nLab volume form

Idea

The volume form on a finite-dimensional oriented (pseudo)-Riemannian manifold $(X,g)$ is the differential form whose integral over pieces of $X$ computes the volume of $X$ as measured by the metric $g$.

If the manifold is unoriented, then we get a volume pseudoform instead, or equivalently a volume density (of weight $1$). We can also consider volume (pseudo)-forms in the absence of a metric, in which case we have a choice of volume forms.

Definition

General

For $X$ a general smooth manifold of finite dimension $n$, a volume form on $X$ is a nondegenerate (nowhere vanishing) differential $n$-form on $X$, equivalently a nondegenerate section of the canonical line bundle on $X$. A volume pseudoform or volume element on $X$ is a positive definite density (of rank $1$) on $X$, or equivalently a positive definite differential $n$-pseudoform on $X$.

A volume form defines an orientation on $X$, the one relative to which it is positive definite. If $X$ is already oriented, then we require the orientations to agree (to have a volume form on $X$ qua oriented manifold); that is, a volume form on an oriented manifold must be positive definite (just as a volume pseudoform on any manifold must be). In this situation, there is essentially no difference between a form and a pseudoform, hence no difference between a volume form and a volume pseudoform or volume element.

For Riemannian manifolds

More specifically, for $(X,g)$ a (pseudo)-Riemannian manifold of dimension $n$, the volume pseudoform or volume element $vol_g$ is a specific differential $n$-pseudoform that measures the volume as seen by the metric $g$. If $X$ is oriented, then we may interpret $vol_g \in \Omega^n(X)$ as a differential $n$-form, also denoted $vol_g$.

This $vol_g$ is characterized by any of the following equivalent statements:

• The symmetric square $vol_g \cdot vol_g$ is equal to the $n$-fold wedge product $g \wedge \cdots \wedge g$, as elements of $\Omega^n(X) \otimes \Omega^n(X)$, and $vol_g$ is positive (meaning that its integral on any open submanifold? is nonnegative).

• The volume form is the image under the Hodge star operator $\star_g\colon \Omega^k(X) \to \Omega^{n-k}(X)$ of the smooth function $1 \in \Omega^0(X)$

  $$vol_g = \star_g 1 \,.$$

• In local oriented coordinates, $vol_g = \sqrt{|det(g)|}$, where $det(g)$ is the determinant of the matrix of the coordinates of $g$. In the case of a Riemannian (not pseudo-Riemannian) metric, this simplifies to $vol_g = \sqrt{\det(g)}$. (Note that local coordinates for a pseudoform include a local orientation, so this makes sense regardless of whether $X$ is oriented.)

• For $(E, \Omega)\colon T X \to iso(n)$ the Lie algebra valued differential form on $X$ with values in the Poincare Lie algebra $iso(n)$ that encodes the metric and orientation (the spin connection $\Omega$ with the vielbein $E$), the volume form is the image of $(E,\Omega)$ under the canonical volume Lie algebra cocycle $vol \in CE(iso(n))$:

  $$vol_g = vol(E) \,.$$

  See Poincare Lie algebra for more on this.

Degenerate cases

If we allow a volume (pseudo)form to be degenerate, then most of this goes through unchanged. In particular, a degenerate (pseudo)-Riemannian metric defines a degenerate volume pseudoform (and hence a degenerate volume form on an oriented manifold).

However, a degenerate $n$-form $\omega$ does not specify an orientation in general, so there is not necessarily a good notion of volume form on an unoriented manifold. On the other hand, if the open submanifold on which $\omega \ne 0$ is dense, then there is at most one compatible orientation, although there still may be none. Of course, on an oriented manifold, forms are equivalent to pseudoforms, so we still know what a degenerate volume form is there.

To remove the requirement of positivity is much more drastic; an arbitrary $n$-pseudoform is simply a $1$-density (and an arbitrary $n$-form is a $1$-pseudodensity). This is at best a notion of signed volume, rather than volume.

Properties

Since an $n$-(pseudo)form is positive iff its integral on any open submanifold? is nonnegative and nondegenerate iff its integral on sufficiently small inhabited? open submanifolds is nonzero, a volume (pseudo)form may be defined as one whose integral on any inhabited open submanifold is (strictly) positive.

A volume (pseudo)form is also equivalent to an absolutely continuous positive Radon measure on $X$. Here, nondegeneracy corresponds precisely to absolute continuity.

If I remember correctly, every volume (pseudo)form comes from a metric, which is unique iff $n \leq 1$.

Related concepts

• density

• divergence