Germs of Engel structures along 3-manifolds (Q1766327)

A tangent distribution of rank 2, maximally non-integrable, on a 4-manifold is called an Engel structure. It has the property that all Engel structures are locally equivalent. This is characteristic of only line fields, contact structures in addition to Engel structures. Contact structures possess a global stability with respect to deformations while Engel structures do not have this property. If an Engel structure is deformed, the Engel line field is not always diffeomorphic to the original one. Thus, a homotopy of Engel structures is not always represented by isotopy. In this work a sufficient condition determining the existence of the germ of an Engel structure along a hypersurface(actually, the 3-manifold) is obtained. It is given by the singular line field on the 3-manifold traced by the Engel structure.