Riemannian Volume Form under Orientation-Preserving Isometry

Theorem
Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be oriented Riemannian manifolds.

Let $\phi : M \to \tilde M$ be an orientation-preserving isometry.

Let $\rd V_g$ be the Riemannian volume form.

Then:


 * $\phi^* \rd V_{\tilde g} = \rd V_g$