View source for 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 [[integral|integrated]] in an intrinsic manner. Abstractly, a density is a [[section (fiber bundle)|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 [[parallelepiped|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 chart]]s 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 [[differential form|''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''<sub>1</sub>, ..., ''v<sub>n</sub>''}} in a ''n''-dimensional vector space ''V''. However, if one wishes to define a function {{nowrap|''&mu;'' : ''V'' × ... × ''V'' → '''R'''}} that assigns a volume for any such parallelotope, it should satisfy the following properties:

* If any of the vectors ''v<sub>k</sub>'' is multiplied by {{nowrap|''λ'' ∈ '''R'''}}, the volume should be multiplied by |''λ''|.
* If any linear combination of the vectors ''v''<sub>1</sub>, ..., ''v''<sub>''j''−1</sub>, ''v''<sub>''j''+1</sub>, ..., ''v<sub>n</sub>'' is added to the vector ''v<sub>j</sub>'', the volume should stay invariant.

These conditions are equivalent to the statement that ''&mu;'' 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|''&mu;'' : ''V'' × ... × ''V'' → '''R'''}} is called a '''density''' on the vector space ''V''. Note that if (''v<sub>1</sub>'', ..., ''v<sub>n</sub>'') is any basis for ''V'', then fixing ''&mu;''(''v<sub>1</sub>'', ..., ''v<sub>n</sub>'') will fix ''&mu;'' entirely; it follows that the set Vol(''V'') of all densities on ''V'' forms a one-dimensional vector space. Any ''n''-form ''&omega;'' on ''V'' defines a density {{abs|''&omega;''}} 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''<sub>1</sub>, ..., ''v<sub>n</sub>'')}} = 1}} for any linearly independent {{nowrap|''v''<sub>1</sub>, ..., ''v<sub>n</sub>''}}. Any non-zero ''n''-form ''&omega;'' 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|''&mu;'' ∈ Vol(''V'')}} define an ''n''-form ''&omega;'' 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|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|''&mu;'' : ''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 ''Vol<sup>s</sup>''(''V''), and any ''n''-form ''&omega;'' on ''V'' defines an ''s''-density |''&omega;''|<sup>''s''</sup> on ''V'' by

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

The product of ''s''<sub>1</sub>- and ''s''<sub>2</sub>-densities ''&mu;''<sub>1</sub> and ''&mu;''<sub>2</sub> form an  (''s''<sub>1</sub>+''s''<sub>2</sub>)-density ''&mu;'' 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|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 ''Vol<sup>s</sup>''(''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''<sub>α</sub>,φ<sub>α</sub>) is an [[atlas (topology)|atlas]] of [[coordinate chart]]s 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''<sub>α</sub> 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 [[integral|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''<sub>α</sub>, 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'''<sup>''n''</sup>.  The transformation law for 1-densities together with the [[integration by substitution|Jacobian change of variables]] ensures compatibility on the overlaps of different coordinate charts, and so the integral of a general [[compact support|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 measure]]s as [[distribution (mathematics)|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|''L<sup>p</sup>'' 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 density|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}}

{{Manifolds}}

[[Category:Differential geometry]]
[[Category:Manifolds]]
[[Category:Lp spaces]]