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 M {\displaystyle M} of dimension n {\displaystyle n} , a volume form is an n {\displaystyle n} -form. It is an element of the space of sections of the line bundle ⋀ n ( T ∗ M ) {\displaystyle \textstyle {\bigwedge }^{n}(T^{*}M)} , denoted as Ω n ( M ) {\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 n {\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.
Orientation The following will only be about orientability of differentiable manifolds (it's a more general notion defined on any topological manifold). 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 . {\displaystyle M.} A volume form ω {\displaystyle \omega } on M {\displaystyle M} gives rise to an orientation in a natural way as the atlas of coordinate charts on M {\displaystyle M} that send ω {\displaystyle \omega } to a positive multiple of the Euclidean volume form d x 1 ∧ ⋯ ∧ d x n . {\displaystyle dx^{1}\wedge \cdots \wedge dx^{n}.}
A volume form also allows for the specification of a preferred class of frames on M . {\displaystyle M.} Call a basis of tangent vectors ( X 1 , … , X n ) {\displaystyle (X_{1},\ldots ,X_{n})} right-handed if
ω ( X 1 , X 2 , … , X n ) > 0. {\displaystyle \omega \left(X_{1},X_{2},\ldots ,X_{n}\right)>0.}
The collection of all right-handed frames is acted upon by the group G L + ( n ) {\displaystyle \mathrm {GL} ^{+}(n)} of general linear mappings in n {\displaystyle n} dimensions with positive determinant. They form a principal G L + ( n ) {\displaystyle \mathrm {GL} ^{+}(n)} sub-bundle of the linear frame bundle of M , {\displaystyle M,} and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of M {\displaystyle M} to a sub-bundle with structure group G L + ( n ) . {\displaystyle \mathrm {GL} ^{+}(n).} That is to say that a volume form gives rise to G L + ( n ) {\displaystyle \mathrm {GL} ^{+}(n)} -structure on M . {\displaystyle M.} More reduction is clearly possible by considering frames that have
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