Parallel vector fields and global injectivity in two dimensions

Francisco Braun and Jaume Llibre

Vol. 333 (2024), No. 1, 1–15

DOI: 10.2140/pjm.2024.333.1

Abstract

Let $U$ be simply connected open subset of ${ℝ}^2$, and let $f:U\to {ℝ}^2$ be a local diffeomorphism. We study the global injectivity of $f$ using the planar vector fields of type annular, radial or strip. Our main result enables the unification of proofs for classical results on global injectivity, such as the Hadamard global invertibility theorem and the condition related to the connectedness of the levels sets of one of the coordinates of $f$.

Keywords

annular vector fields, radial vector field, strip vector field, global injectivity

Mathematical Subject Classification

Primary: 14R15, 34C25

Milestones

Received: 12 March 2024

Revised: 30 October 2024

Accepted: 15 November 2024

Published: 19 December 2024