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 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).
Contents
Motivation (Densities in vector spaces)
In general, there does not exist a natural concept of a "volume" for a parallelotope generated by vectors v_{1}, ..., 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 v_{1}, ..., 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
Any such mapping μ : V × ... × V → R is called a density on the vector space V. Note that if (v_{1}, ..., v_{n}) is any basis for V, then fixing μ(v_{1}, ..., 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
Orientations on a vector space
The set Or(V) of all functions o : V × ... × V → R that satisfy
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 v_{1}, ..., v_{n}. Any non-zero n-form ω on V defines an orientation o ∈ Or(V) such that
and vice versa, any o ∈ Or(V) and any density μ ∈ Vol(V) define an n-form ω on V by
In terms of tensor product spaces,
s-densities on a vector space
The s-densities on V are functions μ : V × ... × V → R such that
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
The product of s_{1}- and s_{2}-densities μ_{1} and μ_{2} form an (s_{1}+s_{2})-density μ by
In terms of tensor product spaces this fact can be stated as
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
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
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
subordinate to the open cover U_{α} such that the associated GL(1)-cocycle satisfies
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
where the latter integral is with respect to the Lebesgue measure on R^{n}. 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 using the Riesz representation theorem.
The set of 1/p-densities such that is a normed linear space whose completion 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
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 is .
- 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