the affine structure).


