Scalar distributions on smooth manifolds

In positive dimension use the Euclidean test-function spaces and their topology from the suppliers below. In dimension zero put ${\mathbb{R}}^0=\{\ast \}$ and declare every complex-valued function on either open subset smooth. Define $\mathcal{D}(\{\ast \})=\mathbb{C}$ by evaluation at $\ast$, with its usual absolute-value norm topology, and $\mathcal{D}(\mathrm{\varnothing })=\{0\}$ with its unique topology. Define ${\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ in this dimension as the continuous complex-linear dual of $\mathcal{D}(\mathrm{\Omega })$: on $\{\ast \}$ its elements are exactly $f\mapsto cf(\ast )$, $c\in \mathbb{C}$, and on the empty set there is only zero. The unique zero-dimensional multi-index has order zero and its derivative is the identity; an empty determinant and the density symbol $\mathrm{\mid }d{x}^1\cdots d{x}^0\mathrm{\mid }$ both equal one.

Let $M$ be a Hausdorff, second-countable smooth $n$-manifold without boundary. In a smooth chart $x=(x^1,\dots ,x^n)$, a smooth density is written ${\varphi }_x(x)\mathrm{\mid }d{x}^1\cdots d{x}^n\mathrm{\mid }$. On a coordinate change $y=\psi (x)$ its coefficient transforms by ${\varphi }_y(y)={\varphi }_x({\psi }^{-1}(y))\mathrm{\mid }\det D{\psi }^{-1}(y)\mathrm{\mid }\mathrm{.}$ A compactly supported smooth density is one whose support (the closure of the set where its coefficient is nonzero) is compact in $M$. For compact $K\subset M$, let ${\mathcal{D}}_K(M;\mathrm{\mid }{\mathrm{\Lambda }}^n{T}^{\ast }M\mathrm{\mid })$ be the complex vector space of smooth densities supported in $K$. For every chart $x:U\to \mathrm{\Omega }$, compact $L\subset U$, and integer $r\ge 0$, give this space the seminorm $p_{x,L,r}(\omega )={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}{\sup }_{z\in x(L)}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{\varphi }_x(z)\mathrm{\mid }\mathrm{.}$ An empty supremum is zero. These seminorms define the fixed-support locally convex topology. On the union $\mathcal{D}(M;\mathrm{\mid }{\mathrm{\Lambda }}^n{T}^{\ast }M\mathrm{\mid })={\bigcup }_{K\text{ compact}}{\mathcal{D}}_K(M;\mathrm{\mid }{\mathrm{\Lambda }}^n{T}^{\ast }M\mathrm{\mid })$, take all seminorms whose restriction to every fixed-support space is continuous; their finite intersections of open seminorm balls and translates define the test-density topology. Thus it is the finest locally convex topology making every fixed-support inclusion continuous, as in the Euclidean test-function topology. This specifies the locally convex inductive limit without choosing an exhaustion.

A scalar distribution on $M$ is a continuous complex-linear functional on this space, with bilinear pairing $⟨u,\omega ⟩$. Write ${\mathcal{D}}^{\mathrm{\prime }}(M)$ for these distributions. The support of $u$ is the complement of the union of open sets on which it annihilates every supported test density; ${\mathcal{E}}^{\mathrm{\prime }}(M)$ denotes those with compact support. For $x:U\to \mathrm{\Omega }$, its coordinate representative is defined for $f\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ by $⟨u_x,f⟩=⟨u,\widetilde{x^{\ast }(f\mathrm{\mid }d{z}^1\cdots d{z}^n\mathrm{\mid })}⟩,$ where the tilde extends the density by zero outside $U$. This extension is smooth because the support is compact in $U$. On each fixed support its derivatives in any other chart are bounded by finitely many derivatives of $f$, by the chain and product rules on compact sets; hence the extension map is continuous for the specified topologies and $u_x$ is a Euclidean distribution. In dimension zero derivatives have order zero and the density determinant is one. On a product manifold use the same definition with product densities. For a smooth diffeomorphism $f:N\to M$, define $f_{\ast }\omega =(f^{-1}{)}^{\ast }\omega$ for test densities, using the displayed absolute-Jacobian rule. Define its distributional pullback by $⟨f^{\ast }u,\omega ⟩=⟨u,f_{\ast }\omega ⟩$. Compact supports are mapped to compact supports, and the same compact derivative estimates prove continuity. This includes coordinate changes and interchange of factors of a product.

Smooth manifolds and their smooth charts

Test function space d of an open set

Test function topology

Distribution