Riemannian Volume Form of Orientable Hypersurface

Theorem

Let $\struct {\tilde M, \tilde g}$ be an oriented Riemannian manifold.

Let $\struct {M, g}$ be an orientable hypersurface with an induced metric.

Let $N$ be a unit global normal vector for $M$.

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

Then Riemannian volume form is given by:

$\rd V_g = \valueat {\paren {N \lrcorner \rd V_{\tilde g} } } M$

where $\lrcorner$ denotes the interior multiplicaiton.

Proof

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Sources

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