Definition:Riemannian Volume Form

Definition

Definition 1

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:

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

Definition 2

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:

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

Definition 3

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

Let $\tuple {x_1, \ldots, x_n}$ be a set of local oriented coordinates.

Let $g_{i j}$ be a local form of metric $g$.

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

$\d V_g = \sqrt {\map \det {g_{i j} } } \rd x^1 \wedge \ldots \wedge \rd x^n$

Notes

$\rd$ in $\rd V_g$ does not mean that $\rd V_g$ is an exact differential.

The notation $\rd V_g$ is chosen to emphasize the similarity of the integral $\int_M f \rd V_g$ with the standard integral of a function over an open subset of $\R^n$.

See also

• Equivalence of Definitions of Riemannian Volume Form

Sources

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