Definition:Riemannian Volume Form/Definition 2

Definition

Let $\struct {M, g}$ be an oriented $n$-dimensional Riemannian manifold.

Let $T M$ be the tangent bundle of $M$.

Let $\tuple {E_1, \ldots, E_n}$ be a local oriented orthonormal frame of $T M$.

The Riemannian volume form, denoted by $\rd V_g$, is an $n$-form such that:

$\map {\rd V_g} {E_1, \ldots, E_n} = 1$

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Sources

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