# On the dynamics of some vector fields tangent to non-integrable plane fields

## TL;DR

This paper investigates the conditions under which certain 2-plane fields exist within 3-distributions on manifolds, with implications for contact and Engel structures, using geometric and topological methods.

## Contribution

It provides necessary and sufficient conditions for the existence of specific plane fields related to non-integrable distributions, extending understanding of contact and Engel structures.

## Key findings

- Derived necessary conditions for plane field existence.
- Established sufficient conditions under Morse-Smale assumptions.
- Connected the existence of vector fields to Legendrian and Engel structures.

## Abstract

Let $\mathcal{E}^3\subset TM^n$ be a smooth $3$-distribution on a smooth manifold of dimension $n$ and $\mathcal{W}\subset\mathcal{E}$ a line field such that $[\mathcal{W},\mathcal{E}]\subset\mathcal{E}$. Under some orientability hypothesis, we give a necessary condition for the existence of a plane field $\mathcal{D}^2$ such that $\mathcal{W}\subset\mathcal{D}$ and $[\mathcal{D},\mathcal{D}]=\mathcal{E}$. Moreover we study the case where a section of $\mathcal{W}$ is non-singular Morse-Smale and we get a sufficient condition for the global existence of $\mathcal{D}$. As a corollary we get conditions for a non-singular vector field $W$ on a $3$-manifold to be Legendrian for a contact structure $\mathcal{D}$. Similarly with these techniques we can study when an even contact structure $\mathcal{E}\subset TM^4$ is induced by an Engel structure $\mathcal{D}$.