Let $Y=(f,g,h):\mathbb{ R}^{ 3} \to \mathbb{ R}^{ 3}$ be a $C^{ 2}$ map and let $\spec(Y)$ denote the set of eigenvalues of the derivative $DY_p$, when $p$ varies in $\mathbb{ R}^3$. We begin proving that if, for some $\epsilon>0,$ $\spec(Y)\cap (-\epsilon,\epsilon)=\emptyset,$ then the foliation $\mathcal{ F}(k),$ with $k\in \{ f,g,h\},$ made up by the level surfaces $\{ k={ \rm constant}\},$ consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of $\mathbb{ R}^n.$
Categories: math.DS