Reconstruction theorem for germs of distributions on smooth manifolds

The reconstruction theorem is a cornerstone of the theory of regularity structures [10]. In [4] the authors formulate and prove this result in the language of distributions theory on the Euclidean space Rd, 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.