Inmathematics, 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-formsonM. 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:
If any of the vectors vk is multiplied by λ ∈ R, the volume should be multiplied by |λ|.
If any linear combination of the vectors v1, ..., vj−1, vj+1, ..., vn is added to the vector vj, the volume should stay invariant.
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 ωonV defines a density |ω|onVby
The set Or(V) of all functions o : V × ... × V → R that satisfy
forms a one-dimensional vector space, and an orientationonV is one of the two elements o ∈ Or(V) such that |o(v1, ..., vn)| = 1 for any linearly independent v1, ..., vn. Any non-zero n-form ωonV 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 ωonVby
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
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 (Folland 1999, Section 11.4, pp. 361-362).
Given a 1-density ƒ supported in a coordinate chart Uα, the integral is defined by
where the latter integral is with respect to the Lebesgue measureonRn. 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 measuresasdistributional 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 Lp spaceofM.
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.
Berline, Nicole; Getzler, Ezra; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag, ISBN978-3-540-20062-8.
Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications (Second ed.), ISBN978-0-471-31716-6, provides a brief discussion of densities in the last section.{{citation}}: CS1 maint: postscript (link)
Nicolaescu, Liviu I. (1996), Lectures on the geometry of manifolds, River Edge, NJ: World Scientific Publishing Co. Inc., ISBN978-981-02-2836-1, MR1435504
Lee, John M (2003), Introduction to Smooth Manifolds, Springer-Verlag