Flexibility for tangent and transverse immersions in Engel manifolds

Abstract

We study the space of immersions of ${\mathbb {S}}^1$ that are tangent to an Engel structure ${\mathcal {D}}$. We show that the full h-principle holds as soon as one excludes the closed orbits of ${\mathcal {W}}$, the characteristic foliation of ${\mathcal {D}}$. This is sharp: we elaborate on work of Bryant and Hsu to show that curves tangent to ${\mathcal {W}}$ sometimes form additional isolated components that cannot be detected at a formal level. We then show that this is an exceptional phenomenon: if ${\mathcal {D}}$ is $C^\infty$-generic, curves tangent to ${\mathcal {W}}$ 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.