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:

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

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Sources

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