Definition:Riemannian Volume Form/Definition 1

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

Let $T^* M$ be the cotangent bundle of $M$.

Let $\tuple {\epsilon^1, \ldots, \epsilon^n}$ be a local oriented orthonormal coframe of $T^* M$.

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


 * $\rd V_g = \epsilon^1 \wedge \ldots \wedge \epsilon^n$