Flexibility for tangent and transverse immersions in Engel manifolds

TL;DR

This paper investigates the flexibility of tangent and transverse immersions in Engel manifolds, establishing an h-principle for most cases and highlighting exceptional behaviors related to the kernel of the structure.

Contribution

It demonstrates the existence of a full h-principle for tangent immersions excluding closed orbits and shows generic conditions eliminate isolated tangent components, also establishing the principle for transverse immersions.

Findings

Full h-principle for tangent immersions excluding closed orbits

Generic conditions remove isolated tangent components

Full h-principle for transverse immersions

Abstract

In this article we study immersions of the circle that are tangent to an Engel structure $\mathcal{D}$. We show that a full $h$-principle does exist as soon as one excludes the closed orbits of $\mathcal{W}$, the kernel of $\mathcal{D}$. This is sharp: we elaborate on work of Bryant and Hsu to show that curves tangent to $\mathcal{W}$ often conform additional isolated components that cannot be detected at a formal level. We then show that this is an exceptional phenomenon: if $\mathcal{D}$ is generic, curves tangent to $\mathcal{W}$ are not isolated anymore. We then go on to show that a full $h$-principle holds for immersions transverse to the Engel structure.