Density on a manifold - Alchetron, The Free Social Encyclopedia

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold which can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain trivial 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 v₁, ..., v_n in a n-dimensional vector space V. However, if one wishes to define a function μ : V × ... × V → R that assigns a volume for any such parallelotope, it should satisfy the following properties:

If any of the vectors v_k is multiplied by λ ∈ R, the volume should be multiplied by |λ|.

If any linear combination of the vectors v₁, ..., 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

μ ( A v 1 , … , A v n ) = | det A | μ ( v 1 , … , v n ) , A ∈ GL ⁡ ( V ) .

Any such mapping μ : V × ... × V → R is called a density on the vector space V. The set Vol(V) of all densities on V forms a one-dimensional vector space, and any n-form ω on V defines a density | ω | on V by

| ω | ( v 1 , … , v n ) := | ω ( v 1 , … , v n ) | .

Orientations on a vector space

The set Or(V) of all functions o : V × ... × V → R that satisfy

o ( A v 1 , … , A v n ) = sign ⁡ ( det A ) o ( v 1 , … , v n ) , A ∈ GL ⁡ ( V )

forms a one-dimensional vector space, and an orientation on V is one of the two elements o ∈ Or(V) such that | o(v₁, ..., v_n) | = 1 for any linearly independent v₁, ..., v_n. Any non-zero n-form ω on V defines an orientation o ∈ Or(V) such that

o ( v 1 , … , v n ) | ω | ( v 1 , … , v n ) = ω ( v 1 , … , v n ) ,

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

ω ( v 1 , … , v n ) = o ( v 1 , … , v n ) μ ( v 1 , … , v n ) .

In terms of tensor product spaces,

Or ⁡ ( V ) ⊗ Vol ⁡ ( V ) = ⋀ n V ∗ , Vol ⁡ ( V ) = Or ⁡ ( V ) ⊗ ⋀ n V ∗ .

s-densities on a vector space

The s-densities on V are functions μ : V × ... × V → R such that

μ ( A v 1 , … , A v n ) = | det A | s μ ( v 1 , … , v n ) , A ∈ GL ⁡ ( 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

| ω | s ( v 1 , … , v n ) := | ω ( v 1 , … , v n ) | s .

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

μ ( v 1 , … , v n ) := μ 1 ( v 1 , … , v n ) μ 2 ( v 1 , … , v n ) .

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

Vol s 1 ⁡ ( V ) ⊗ Vol s 2 ⁡ ( V ) = Vol s 1 + s 2 ⁡ ( 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

ρ ( A ) = | det A | − s , A ∈ GL ⁡ ( 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

| Λ | M s = | Λ | s ( T M ) .

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 | Λ | M s

t α : | Λ | M s | U α → ϕ α ( U α ) × R

subordinate to the open cover U_α such that the associated GL(1)-cocycle satisfies

t α β = | det ( d ϕ α ∘ d ϕ β − 1 ) | − s .

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 U_α, the integral is defined by

∫ U α f = ∫ ϕ α ( U α ) t α ∘ f ∘ ϕ α − 1 d μ

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 | Λ | M 1 using the Riesz representation theorem.

The set of 1/p-densities such that | ϕ | p = ( ∫ | ϕ | p ) 1 / p < ∞ 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

ρ ( A ) = | det A | − s / n .

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 | Λ | M s is | Λ | M − s .

Tensor densities are sections of the tensor product of a density bundle with a tensor bundle.

References

Density on a manifold Wikipedia

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