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.
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 v1, ..., vn in a n-dimensional vector space V. However, if one wishes to define a function μ : V × ... × V → R 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 μ : V × ... × V → R is called a density on the vector space V. Note that if (v1, ..., vn) is any basis for V, then fixing μ(v1, ..., vn) 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 |ω| on V by The set Or(V) of all functions o : V × ... × V → R that satisfy if
Any non-zero n-form ω on V defines an orientation o ∈ Or(V) such that and vice versa, any o ∈ Or(V) and any density μ ∈ Vol(V) define an n-form ω on V by In terms of tensor product spaces, The s-densities on V are functions μ : V × ... × V → R such that Just like densities, s-densities form a one-dimensional vector space Vols(V), and any n-form ω on V defines an s-density |ω|s on V by The product of s1- and s2-densities μ1 and μ2 form an (s1+s2)-density μ by In terms of tensor product spaces this fact can be stated as Formally, the s-density bundle Vols(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 (Uα,φα) is an atlas of coordinate charts on M, then there is associated a local trivialization of
subordinate to the open cover Uα such that the associated GL(1)-cocycle satisfies Densities play a significant role in the theory of integration on manifolds.
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.
is called the intrinsic Lp 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.