# 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 *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*is multiplied by_{k}*λ*∈**R**, the volume should be multiplied by |*λ*|. - If any linear combination of the vectors
*v*_{1}, ...,*v*−1,_{j}*v*+1, ...,_{j}*v*is added to the vector_{n}*v*, the volume should stay invariant._{j}

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*. 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

### 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*. Any non-zero

_{n}*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