Density on a manifold

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 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(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. 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 × ... × 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(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,\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

\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 × ... × 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 Vol^s(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 Vol^s(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 (U_α,φ_α) 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 U_α 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 (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

\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 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) = \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.

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