Density on a manifold

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 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 :\mu(Av_1,\ldots,Av_n)=\left|\det A\right|\mu(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(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{sign}(\det A)o(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(V) if v_1,\ldots,v_n are linearly independent and o(v_1,\ldots,v_n) = 0 otherwise 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{Or}(V)\otimes \operatorname{Vol}(V) = \bigwedge^n V^*, \quad \operatorname{Vol}(V) = \operatorname{Or}(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\right|^s\mu(v_1,\ldots,v_n), \quad A\in \operatorname{GL}(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 s1- and s2-densities μ1 and μ2 form an (s1+s2)-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{Vol}^{s_1}(V)\otimes \operatorname{Vol}^{s_2}(V) = \operatorname{Vol}^{s_1+s_2}(V). ==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 :\rho(A) = \left|\det A\right|^{-s},\quad A\in \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 :\left|\Lambda\right|^s_M = \left|\Lambda\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 (Uα,φα) is an atlas of coordinate charts on M, then there is associated a local trivialization of \left|\Lambda\right|^s_M :t_\alpha : \left|\Lambda\right|^s_M|_{U_\alpha} \to \phi_\alpha(U_\alpha)\times\mathbb{R} subordinate to the open cover Uα such that the associated GL(1)-cocycle satisfies :t_{\alpha\beta} = \left|\det (d\phi_\alpha\circ d\phi_\beta^{-1})\right|^{-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 . Given a 1-density ƒ supported in a coordinate chart Uα, the integral is defined by :\int_{U_\alpha} f = \int_{\phi_\alpha(U_\alpha)} t_\alpha\circ f\circ\phi_\alpha^{-1}d\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 |\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)^{1/p} 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, a different weighting convention is used: the bundle of s-densities is instead associated with the character :\rho(A) = \left|\det A\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. ==Properties==

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