In mathematics, a volume form on a differentiable manifold is a nowhere vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn(M) = Λn(T∗M), that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable, and orientable manifolds have infinitely many.
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.
Many classes of manifolds come with canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented Riemannian manifolds and pseudo-Riemannian manifolds have a canonical volume form associated with them. For instance Kähler manifolds, being complex manifolds, are naturally oriented, and so possess a volume form. More generally, the nth exterior power of the symplectic form on a symplectic manifold is a volume form.
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On Form Yet to come Performance Research 2005 Volume 10 No 2 June Issue Editor Ric Allsopp with Emil Hrvatin amd Goran Sergej Pristas
