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

${\displaystyle \mu (Av_{1},\ldots ,Av_{n})=\left|\det A\right|\mu (v_{1},\ldots ,v_{n}),\quad A\in \operatorname {GL} (V).}$

Any such mapping 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

${\displaystyle |\omega |(v_{1},\ldots ,v_{n}):=|\omega (v_{1},\ldots ,v_{n})|.}$

Orientations on a vector space

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

${\displaystyle o(Av_{1},\ldots ,Av_{n})=\operatorname {sign} (\det A)o(v_{1},\ldots ,v_{n}),\quad A\in \operatorname {GL} (V)}$

if ${\displaystyle v_{1},\ldots ,v_{n}}$ are linearly independent and ${\displaystyle o(v_{1},\ldots ,v_{n})=0}$ 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

${\displaystyle o(v_{1},\ldots ,v_{n})|\omega |(v_{1},\ldots ,v_{n})=\omega (v_{1},\ldots ,v_{n}),}$

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

${\displaystyle \omega (v_{1},\ldots ,v_{n})=o(v_{1},\ldots ,v_{n})\mu (v_{1},\ldots ,v_{n}).}$

In terms of tensor product spaces,

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

s-densities on a vector space

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

${\displaystyle \mu (Av_{1},\ldots ,Av_{n})=\left|\det A\right|^{s}\mu (v_{1},\ldots ,v_{n}),\quad A\in \operatorname {GL} (V).}$

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

${\displaystyle |\omega |^{s}(v_{1},\ldots ,v_{n}):=|\omega (v_{1},\ldots ,v_{n})|^{s}.}$

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

${\displaystyle \mu (v_{1},\ldots ,v_{n}):=\mu _{1}(v_{1},\ldots ,v_{n})\mu _{2}(v_{1},\ldots ,v_{n}).}$

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

${\displaystyle \operatorname {Vol} ^{s_{1}}(V)\otimes \operatorname {Vol} ^{s_{2}}(V)=\operatorname {Vol} ^{s_{1}+s_{2}}(V).}$

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

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

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

${\displaystyle \left|\Lambda \right|_{M}^{s}=\left|\Lambda \right|^{s}(TM).}$

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

${\displaystyle t_{\alpha }:\left|\Lambda \right|_{M}^{s}|_{U_{\alpha }}\to \phi _{\alpha }(U_{\alpha })\times \mathbb {R} }$

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

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

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 ( 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

${\displaystyle \int _{U_{\alpha }}f=\int _{\phi _{\alpha }(U_{\alpha })}t_{\alpha }\circ f\circ \phi _{\alpha }^{-1}d\mu }$

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 ${\displaystyle |\Lambda |_{M}^{1}}$ using the Riesz-Markov-Kakutani representation theorem.

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

${\displaystyle \rho (A)=\left|\det A\right|^{-s/n}.}$

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 ${\displaystyle |\Lambda |_{M}^{s}}$ is ${\displaystyle |\Lambda |_{M}^{-s}}$.
• 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.
• 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