Volume form
In mathematics, a volume form on a differentiable manifold is a topdimensional form (i.e., a differential form of top degree). Thus on a manifold M of dimension n, a volume form is an nform, a section of the line bundle Ω^{n}(M) = ⋀^{n}(T^{∗}M). A manifold admits a nowherevanishing 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 nonorientable 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 pseudovolume 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 pseudoRiemannian manifolds have an associated canonical volume form.
OrientationEdit
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 .
A volume form also allows for the specification of a preferred class of frames on M. Call a basis of tangent vectors (X_{1}, ..., X_{n}) righthanded if
The collection of all righthanded 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) subbundle 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 subbundle 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

(1)
Thus a volume form gives rise to an SL(n)structure as well. Conversely, given an SL(n)structure, one can recover a volume form by imposing (1) for the special linear frames and then solving for the required nform ω by requiring homogeneity in its arguments.
A manifold is orientable if and only if it has a volume form. Indeed, SL(n) → GL^{+}(n) is a deformation retract since GL^{+} = SL × R^{+}, where the positive reals are embedded as scalar matrices. Thus every GL^{+}(n)structure is reducible to an SL(n)structure, and GL^{+}(n)structures coincide with orientations on M. More concretely, triviality of the determinant bundle is equivalent to orientability, and a line bundle is trivial if and only if it has a nowherevanishing section. Thus the existence of a volume form is equivalent to orientability.
Relation to measuresEdit
Given a volume form ω on an oriented manifold, the density ω is a volume pseudoform on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on nonorientable manifolds.
Any volume pseudoform ω (and therefore also any volume form) defines a measure on the Borel sets by
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 considers as a volume form, not simply a measure, and indicates "integrate over the cell with the opposite orientation, sometimes denoted ".
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.
DivergenceEdit
Given a volume form ω on M, one can define the divergence of a vector field X as the unique scalarvalued function, denoted by div X, satisfying
where L_{X} denotes the Lie derivative along X and denotes the interior product or the left contraction of ω along X. If X is a compactly supported vector field and M is a manifold with boundary, then Stokes' theorem implies
which is a generalization of the divergence theorem.
The solenoidal vector fields are those with div X = 0. 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 volumepreserving flows. This fact is wellknown, 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 casesEdit
Lie groupsEdit
For any Lie group, a natural volume form may be defined by translation. That is, if ω_{e} is an element of , then a leftinvariant form may be defined by , where L_{g} is lefttranslation. 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 manifoldsEdit
Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. If M is a 2ndimensional manifold with symplectic form ω, then ω^{n} 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 formEdit
Any oriented pseudoRiemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as
where the are 1forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.
The volume form is denoted variously by
Here, the is the Hodge star, thus the last form, , emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the LeviCivita tensor ε.
Although the Greek letter ω is frequently used to denote the volume form, this notation is not universal; the symbol ω often carries many other meanings in differential geometry (such as a symplectic form).
Invariants of a volume formEdit
Volume forms are not unique; they form a torsor over nonvanishing functions on the manifold, as follows. Given a nonvanishing function f on M, and a volume form , is a volume form on M. Conversely, given two volume forms , their ratio is a nonvanishing function (positive if they define the same orientation, negative if they define opposite orientations).
In coordinates, they are both simply a nonzero 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 of with respect to . 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 structureEdit
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 (Kobayashi 1972). That is, for every point p in M, there is an open neighborhood U of p and a diffeomorphism φ of U onto an open set in R^{n} such that the volume form on U is the pullback of along φ.
As a corollary, if M and N are two manifolds, each with volume forms , then for any points , there are open neighborhoods U of m and V of n and a map such that the volume form on N restricted to the neighborhood V pulls back to volume form on M restricted to the neighborhood U: .
In one dimension, one can prove it thus: given a volume form on , define
Then the standard Lebesgue measure pulls back to under f: . Concretely, . In higher dimensions, given any point , it has a neighborhood locally homeomorphic to , and one can apply the same procedure.
Global structure: volumeEdit
A volume form on a connected manifold M has a single global invariant, namely the (overall) volume (denoted ), which is invariant under volumeform preserving maps; this may be infinite, such as for Lebesgue measure on . On a disconnected manifold, the volume of each connected component is the invariant.
In symbols, if is a homeomorphism of manifolds that pulls back to , then
and the manifolds have the same volume.
Volume forms can also be pulled back under covering maps, in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber). In the case of an infinite sheeted cover (such as ), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.
See alsoEdit
 Cylindrical coordinate system § Line and volume elements
 Measure (mathematics)
 Poincaré metric provides a review of the volume form on the complex plane
 Spherical coordinate system § Integration and differentiation in spherical coordinates
ReferencesEdit
 Kobayashi, S. (1972), Transformation Groups in Differential Geometry, Classics in Mathematics, Springer, ISBN 3540586598, OCLC 31374337.
 Spivak, Michael (1965), Calculus on Manifolds, Reading, Massachusetts: W.A. Benjamin, Inc., ISBN 0805390219.