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

Abstract

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 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.

Keywords

• affine manifold
• manifold over algebra of dual numbers
• foliation
• foliated bundle
• tangent bundle
• tangent manifold
• torus over the algebra of dual numbers
• weil bundle