Density on a manifold

In general, there does not exist a natural concept of a "volume" for a parallelotope generated by vectors {{nowrap|v₁, ..., v_n}} 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 v_k is multiplied by {{nowrap|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

:$\backslash mu(Av\_1,\backslash ldots,Av\_n)=\backslash left|\backslash det\; A\backslash right|\backslash mu(v\_1,\backslash ldots,v\_n),\; \backslash quad\; A\backslash in\; \backslash operatorname\{GL\}(V).$

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

:$|\backslash omega|(v\_1,\backslash ldots,v\_n)\; :=\; |\backslash omega(v\_1,\backslash ldots,v\_n)|.$

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

:$o(Av\_1,\backslash ldots,Av\_n)=\backslash operatorname\{sign\}(\backslash det\; A)o(v\_1,\backslash ldots,v\_n),\; \backslash quad\; A\backslash in\; \backslash operatorname\{GL\}(V)$

if $v\_1,\backslash ldots,v\_n$ are linearly independent and $o(v\_1,\backslash ldots,v\_n)\; =\; 0$ 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(v₁, ..., v_n)}} = 1}} for any linearly independent {{nowrap|v₁, ..., v_n}}. Any non-zero n-form ω on V defines an orientation {{nowrap|o ∈ Or(V)}} such that

:$o(v\_1,\backslash ldots,v\_n)|\backslash omega|(v\_1,\backslash ldots,v\_n)\; =\; \backslash omega(v\_1,\backslash ldots,v\_n),$

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

:$\backslash omega(v\_1,\backslash ldots,v\_n)=\; o(v\_1,\backslash ldots,v\_n)\backslash mu(v\_1,\backslash ldots,v\_n).$

In terms of tensor product spaces,

:$\backslash operatorname\{Or\}(V)\backslash otimes\; \backslash operatorname\{Vol\}(V)\; =\; \backslash bigwedge^n\; V^*,\; \backslash quad\; \backslash operatorname\{Vol\}(V)\; =\; \backslash operatorname\{Or\}(V)\backslash otimes\; \backslash bigwedge^n\; V^*.$

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

:$\backslash mu(Av\_1,\backslash ldots,Av\_n)=\backslash left|\backslash det\; A\backslash right|^s\backslash mu(v\_1,\backslash ldots,v\_n),\; \backslash quad\; A\backslash in\; \backslash operatorname\{GL\}(V).$

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

:$|\backslash omega|^s(v\_1,\backslash ldots,v\_n)\; :=\; |\backslash omega(v\_1,\backslash ldots,v\_n)|^s.$

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

:$\backslash mu(v\_1,\backslash ldots,v\_n)\; :=\; \backslash mu\_1(v\_1,\backslash ldots,v\_n)\backslash mu\_2(v\_1,\backslash ldots,v\_n).$

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

:$\backslash operatorname\{Vol\}^\{s\_1\}(V)\backslash otimes\; \backslash operatorname\{Vol\}^\{s\_2\}(V)\; =\; \backslash operatorname\{Vol\}^\{s\_1+s\_2\}(V).$

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

:$\backslash rho(A)\; =\; \backslash left|\backslash det\; A\backslash right|^\{-s\},\backslash quad\; A\backslash in\; \backslash operatorname\{GL\}(n)$

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

:$\backslash left|\backslash Lambda\backslash right|^s\_M\; =\; \backslash left|\backslash Lambda\backslash right|^s(TM).$

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 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 $\backslash left|\backslash Lambda\backslash right|^s\_M$

:$t\_\backslash alpha\; :\; \backslash left|\backslash Lambda\backslash right|^s\_M|\_\{U\_\backslash alpha\}\; \backslash to\; \backslash phi\_\backslash alpha(U\_\backslash alpha)\backslash times\backslash mathbb\{R\}$

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

:$t\_\{\backslash alpha\backslash beta\}\; =\; \backslash left|\backslash det\; (d\backslash phi\_\backslash alpha\backslash circ\; d\backslash phi\_\backslash beta^\{-1\})\backslash right|^\{-s\}.$

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

:$\backslash int\_\{U\_\backslash alpha\}\; f\; =\; \backslash int\_\{\backslash phi\_\backslash alpha(U\_\backslash alpha)\}\; t\_\backslash alpha\backslash circ\; f\backslash circ\backslash phi\_\backslash alpha^\{-1\}d\backslash mu$

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 $|\backslash Lambda|^1\_M$ using the Riesz-Markov-Kakutani representation theorem.

The set of 1/p-densities such that is a normed linear space whose completion $L^p(M)$ is called the intrinsic L^p space of M.

In some areas, particularly conformal geometry, a different weighting convention is used: the bundle of s-densities is instead associated with the character

:$\backslash rho(A)\; =\; \backslash left|\backslash det\; A\backslash right|^\{-s/n\}.$

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.

• The dual vector bundle of $|\backslash Lambda|^s\_M$ is $|\backslash Lambda|^\{-s\}\_M$.
• Tensor densities are sections of the tensor product of a density bundle with a tensor bundle.

• {{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