Reconstruction Theorem for Germs of Distributions on Smooth Manifolds

The reconstruction theorem is a cornerstone of the theory of regularity structures [Hai14]. In [CZ20] the authors formulate and prove this result in the language of distributions theory on the Euclidean space $\mathbb{R}^d$, without any reference to the original framework. In this paper we generalize their constructions to the case of distributions over a generic $d$-dimensional smooth manifold $M$, proving the reconstruction theorem in this setting. This is done having in mind the extension of the theory of regularity structures to smooth manifolds.