In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold-form. It is an element of the space of sections of the line bundle. 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 manifoldsth 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 volume form also allows for the specification of a preferred class of frames

The collection of all right-handed frames is acted upon by the group of general linear dimensions with positive determinant. They form a \mathrm{GL}^+(n) sub-bundle of the linear frame bundle That is to say that a volume form gives rise to \mathrm{GL}^+(n)-structure More reduction is clearly possible by considering frames that have

-structure, one can recover a volume form by imposing ( by requiring homogeneity in its arguments.

is a deformation retract where the positive reals 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

on an oriented manifold, the density 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.

(and therefore also any volume form) defines a measure on the Borel sets

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".

Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form need not be absolutely continuous.

Divergence

one can define the divergence of a vector field denotes the Lie derivative denotes the interior product or the left contraction is a compactly supported is a manifold with boundary, then Stokes' theorem which is a generalization of the divergence theorem.

The solenoidal It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, in fluid mechanics where the divergence of a velocity field measures the compressibility of a fluid, which in turn represents the extent to which volume is preserved along flows of the fluid.

Special cases

Lie groups

For any Lie group is left-translation. As a corollary, every Lie group is orientable. This volume form is unique up to a scalar, and the corresponding measure is known as the Haar measure.

Symplectic manifolds

Any symplectic manifold (or indeed any almost symplectic manifold-dimensional manifold with symplectic form is nowhere zero as a consequence of the nondegeneracy of the symplectic form. As a corollary, any symplectic manifold is orientable (indeed, oriented). If the manifold is both symplectic and Riemannian, then the two volume forms agree if the manifold is Kähler.

Riemannian volume form

Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates are 1-forms that form a positively oriented basis for the cotangent bundle is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.

is the Hodge star emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita tensor

often carries many other meanings in differential geometry (such as a symplectic form).

Invariants of a volume form

Volume forms are not unique; they form a torsor their ratio is a non-vanishing function (positive if they define the same orientation, negative if they define opposite orientations).

In coordinates, they are both simply a non-zero function times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is the Radon–Nikodym derivative On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of the Radon–Nikodym theorem.

No local structure

A volume form on a manifold has no local structure in the sense that it is not possible on small open sets to distinguish between the given volume form and the volume form on Euclidean space and a diffeomorphism is the pullback

Then the standard Lebesgue measure pulls back and one can apply the same procedure.

Global structure: volume

On a disconnected manifold, the volume of each connected component is the invariant.

and the manifolds have the same volume.

Volume forms can also be pulled back under covering maps), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.

See also

• Poincaré metric provides a review of the volume form on the complex plane

References

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