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

Let $\mathcal{E}^3 \subset TM^n$ be a smooth $3$-distribution on a smooth $n$-manifold, and $W \subset \mathcal{E}$ a line field such that $[\mathcal{W}, \mathcal{E}] \subset \mathcal{E}$.We give a condition for the existence of a plane field $\mathcal{D}^2$ such that $\mathcal{W} \subset\mathcal{D}$ and $[\mathcal{D},\mathcal{D}] = \mathcal{E}$ near a closed orbit of $\mathcal{W}$. If $\mathcal{W}$ has a non-singular Morse–Smale section, we get a condition for the global existence of $\mathcal{D}$. As a corollary we obtain conditions for a non-singular vector field $\mathcal{W}$ on a $3$-manifold to be Legendrian, and for an even contact structure $\mathcal{E} \subset TM^4$ to be induced by an Engel structure $\mathcal{D}$.

Full Text (PDF format)

The author is supported by the DAAD program Research Grants for Doctoral Candidates and Young Academics and Scientists (more than 6 months) No. 57381410, 2018/19.

Received 12 June 2020

Accepted 13 August 2020

Published 27 May 2021