Density on a manifold

{{short description|Section of a certain line bundle}} 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''{{sup|∗}}''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 {{nowrap|''v''1, ..., ''vn''}} in a ''n''-dimensional vector space ''V''. However, if one wishes to define a function {{nowrap|''μ'' : ''V'' × ... × ''V'' → '''R'''}} that assigns a volume for any such parallelotope, it should satisfy the following properties:

* If any of the vectors ''vk'' is multiplied by {{nowrap|''λ'' ∈ '''R'''}}, the volume should be multiplied by |''λ''|. * If any linear combination of the vectors ''v''1, ..., ''v''''j''−1, ''v''''j''+1, ..., ''vn'' is added to the vector ''vj'', 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

:<math>\mu(Av_1,\ldots,Av_n)=\left|\det A\right|\mu(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(V).</math>

Any such mapping {{nowrap|''μ'' : ''V'' × ... × ''V'' → '''R'''}} is called a '''density''' on the vector space ''V''. Note that if (''v1'', ..., ''vn'') is any basis for ''V'', then fixing ''μ''(''v1'', ..., ''vn'') 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 {{abs|''ω''}} on ''V'' by

:<math>|\omega|(v_1,\ldots,v_n) := |\omega(v_1,\ldots,v_n)|.</math>

===Orientations on a vector space===

The set Or(''V'') of all functions {{nowrap|''o'' : ''V'' × ... × ''V'' → '''R'''}} that satisfy

:<math>o(Av_1,\ldots,Av_n)=\operatorname{sign}(\det A)o(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(V) </math> if <math>v_1,\ldots,v_n</math> are linearly independent and <math>o(v_1,\ldots,v_n) = 0</math> otherwise

forms a one-dimensional vector space, and an '''orientation''' on ''V'' is one of the two elements {{nowrap|''o'' ∈ Or(''V'')}} such that {{nowrap|1={{abs|''o''(''v''1, ..., ''vn'')}} = 1}} for any linearly independent {{nowrap|''v''1, ..., ''vn''}}. Any non-zero ''n''-form ''ω'' on ''V'' defines an orientation {{nowrap|''o'' ∈ Or(''V'')}} such that

:<math>o(v_1,\ldots,v_n)|\omega|(v_1,\ldots,v_n) = \omega(v_1,\ldots,v_n),</math>

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

:<math>\omega(v_1,\ldots,v_n)= o(v_1,\ldots,v_n)\mu(v_1,\ldots,v_n).</math>

In terms of tensor product spaces,

:<math> \operatorname{Or}(V)\otimes \operatorname{Vol}(V) = \bigwedge^n V^*, \quad \operatorname{Vol}(V) = \operatorname{Or}(V)\otimes \bigwedge^n V^*. </math>

===''s''-densities on a vector space===

The ''s''-densities on ''V'' are functions {{nowrap|''μ'' : ''V'' × ... × ''V'' → '''R'''}} such that

:<math>\mu(Av_1,\ldots,Av_n)=\left|\det A\right|^s\mu(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(V).</math>

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

:<math>|\omega|^s(v_1,\ldots,v_n) := |\omega(v_1,\ldots,v_n)|^s.</math>

The product of ''s''1- and ''s''2-densities ''μ''1 and ''μ''2 form an (''s''1+''s''2)-density ''μ'' by

:<math>\mu(v_1,\ldots,v_n) := \mu_1(v_1,\ldots,v_n)\mu_2(v_1,\ldots,v_n).</math>

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

:<math> \operatorname{Vol}^{s_1}(V)\otimes \operatorname{Vol}^{s_2}(V) = \operatorname{Vol}^{s_1+s_2}(V). </math>

==Definition==

Formally, the ''s''-density bundle ''Vols''(''M'') of a differentiable manifold ''M'' is obtained by an associated bundle construction, intertwining the one-dimensional group representation

:<math>\rho(A) = \left|\det A\right|^{-s},\quad A\in \operatorname{GL}(n)</math>

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

:<math>\left|\Lambda\right|^s_M = \left|\Lambda\right|^s(TM).</math> 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,φα) is an atlas of coordinate charts on ''M'', then there is associated a local trivialization of <math>\left|\Lambda\right|^s_M</math>

:<math>t_\alpha : \left|\Lambda\right|^s_M|_{U_\alpha} \to \phi_\alpha(U_\alpha)\times\mathbb{R}</math>

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

:<math>t_{\alpha\beta} = \left|\det (d\phi_\alpha\circ d\phi_\beta^{-1})\right|^{-s}.</math>

== 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 {{Harv |Folland |1999 |loc = Section 11.4, pp. 361-362}}.

Given a 1-density ƒ supported in a coordinate chart ''U''α, the integral is defined by :<math>\int_{U_\alpha} f = \int_{\phi_\alpha(U_\alpha)} t_\alpha\circ f\circ\phi_\alpha^{-1}d\mu</math> 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 <math>|\Lambda|^1_M</math> using the Riesz-Markov-Kakutani representation theorem.

The set of ''1/p''-densities such that <math>|\phi|_p = \left( \int|\phi|^p \right)^{1/p} < \infty</math> is a normed linear space whose completion <math>L^p(M)</math> is called the '''intrinsic ''Lp'' 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 :<math>\rho(A) = \left|\det A\right|^{-s/n}.</math> 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 <math>|\Lambda|^s_M</math> is <math>|\Lambda|^{-s}_M</math>. * Tensor densities are sections of the tensor product of a density bundle with a tensor bundle.

==References==

* {{Citation|last1=Berline|first1=Nicole|last2=Getzler|first2=Ezra|last3=Vergne|first3=Michèle|title=Heat Kernels and Dirac Operators|isbn=978-3-540-20062-8|year=2004|publisher=Springer-Verlag|location=Berlin, New York}}. * {{Citation|first=Gerald B.|last=Folland|authorlink=Gerald Folland|title=Real Analysis: Modern Techniques and Their Applications|edition=Second|isbn=978-0-471-31716-6|year=1999|postscript=, provides a brief discussion of densities in the last section.}} * {{Citation|last1=Nicolaescu|first1=Liviu I.|title=Lectures on the geometry of manifolds|publisher=World Scientific Publishing Co. Inc.|location=River Edge, NJ|isbn=978-981-02-2836-1|mr=1435504|year=1996}} * {{Citation|last1=Lee|first1=John M|title=Introduction to Smooth Manifolds|publisher=Springer-Verlag|year=2003}}

Category:Differential geometry Category:Manifolds Category:Lp spaces