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:

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

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds