To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Density on a manifold

From Wikipedia, the free encyclopedia

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 TM (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 v1, ..., vn in a n-dimensional vector space V. However, if one wishes to define a function μ : V × ... × VR that assigns a volume for any such parallelotope, it should satisfy the following properties:

  • If any of the vectors vk is multiplied by λR, the volume should be multiplied by |λ|.
  • If any linear combination of the vectors v1, ..., vj−1, vj+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

Any such mapping μ : V × ... × VR 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 |ω| on V by

Orientations on a vector space

The set Or(V) of all functions o : V × ... × VR 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(v1, ..., vn)| = 1 for any linearly independent v1, ..., vn. Any non-zero 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 × ... × VR such that

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

The product of s1- and s2-densities μ1 and μ2 form an (s1+s2)-density μ by

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

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

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 (Uαα) is an atlas of coordinate charts on M, then there is associated a local trivialization of

subordinate to the open cover Uα 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 Uα, the integral is defined by

where the latter integral is with respect to the Lebesgue measure on Rn. 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-Markov-Kakutani representation theorem.

The set of 1/p-densities such that is a normed linear space whose completion 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

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

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.CS1 maint: postscript (link)
  • 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
  • Lee, John M (2003), Introduction to Smooth Manifolds, Springer-Verlag
This page was last edited on 29 March 2021, at 14:59
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.