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 in a ''n''-dimensional vector space ''V''. However, if one wishes to define a function that assigns a volume for any such parallelotope, it should satisfy the following properties:

* If any of the vectors ''v_k'' is multiplied by , 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

:$\mu(Av_1,\ldots,Av_n)=\left, \det A\\mu(v_1,\ldots,v_n), \quad A\in \operatorname(V).$

Any such mapping 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 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 that satisfy

:$o(Av_1,\ldots,Av_n)=\operatorname(\det A)o(v_1,\ldots,v_n), \quad A\in \operatorname(V)$
if $v_1,\ldots,v_n$ are linearly independent and $o(v_1,\ldots,v_n) = 0$ otherwise

forms a one-dimensional vector space, and an orientation on ''V'' is one of the two elements such that for any linearly independent . Any non-zero ''n''-form ''ω'' on ''V'' defines an orientation such that

:$o(v_1,\ldots,v_n), \omega, (v_1,\ldots,v_n) = \omega(v_1,\ldots,v_n),$

and vice versa, any and any density 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,

:$\operatorname(V)\otimes \operatorname(V) = \bigwedge^n V^*, \quad \operatorname(V) = \operatorname(V)\otimes \bigwedge^n V^*.$

''s''-densities on a vector space

The ''s''-densities on ''V'' are functions such that

:$\mu(Av_1,\ldots,Av_n)=\left, \det A\^s\mu(v_1,\ldots,v_n), \quad A\in \operatorname(V).$

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 ''s''₁- and ''s''₂-densities ''μ''₁ and ''μ''₂ form an (''s''₁+''s''₂)-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

:$\operatorname^(V)\otimes \operatorname^(V) = \operatorname^(V).$

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

:$\rho(A) = \left, \det A\^,\quad A\in \operatorname(n)$

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

:$\left, \Lambda\^s_M = \left, \Lambda\^s(TM).$
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 $\left, \Lambda\^s_M$

:$t_\alpha : \left, \Lambda\^s_M, _ \to \phi_\alpha(U_\alpha)\times\mathbb$

subordinate to the open cover ''U''_α such that the associated GL(1)-cocycle satisfies

:$t_ = \left, \det (d\phi_\alpha\circ d\phi_\beta^)\^.$

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 .

Given a 1-density ƒ supported in a coordinate chart ''U''_α, the integral is defined by
:$\int_ f = \int_ t_\alpha\circ f\circ\phi_\alpha^d\mu$
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 $, \Lambda, ^1_M$ using the Riesz-Markov-Kakutani representation theorem.

The set of ''1/p''-densities such that $, \phi, _p = \left( \int, \phi, ^p \right)^ < \infty$ 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
:$\rho(A) = \left, \det A\^.$
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 $, \Lambda, ^_M$.
* Tensor densities are sections of the tensor product of a density bundle with a tensor bundle.

References

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{{Manifolds

Differential geometry
Manifolds
Lp spaces