Foliations, solvability and global injectivity

TL;DR

This paper investigates conditions under which smooth, locally invertible maps from ^n to ^n are globally injective, focusing on the two-dimensional case and extending considerations to higher dimensions.

Contribution

It revisits assumptions ensuring global injectivity of smooth maps with invertible derivatives, emphasizing the two-dimensional case and exploring higher-dimensional scenarios.

Findings

Analysis of foliation structures related to injectivity

Discussion of conditions for global injectivity in 2D

Extension considerations to higher dimensions

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 $n\geq2$. 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.