In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a volume form or top-dimensional form is a
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
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
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
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 function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
.
A volume form provides a means to define the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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 manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
s, being
complex manifolds, are naturally oriented, and so possess a volume form. More generally, the
th
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
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
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
frames on
Call a basis of tangent vectors
right-handed if
The collection of all right-handed frames is
acted upon by the
group of
general linear mappings in
dimensions with positive determinant. They form a
principal sub-bundle of the
linear frame bundle of
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
-structure on
More reduction is clearly possible by considering frames that have
Thus a volume form gives rise to an
-structure as well. Conversely, given an
-structure, one can recover a volume form by imposing () 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.
Relation to measures
Given a volume form
on an oriented manifold, the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
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 set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
s 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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, writing
considers
as a volume form, not simply a measure, and
indicates "integrate over the cell