  Let $F: \mathbb{R}^n\to\mathbb{R}^n$ be a $C^{\infty}$ map such that $DF(x)$
is invertible for every $x\in\mathbb{R}^n$. Although being a local
diffeomorphism, $F$ is not necessarily globally injective if $n\geq2$. Finding
additional assumptions implying the global injectivity of $F$ for $n\geq 2$ is
object of intense study in several areas of Mathematics. In this paper we
revisit some assumptions and relations between them in the bidimensional case
and discuss the natural higher dimensional situation.
