density bundle
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and specifically
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, a density is a spatially varying quantity on a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
that can be integrated in an intrinsic manner. Abstractly, a density is a
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of a certain
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
, 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 vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
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 In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordi ...
).


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 ''vk'' 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, ..., ''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 :\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'', ..., ''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 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) 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 ''Vols''(''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 ''Vols''(''M'') of a differentiable manifold ''M'' is obtained by an
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
construction, intertwining the one-dimensional
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
:\rho(A) = \left, \det A\^,\quad A\in \operatorname(n) of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
with the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
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 In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
''E'' on ''M''. In detail, if (''U''αα) is an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
of coordinate charts on ''M'', then there is associated a
local trivialization In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
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 In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0 ...
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 In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s 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 ''Lp'' space of ''M''.


Conventions

In some areas, particularly
conformal geometry In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
, 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 In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is ...
of weight 2.


Properties

* The dual vector bundle of , \Lambda, ^s_M is , \Lambda, ^_M. * Tensor densities are sections of the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of a density bundle with a tensor bundle.


References

* . * * * {{Manifolds Differential geometry Manifolds