Engel structures with trivial characteristic foliations

Abstract

Engel structures on $M\times S^1$ and $M\times I$ are studied in this paper, where $M$ is a $3$–dimensional manifold. We suppose that these structures have characteristic line fields parallel to the fibres, $S^1$ or $I$. It is proved that they are characterized by contact structures on the cross section $M$, the twisting numbers, and Legendrian foliations on both ends $M\times \partial I$ in the case of $M\times I$.