Foliations, solvability and global injectivity

Abstract

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 $n\ge 2$. Finding additional assumptions implying the global injectivity of $F$ for $n\ge 2$ 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.

References in corpus (1)

• On polynomial submersions of degree $4$ and the real Jacobian conjecture in ${\mathbb{R}}^2$

Cited by in corpus (1)

• Surjectivity of linear operators and semialgebraic global diffeomorphisms