Holonomy pseudogroups as obstructions to equivalence of manifolds over the algebra of dual numbers

A smooth manifold over the algebra of dual numbers $\mathbb{D}$ (a $\mathbb{D}$-smooth manifold) carries the canonical foliation whose leaves are affine manifolds. Extension of charts on a $\mathbb{D}$-smooth manifold along leaf paths allows ones to associate with an immersed transversal of the canonical foliation a pseudogroup of local $\mathbb{D}$-diffeomorphisms called the holonomy pseudogroup. In the present paper, holonomy pseudogroups are applied to the study of $\mathbb{D}$-diffeomorphisms between quotient manifolds of the algebra $\mathbb{D}$ by lattices. In particular, it is shown that a $\mathbb{D}$-diffeomorphism between two such manifolds exists if and only if one of the lattices is obtained from the other by the multiplication by a dual number. In addition, it is shown that some $\mathbb{D}$-smooth manifolds naturally associated with an affine manifold are $\mathbb{D}$-diffeomorphic if and only if this manifold is radiant.