Flexibility for tangent and transverse immersions in Engel manifolds

Abstract

We study the space of immersions of S1 that are tangent to an Engel structure D. We show that the full h-principle holds as soon as one excludes the closed orbits of W, the characteristic foliation of D. This is sharp: we elaborate on work of Bryant and Hsu to show that curves tangent to W sometimes form additional isolated components that cannot be detected at a formal level. We then show that this is an exceptional phenomenon: if D is C∞-generic, curves tangent to W are not isolated anymore. These results, in conjunction with an argument due to M. Gromov, prove that a full h-principle holds for immersions transverse to the Engel structure.