$S$-Density Bundle | Itinerarium Mentis

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.