# $s$-density Bundle

## Definition

Definition($s-density bundle) . The$s$-density bundle$|\Lambda|^s \to X$over a manifold$X$of dimension$n$, is a line bundle associated to the frame bundle for the tangent bundle, which is a principal$GL(n)$-bundle, over a manifold$X$via the 1-dimensional representation, or character, $$GL(n)\to GL(1)\cong \operatorname{Aut}_{\text{Vect}}(\mathbb{R}): A\mapsto |\operatorname{det}(A)|^{-s}.$$ A section$\Gamma(X,|\Lambda|^s)$of a$s$-density bundle$|\Lambda|^s\to X$is called a$s$-density on$X$. The parameter$s$is called the weight or order of the density. Using a prtition of unity a nowhere-vanishing section of$|\Lambda|^s$can be constructued, so$|\Lambda|^s$is trivializable. For the 1-density bundle, or simply density bundle, there is a result Proposition. Let$X$be a manifold of dimension$n$and$|\Lambda|$its density bundle. For an open set$U\subset \mathbb{R}^n$, denote by$|dx|$the 1-density s.t. $$|dx|(\partial_1\wedge\ldots\wedge\partial_n) = 1.$$ There is a unique linear form, called the integral, denoted by $$\int_X:\Gamma_c(X,|\Lambda|)\to\mathbb{R}$$ that is invariant under diffeomorphisms and locally agrees with the Lebesgue measure $$\int_X f(x) |dx| = \int_{\mathbb{R}^n}f(x) d x_1 ...dx_n.$$ Obviously$|\Lambda|$is closely related to the canonical bundle$\wedge^n T^\ast X$(the bundle of volume forms).$\wedge^n T^\ast X$is trivializable iff$X$is orientable, and when orientable the fram bundle has two components, upon choosing one component (choosing an orientation of$M$) of the frame bundle (associated to the tangent bundle) of$X$, that is, a principal$GL^+ (n)$-bundle. An orientation of a manifold$X$is equivalent to the choice of an isomorphism between$\wedge^n T^\ast X$and$|\Lambda|_M$. In other words, a positive definite density on an oriented manifold is equivalently a volume form on an oriented manifold. Let$C^\infty(X,|\Lambda|^s)$be the space of$s$-densities on$X$. For$\rho\in C^\infty(X,|\Lambda|^a)$and$\sigma\in C^\infty(X,|\Lambda|^b)$, the pointwise multiplication leas to a product$\rho\cdot \sigma \in C^\infty(X,|\Lambda|^{a+b})$. For$a=s$and$b=(1-s)$the integral then defines a continuous bilinear form on$C^\infty(X,|\Lambda|^s)\times C^\infty(X,|\Lambda|^{1-s})$, $$(\rho,\sigma) \mapsto \int_X \rho\cdot\sigma$$ Hence$\rho$can be viewed as a linear functional$\rho:\sigma\mapsto \int \rho\cdot \sigma \in \mathbb{R}$, i.e.$\rho\in [C^\infty_0(X,|\Lambda|^{1-s})]’$. Now since there is a continuous embedding$C^\infty(X,|\Lambda|^s) \hookrightarrow [C^\infty_0(X,|\Lambda|^{1-s})]’$, the space$[C^\infty_0(X,|\Lambda|^{1-s})]’$is also denoted$\mathcal{D}’(X,|\Lambda|^{s})$and called the space of$s$-distribution densities. A class of$\frac{1}{2}\$-distribution densities on a conic Lagrangian submanifold in the cotangent bundle are global Fourier integrals.