In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x.
From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into s-densities, whose coordinate representations become multiplied by the s-th power of the absolute value of the jacobian determinant. On an oriented manifold, 1-densities can be canonically identified with the n-forms on M. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of M and the n-th exterior product bundle of T'M (see pseudotensor).
In general, there does not exist a natural concept of a "volume" for a parallelotope generated by vectors in a n-dimensional vector space V. However, if one wishes to define a function that assigns a volume for any such parallelotope, it should satisfy the following properties:
These conditions are equivalent to the statement that μ is given by a translation-invariant measure on V, and they can be rephrased as
Any such mapping is called a density on the vector space V. Note that if (v<sub>1</sub>, ..., v<sub>n</sub>) is any basis for V, then fixing μ(v<sub>1</sub>, ..., v<sub>n</sub>) will fix μ entirely; it follows that the set Vol(V) of all densities on V forms a one-dimensional vector space. Any n-form ω on V defines a density on V by
The set Or(V) of all functions that satisfy
if are linearly independent and otherwise
forms a one-dimensional vector space, and an orientation on V is one of the two elements such that for any linearly independent . Any non-zero n-form ω on V defines an orientation such that
and vice versa, any and any density define an n-form ω on V by
In terms of tensor product spaces,
The s-densities on V are functions such that
Just like densities, s-densities form a one-dimensional vector space Vol<sup>s</sup>(V), and any n-form ω on V defines an s-density |ω|<sup>s</sup> on V by
The product of s<sub>1</sub>- and s<sub>2</sub>-densities μ<sub>1</sub> and μ<sub>2</sub> form an (s<sub>1</sub>+s<sub>2</sub>)-density μ by
In terms of tensor product spaces this fact can be stated as
Formally, the s-density bundle Vol<sup>s</sup>(M) of a differentiable manifold M is obtained by an associated bundle construction, intertwining the one-dimensional group representation
of the general linear group with the frame bundle of M.
The resulting line bundle is known as the bundle of s-densities, and is denoted by
A 1-density is also referred to simply as a density.
More generally, the associated bundle construction also allows densities to be constructed from any vector bundle E on M.
In detail, if (U<sub>ñ</sub>,ÃÂ<sub>ñ</sub>) is an atlas of coordinate charts on M, then there is associated a local trivialization of
subordinate to the open cover U<sub>ñ</sub> such that the associated GL(1)-cocycle satisfies
Densities play a significant role in the theory of integration on manifolds. Indeed, the definition of a density is motivated by how a measure dx changes under a change of coordinates .
Given a 1-density àsupported in a coordinate chart U<sub>ñ</sub>, the integral is defined by
where the latter integral is with respect to the Lebesgue measure on R<sup>n</sup>. The transformation law for 1-densities together with the Jacobian change of variables ensures compatibility on the overlaps of different coordinate charts, and so the integral of a general compactly supported 1-density can be defined by a partition of unity argument. Thus 1-densities are a generalization of the notion of a volume form that does not necessarily require the manifold to be oriented or even orientable. One can more generally develop a general theory of Radon measures as distributional sections of using the Riesz-Markov-Kakutani representation theorem.
The set of 1/p-densities such that is a normed linear space whose completion is called the intrinsic L<sup>p</sup> space of M.
In some areas, particularly conformal geometry, a different weighting convention is used: the bundle of s-densities is instead associated with the character
With this convention, for instance, one integrates n-densities (rather than 1-densities). Also in these conventions, a conformal metric is identified with a tensor density of weight 2.