Given a Riemannian \(n\)-manifold \((M,g)\). Then there exists a unique smooth differential \(n\)-form \(\omega_g\) which satisfies
\begin{equation*} \omega_g(E_1,\ldots, E_n)=1, \end{equation*}for all local orthonormal frame [@lee2013smooth_manifolds, Proposition 15.29].
Remarks
- The Riemannian volume form is very useful for integration . That is why it is also noted by \(dV_g\).