Holonomy pseudogroup of a manifold over the algebra of dual numbers and some its applications

We study properties of the holonomy pseudogroup on a total immersed transversal of the canonical foliation on a smooth manifold over the algebra of dual numbers $\mathbb{D}$. We apply holonomy pseudogroups to the investigation of $\mathbb{D}$-diffeomorphisms between quotient manifolds of the algebra $\mathbb{D}$ by lattices and between $\mathbb{D}$-smooth manifolds naturally associated with an affine manifold.