Half-Reeb components, Palais-Smale condition and global injectivity of local diffeomorphisms in $\mathbb{R}^3$

Abstract

Let $F=(F_1,F_2,F_3)\colon\mathbb{R}^3\to\mathbb{R}^3$ be a $C^\infty$ local diffeomorphism. We prove that each of the following conditions are sufficient to the global injectivity of $F$:

A) The foliations $\mathcal{F}_{F_i}$ made up by the connected components of the level surfaces $F_i=\text{constant}$, consist of leaves without half-Reeb components induced by $F_j$, $j\in\{1,2,3\}\setminus\{i\}$, for $i\in\{1,2,3\}$.

B) For each $i\neq j\in\{1,2,3\}$, $F_i\rvert_L\colon L\to\mathbb{R}$ satisfy the Palais--Smale condition, for all $L\in\mathcal{F}_{F_j}$.

We also prove that B) implies A) and give examples to show that the converse is not true. Further, we give examples showing that none of these conditions is necessary to the global injectivity of F.