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 formula_1 of dimension formula_2, a volume form is an formula_2-form, a section of the line bundle formula_4. 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 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 formula_2th 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 formula_1. A volume form formula_7 on formula_1 gives rise to an orientation in a natural way as the atlas of coordinate charts on formula_1 that send formula_7 to a positive multiple of the Euclidean volume form formula_11.
A volume form also allows for the specification of a preferred class of frames on formula_1. Call a basis of tangent vectors formula_13 right-handed if
The collection of all right-handed frames is acted upon by the group formula_15 of general linear mappings in formula_2 dimensions with positive determinant. They form a principal formula_15 sub-bundle of the linear frame bundle of formula_1, and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of formula_1 to a sub-bundle with structure group formula_15. That is to say that a volume form gives rise to formula_15-structure on formula_1. More reduction is clearly possible by considering frames that have
Thus a volume form gives rise to an formula_23-structure as well. Conversely, given an formula_23-structure, one can recover a volume form by imposing () for the special linear frames and then solving for the required formula_2-form formula_7 by requiring homogeneity in its arguments.
A manifold is orientable if and only if it has a volume form. Indeed, formula_27 is a deformation retract since formula_28, where the positive reals are embedded as scalar matrices. Thus every formula_15-structure is reducible to an formula_23-structure, and formula_15-structures coincide with orientations on formula_1. More concretely, triviality of the determinant bundle formula_33 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 formula_7 on an oriented manifold, the density formula_35 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 formula_7 (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 formula_38 considers formula_39 as a volume form, not simply a measure, and formula_40 indicates "integrate over the cell formula_41 with the opposite orientation, sometimes denoted formula_42".
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.
Given a volume form "ω" on "M", one can define the divergence of a vector field "X" as the unique scalar-valued function, denoted by div "X", satisfying
where "L""X" denotes the Lie derivative along "X" and formula_44 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 . 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, a natural volume form may be defined by translation. That is, if "ω""e" is an element of formula_46, then a left-invariant form may be defined by formula_47, where "L""g" 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) has a natural volume form. If "M" is a 2"n"-dimensional 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 form.
Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as
where the formula_49 are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, formula_50 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 formula_52 is the Hodge star, thus the last form, formula_53, emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita "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 form.
Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. Given a non-vanishing function "f" on "M", and a volume form formula_7, 
formula_55 is a volume form on "M". Conversely, given two volume forms formula_56, 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 of formula_57 with respect to formula_7. 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 . 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 formula_11 along "φ".
As a corollary, if "M" and "N" are two manifolds, each with volume forms formula_60, then for any points formula_61, there are open neighborhoods "U" of "m" and "V" of "n" and a map formula_62 such that the volume form on "N" restricted to the neighborhood "V" pulls back to volume form on "M" restricted to the neighborhood "U": formula_63.
In one dimension, one can prove it thus:
given a volume form formula_7 on formula_65, define
Then the standard Lebesgue measure formula_39 pulls back to formula_7 under "f": formula_69. Concretely, formula_70. In higher dimensions, given any point formula_71, it has a neighborhood locally homeomorphic to formula_72, and one can apply the same procedure.
Global structure: volume.
A volume form on a connected manifold "M" has a single global invariant, namely the (overall) volume (denoted formula_73), which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on formula_74. On a disconnected manifold, the volume of each connected component is the invariant.
In symbols, if formula_75 is a homeomorphism of manifolds that pulls back formula_76 to formula_77, 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 formula_79), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.