Holonomy Pseudogroups as Obstructions to Equivalence of Manifolds over the Algebra of Dual Numbers

A smooth manifold over the algebra of dual numbers D (a D -smooth manifold) carries the canonical foliation whose leaves are affine manifolds. Extension of charts on a D -smooth manifold along leaf paths allows ones to associate with an immersed transversal of the canonical foliation a pseudogroup of local D -diffeomorphisms called the holonomy pseudogroup. In the present paper, holonomy pseudogroups are applied to the study of D -diffeomorphisms between quotient manifolds of the algebra D by lattices. In particular, it is shown that a 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 D -smooth manifolds naturally associated with an affine manifold are D -diffeomorphic if and only if this manifold is radiant.