Inmathematics, a volume formortop-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold of dimension
, a volume form is an
-form. It is an element of the space of sections of the line bundle
, denoted as
. 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 formorpseudo-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 thexterior 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.
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 A volume form
on
gives rise to an orientation in a natural way as the atlas of coordinate charts on
that send
to a positive multiple of the Euclidean volume form
A volume form also allows for the specification of a preferred class of frameson Call a basis of tangent vectors
right-handed if
The collection of all right-handed frames is acted upon by the group ofgeneral linear mappings in
dimensions with positive determinant. They form a principal
sub-bundle of the linear frame bundleof
and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of
to a sub-bundle with structure group
That is to say that a volume form gives rise to
-structureon
More reduction is clearly possible by considering frames that have
(1) |
Thus a volume form gives rise to an -structure as well. Conversely, given an
-structure, one can recover a volume form by imposing (1) for the special linear frames and then solving for the required
-form
by requiring homogeneity in its arguments.
A manifold is orientable if and only if it has a nowhere-vanishing volume form. Indeed, is a deformation retract since
where the positive reals are embedded as scalar matrices. Thus every
-structure is reducible to an
-structure, and
-structures coincide with orientations on
More concretely, triviality of the determinant bundle
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.
Given a volume form 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.
Any volume pseudo-form (and therefore also any volume form) defines a measure on the Borel setsby
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.
Given a volume form on
one can define the divergence of a vector field
as the unique scalar-valued function, denoted by
satisfying
where
denotes the Lie derivative along
and
denotes the interior product or the left contractionof
along
If
is a compactly supported vector field and
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.
For any Lie group, a natural volume form may be defined by translation. That is, if is an element of
then a left-invariant form may be defined by
where
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.
Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. If is a
-dimensional manifold with symplectic form
then
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.
Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as
where the
are 1-forms 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 Levi-Civita tensor
Although the Greek letter is frequently used to denote the volume form, this notation is not universal; the symbol
often carries many other meanings in differential geometry (such as a symplectic form).
Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. Given a non-vanishing function on
and a volume form
is a volume form on
Conversely, given two volume forms
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 derivativeof 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.
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 in
there is an open neighborhood
of
and a diffeomorphism
of
onto an open set in
such that the volume form on
is the pullbackof
along
As a corollary, if and
are two manifolds, each with volume forms
then for any points
there are open neighborhoods
of
and
of
and a map
such that the volume form on
restricted to the neighborhood
pulls back to volume form on
restricted to the neighborhood
:
In one dimension, one can prove it thus:
given a volume form on
define
Then the standard Lebesgue measure
pulls backto
under
:
Concretely,
In higher dimensions, given any point
it has a neighborhood locally homeomorphic to
and one can apply the same procedure.
A volume form on a connected manifold has a single global invariant, namely the (overall) volume, denoted
which is invariant under volume-form 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.
| |||||||||
---|---|---|---|---|---|---|---|---|---|
Basic concepts |
| ||||||||
Main results (list) |
| ||||||||
Maps |
| ||||||||
Types of manifolds |
| ||||||||
Tensors |
| ||||||||
Related |
| ||||||||
Generalizations |
|
| |||||
---|---|---|---|---|---|
Scope |
| ||||
Notation |
| ||||
Tensor definitions |
| ||||
Operations |
| ||||
Related abstractions |
| ||||
Notable tensors |
| ||||
Mathematicians |
|