# Volume form

In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a 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) = ⋀n(TM). A manifold admits a nonzero volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a 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 nth 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.

A manifold is orientable if it has a coordinate atlas all of whose transition functions have positive Jacobian determinants. A selection of a maximal such atlas is an orientation on M. A volume form ω on M gives rise to an orientation in a natural way as the atlas of coordinate charts on M that send ω to a positive multiple of the Euclidean volume form ${\displaystyle dx^{1}\wedge \cdots \wedge dx^{n}}$.

A volume form also allows for the specification of a preferred class of frames on M. Call a basis of tangent vectors (X1, ..., Xn) right-handed if

The collection of all right-handed frames is acted upon by the group GL+(n) of general linear mappings in n dimensions with positive determinant. They form a principal GL+(n) sub-bundle of the linear frame bundle of M, and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of M to a sub-bundle with structure group GL+(n). That is to say that a volume form gives rise to GL+(n)-structure on M. More reduction is clearly possible by considering frames that have

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