Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty\_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions $a\in\mathcal{D}(\mathbb{R})$ that are localized at a point $r\in\mathbb{R}$; $a$ must be a linear combination of distributional derivatives of the dirac $\delta$-distribution, $\delta\_r$.


Let $M$ be a compact manifold and $\Lambda\subset M$ a closed submanifold. Let $\mathcal{D}(M)$ be the continuous dual of $C^\infty\_c(M)$, i.e. $\mathcal{D}$ is the collection of zero-currents (in the sense of de Rham). Is there an analogous characterization of zero-currents that are localized along $\Lambda$?



Zero currents localized along a submanifold
A special case is this:


Let $u\in\mathcal{E}'(\mathbb{R}^{n+m})$. Then $supp(u)\subseteq\mathbb{R}^n\times\{0\}$ iff $u$ is a linear combination of the form $\sum\_{\substack{\alpha\in\{0\}\times\mathbb{N}^m \\ |\alpha|\leq k}} c\_\alpha \partial^\alpha \iota(v\_\alpha)$ for certain $v\_\alpha\in\mathcal{E}'(\mathbb{R}^n)$. Here $\iota(v)$ denotes the distribution on $\mathbb{R}^{n+m}$ defined by $\langle \iota(v),\phi\rangle:=\langle v,\phi\_{|\mathbb{R}^n}\rangle$.


The proof idea is to use a taylor expansion of $\phi$ orthogonal to $\mathbb{R}^n$, to show that $u$ vanishes on those functions whose orthogonal derivates up to the $k$th order vanish on $\mathbb{R}^n\times\{0\}$ and deduce the result.


(I suspect that a similar thing could be true for $u\in\mathcal{D}'$. In that case the order of $u$ is only locally bounded, to one would expect that that linear combination has to be replaced by some infinite (but locally finite) linear combination of derivatives)


By localizing and some application of change of coordinates I think this should extend to the following general result: If $u\in\mathcal{E}'(M,E)$ is a distributional section of a vector bundle $E$ on the riemannian manifold $M$ (in your case $E$ is the bundle of $n$-forms) and $S\subseteq M$ an embedded submanifold, then $supp(u)\subseteq S$ iff it is of the form $\sum\_{a=0}^k \mathcal{L}\_{X\_{i1}} \cdots \mathcal{L}\_{X\_{ij\_a}}\iota(v\_a)$ for certain $v\_a\in\mathcal{E}'(S,E\_{|S})$ and certain vector fields $X\_{ij} \in \Gamma(NS)$ where $NS\subseteq TM$ is the normal bundle of $S$ and $\iota:\mathcal{E}'(S,E\_{|S})\to\mathcal{E}'(M,E)$ is the analogue embedding. Again, it might be the case that some version of this also holds for $\mathcal{D}'$ if one replace the finite sum by an locally finite series.


