Horizontal gradient of polynomial functions for the standard Engel structure on ${\mathbb{R}}^4$

Abstract

We investigate the set $V_f$ of horizontal critical points of a polynomial function $f$ for the standard Engel structure defined by the 1-forms ${\omega }_3=d{x}_3-x_{1}d{x}_2,$ ${\omega }_4=d{x}_4-x_{3}d{x}_2$, endowed with the sub-Riemannian metric $g_{SR}=d{x}_1^2+d{x}_2^2$. For a generic polynomial, we show that the intersection of any fiber of $f$ and $V_f$ does not contain a horizontal curve. Then we prove that each trajectory of the horizontal gradient of $f$ approaching the set $V_f$ has a limit.

arXiv comment19 pages