Volume form

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold ${\displaystyle M}$ of dimension ${\displaystyle n}$, a volume form is an ${\displaystyle n}$-form. It is an element of the space of sections of the line bundle ${\displaystyle \textstyle {\bigwedge }^{n}(T^{*}M)}$, denoted as ${\displaystyle \Omega ^{n}(M)}$. A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.

A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. It also defines a measure, but exists on any differentiable manifold, orientable or not.

Kähler manifolds, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the ${\displaystyle n}$th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented pseudo-Riemannian manifolds have an associated canonical volume form.