Relation between Volume Forms of Conformally Related Metrics on Oriented Riemannian Manifold

Theorem

Let $M$ be an $n$-dimensional oriented Riemannian manifold.

Let $g_1$ and $g_2$ be conformally related Riemannian metrics.

That is, suppose:

$g_2 = f g_1$

where $f \in \map {C^\infty} M$ is a positive smooth real function.

Let $\rd V_{g_1}$ and $\rd V_{g_2}$ be Riemannian volume forms associated with $g_1$ and $g_2$.

Then:

$\ds \rd V_{g_2} = f^{\frac n 2} \rd V_{g_1}$

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): $\S 3$: Model Riemannian Manifolds. Euclidean Spaces