Density on a manifold

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In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold which can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain trivial 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 parallelotype generated by vectors v1,...,vn in a n-dimensional vector space V. However, if one wishes to define a function μ:V×...×VR that assigns a volume for any such parallelotype, it should satisfy the following properties:

  • If any of the vectors vk is multiplied by λR, 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

\mu(Av_1,\ldots,Av_n)=|\det A|\mu(v_1,\ldots,v_n), \quad A\in GL(V).

Any such mapping μ:V×...×VR is called a density on the vector space V. The set Vol(V) of all densities on V forms a one-dimensional vector space, and any n-form ω on V defines a density |ω| 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×...×VR that satisfy

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

forms a one-dimensional vector space, and an orientation on V is one of the two elements oOr(V) such that |o(v1,...,vn)|=1 for any linearly independent v1,...,vn. Any non-zero n-form ω on V defines an orientation oOr(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 oOr(V) and any density μVol(V) define an n-form ω 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 μ:V×...×VR such that

\mu(Av_1,\ldots,Av_n)=|\det A|^s\mu(v_1,\ldots,v_n), \quad A\in GL(V).

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

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

The product of s1- and s2-densities μ1 and μ2 form an (s1+s2)-density μ 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) = |\det A|^{-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

|\Lambda|^s_M = |\Lambda|^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 (Uαα) is an atlas of coordinate charts on M, then there is associated a local trivialization of |\Lambda|^s_M

t_\alpha : |\Lambda|^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} = |\det (d\phi_\alpha\circ d\phi_\beta^{-1})|^{-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 (Folland 1999, Section 11.4, pp. 361-362).

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 representation theorem.

The set of 1/p-densities such that |\phi|_p = (\int|\phi|^p)^{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) = |\det A|^{-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

References








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