Density on a manifold

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).

Motivation (densities in vector spaces)
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 &mu; : 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 &mu; is given by a translation-invariant measure on V, and they can be rephrased as


 * $$\mu(Av_1,\ldots,Av_n)=\left|\det A\right|\mu(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(V).$$

Any such mapping &mu; : V × ... × V → R is called a density on the vector space V. Note that if (v1, ..., vn) is any basis for V, then fixing &mu;(v1, ..., vn) will fix &mu; entirely; it follows that the set Vol(V) of all densities on V forms a one-dimensional vector space. Any n-form &omega; on V defines a density $|&omega;|$ on V by


 * $$|\omega|(v_1,\ldots,v_n) := |\omega(v_1,\ldots,v_n)|.$$

Orientations on a vector space
The set Or(V) of all functions o : V × ... × V → R that satisfy


 * $$o(Av_1,\ldots,Av_n)=\operatorname{sign}(\det A)o(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(V)$$

forms a one-dimensional vector space, and an orientation on V is one of the two elements o ∈ Or(V) such that $|o(v_{1}, ..., v_{n})|$ = 1 for any linearly independent v1, ..., vn. Any non-zero n-form &omega; on V defines an orientation o ∈ Or(V) such that


 * $$o(v_1,\ldots,v_n)|\omega|(v_1,\ldots,v_n) = \omega(v_1,\ldots,v_n),$$

and vice versa, any o ∈ Or(V) and any density &mu; ∈ Vol(V) define an n-form &omega; on V by


 * $$\omega(v_1,\ldots,v_n)= o(v_1,\ldots,v_n)\mu(v_1,\ldots,v_n).$$

In terms of tensor product spaces,


 * $$ \operatorname{Or}(V)\otimes \operatorname{Vol}(V) = \bigwedge^n V^*, \quad \operatorname{Vol}(V) = \operatorname{Or}(V)\otimes \bigwedge^n V^*. $$

s-densities on a vector space
The s-densities on V are functions &mu; : V × ... × V → R such that


 * $$\mu(Av_1,\ldots,Av_n)=\left|\det A\right|^s\mu(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(V).$$

Just like densities, s-densities form a one-dimensional vector space Vols(V), and any n-form &omega; on V defines an s-density |&omega;|s on V by


 * $$|\omega|^s(v_1,\ldots,v_n) := |\omega(v_1,\ldots,v_n)|^s.$$

The product of s1- and s2-densities &mu;1 and &mu;2 form an (s1+s2)-density &mu; by


 * $$\mu(v_1,\ldots,v_n) := \mu_1(v_1,\ldots,v_n)\mu_2(v_1,\ldots,v_n).$$

In terms of tensor product spaces this fact can be stated as


 * $$ \operatorname{Vol}^{s_1}(V)\otimes \operatorname{Vol}^{s_2}(V) = \operatorname{Vol}^{s_1+s_2}(V). $$

Definition
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


 * $$\rho(A) = \left|\det A\right|^{-s},\quad A\in \operatorname{GL}(n)$$

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


 * $$\left|\Lambda\right|^s_M = \left|\Lambda\right|^s(TM).$$

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 $$\left|\Lambda\right|^s_M$$


 * $$t_\alpha : \left|\Lambda\right|^s_M|_{U_\alpha} \to \phi_\alpha(U_\alpha)\times\mathbb{R}$$

subordinate to the open cover Uα such that the associated GL(1)-cocycle satisfies


 * $$t_{\alpha\beta} = \left|\det (d\phi_\alpha\circ d\phi_\beta^{-1})\right|^{-s}.$$

Integration
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 ƒ supported in a coordinate chart Uα, the integral is defined by
 * $$\int_{U_\alpha} f = \int_{\phi_\alpha(U_\alpha)} t_\alpha\circ f\circ\phi_\alpha^{-1}d\mu$$

where the latter integral is with respect to the Lebesgue measure on Rn. 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 $$|\Lambda|^1_M$$ using the Riesz-Markov-Kakutani representation theorem.

The set of 1/p-densities such that $$|\phi|_p = \left( \int|\phi|^p \right)^{1/p} < \infty$$ is a normed linear space whose completion $$L^p(M)$$ is called the intrinsic Lp space of M.

Conventions
In some areas, particularly conformal geometry, a different weighting convention is used: the bundle of s-densities is instead associated with the character
 * $$\rho(A) = \left|\det A\right|^{-s/n}.$$

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.

Properties

 * The dual vector bundle of $$|\Lambda|^s_M$$ is $$|\Lambda|^{-s}_M$$.
 * Tensor densities are sections of the tensor product of a density bundle with a tensor bundle.