Canonical Cartan connection for $4$-dimensional CR-manifolds belonging to general class $\mathsf{II}$

Abstract

We study the equivalence problem for $4$-dimensional CR-manifolds of CR-dimension $1$ and codimension $2$ which are referred to as Engel CR-manifolds. We construct a canonical Cartan connection on such CR-manifolds through Cartan equivalence's method. In particular, we give the explicit expression of $4$ biholomorphic invariants, the annulation of which is a necessary and sufficient condition for an Engel manifold to be locally biholomorphic to Beloshapka's cubic in ${\mathbb{C}}^3$.

17 pages. arXiv admin note: substantial text overlap with arXiv:1405.1342