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, ..., v_n 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 v_k is multiplied by λ ∈ R, the volume should be multiplied by |λ|.
• If any linear combination of the vectors v₁, ..., v_j−1, v_j+1, ..., v_n is added to the vector v_j, 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

$\mu (A{v}_1,\dots ,A{v}_n)=|\det A|\mu (v_1,\dots ,v_n),A\in \mathrm{GL}(V).$

Any such mapping μ : V × ... × V → R is called a density on the vector space V. Note that if (v₁, ..., v_n) is any basis for V, then fixing μ(v₁, ..., v_n) 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

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

Orientations on a vector space

The set Or(V) of all functions o : V × ... × V → R that satisfy

$o(A{v}_1,\dots ,A{v}_n)=\mathrm{sign}(\det A)o(v_1,\dots ,v_n),A\in \mathrm{GL}(V)$

if $v_1,\dots ,v_n$ are linearly independent and $o(v_1,\dots ,v_n)=0$ otherwise forms a one-dimensional vector space, and an orientation on V is one of the two elements o ∈ Or(V) such that |o(v₁, ..., v_n)| = 1 for any linearly independent v₁, ..., v_n. Any non-zero n-form ω on V defines an orientation o ∈ Or(V) such that

$o(v_1,\dots ,v_n)|\omega |(v_1,\dots ,v_n)=\omega (v_1,\dots ,v_n),$

and vice versa, any o ∈ Or(V) and any density μ ∈ Vol(V) define an n-form ω on V by

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

In terms of tensor product spaces,

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

s-densities on a vector space

The s-densities on V are functions μ : V × ... × V → R such that

$\mu (A{v}_1,\dots ,A{v}_n)={|\det A|}^s\mu (v_1,\dots ,v_n),A\in \mathrm{GL}(V).$

Just like densities, s-densities form a one-dimensional vector space Vol^s(V), and any n-form ω on V defines an s-density |ω|^s on V by

$|\omega {|}^s(v_1,\dots ,v_n):=|\omega (v_1,\dots ,v_n){|}^s.$

The product of s₁- and s₂-densities μ₁ and μ₂ form an (s₁+s₂)-density μ by

$\mu (v_1,\dots ,v_n):={\mu }_1(v_1,\dots ,v_n){\mu }_2(v_1,\dots ,v_n).$

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

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

Definition

Formally, the s-density bundle Vol^s(M) of a differentiable manifold M is obtained by an associated bundle construction, intertwining the one-dimensional group representation

$\rho (A)={|\det A|}^{-s},A\in \mathrm{GL}(n)$

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

${\left|\mathrm{\Lambda }\right|}_M^s={\left|\mathrm{\Lambda }\right|}^s(TM).$

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 (U_α,φ_α) is an atlas of coordinate charts on M, then there is associated a local trivialization of ${\left|\mathrm{\Lambda }\right|}_M^s$

$t_{\alpha }:{\left|\mathrm{\Lambda }\right|}_M^s{|}_{U_{\alpha }}\to {\varphi }_{\alpha }(U_{\alpha })\times \mathbb{ℝ}$

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

$t_{\alpha \beta }={|\det (d{\varphi }_{\alpha }\circ d{\varphi }_{\beta }^{-1})|}^{-s}.$

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 U_α, the integral is defined by

$\underset{U_{\alpha }}{\int }f=\underset{{\varphi }_{\alpha }(U_{\alpha })}{\int }{t}_{\alpha }\circ f\circ {\varphi }_{\alpha }^{-1}d\mu$

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 $|\mathrm{\Lambda }{|}_M^1$ using the Riesz-Markov-Kakutani representation theorem. The set of 1/p-densities such that $|\varphi {|}_p={(\int |\varphi {|}^p)}^{1/p}<\mathrm{\infty }$ is a normed linear space whose completion $L^p(M)$ is called the intrinsic L^p 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

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

References

• Berline, Nicole; Getzler, Ezra; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag, ISBN 978-3-540-20062-8.
• Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications (Second ed.), ISBN 978-0-471-31716-6, 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., ISBN 978-981-02-2836-1, MR 1435504
• Lee, John M (2003), Introduction to Smooth Manifolds, Springer-Verlag