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.

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 ${\displaystyle M.}$ A volume form ${\displaystyle \omega }$ on ${\displaystyle M}$ gives rise to an orientation in a natural way as the atlas of coordinate charts on ${\displaystyle M}$ that send ${\displaystyle \omega }$ 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 ${\displaystyle M.}$ Call a basis of tangent vectors ${\displaystyle (X_{1},\ldots ,X_{n})}$ right-handed if $${\displaystyle \omega \left(X_{1},X_{2},\ldots ,X_{n}\right)>0.}$$

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

${\displaystyle \omega \left(X_{1},X_{2},\ldots ,X_{n}\right)=1.}$

(1)

Thus a volume form gives rise to an ${\displaystyle \mathrm {SL} (n)}$-structure as well. Conversely, given an ${\displaystyle \mathrm {SL} (n)}$-structure, one can recover a volume form by imposing (1) for the special linear frames and then solving for the required ${\displaystyle n}$-form ${\displaystyle \omega }$ by requiring homogeneity in its arguments.

A manifold is orientable if and only if it has a nowhere-vanishing volume form. Indeed, ${\displaystyle \mathrm {SL} (n)\to \mathrm {GL} ^{+}(n)}$ is a deformation retract since ${\displaystyle \mathrm {GL} ^{+}=\mathrm {SL} \times \mathbb {R} ^{+},}$ where the positive reals are embedded as scalar matrices. Thus every ${\displaystyle \mathrm {GL} ^{+}(n)}$-structure is reducible to an ${\displaystyle \mathrm {SL} (n)}$-structure, and ${\displaystyle \mathrm {GL} ^{+}(n)}$-structures coincide with orientations on ${\displaystyle M.}$ More concretely, triviality of the determinant bundle ${\displaystyle \Omega ^{n}(M)}$ is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus, the existence of a volume form is equivalent to orientability.

Relation to measures

Given a volume form ${\displaystyle \omega }$ on an oriented manifold, the density ${\displaystyle |\omega |}$ is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds.

Any volume pseudo-form ${\displaystyle \omega }$ (and therefore also any volume form) defines a measure on the Borel sets by $${\displaystyle \mu _{\omega }(U)=\int _{U}\omega .}$$

The difference is that while a measure can be integrated over a (Borel) subset, a volume form can only be integrated over an oriented cell. In single variable calculus, writing ${\textstyle \int _{b}^{a}f\,dx=-\int _{a}^{b}f\,dx}$ considers ${\displaystyle dx}$ as a volume form, not simply a measure, and ${\textstyle \int _{b}^{a}}$ indicates "integrate over the cell ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Volume_form
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