The riemannian volume form is the unique positive unit top form

Given: An oriented Riemannian manifold.

Riemannian volume form on an oriented manifold: On an oriented Riemannian $n$-manifold, the Riemannian volume form is ${\mathrm{vol}}_g=\sqrt{\det G_x}d{x}^1\wedge \cdots \wedge d{x}^n$ in positively oriented charts for $n\ge 1$. For $n=0$ it is the supplied orientation sign $\epsilon (p)\in \{1,-1\}$ at each point. def-oriented-smooth-manifold-and-oriented-chart supplies the orientation. On positive-chart overlaps the Jacobian determinant is positive, so the density calculation in lem-the-riemannian-volume-density-is-coordinate-independent is also the top-form transformation law. Thus the formula glues, and $\mathrm{\mid }{\mathrm{vol}}_g\mathrm{\mid }={\mu }_g$. Reversing orientation negates the form but leaves the density unchanged, also in dimension zero.

Riemannian metrics induce metrics on dual tensor and exterior bundles: A Riemannian metric induces smooth metrics on dual, tensor and exterior bundles. On decomposable covectors, $⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k⟩=\det (⟨{\alpha }_i,{\beta }_j⟩)$; increasing orthonormal wedge monomials have norm one.