Metric regularity, pseudo-Jacobians and global inversion theorems on Finsler manifolds

TL;DR

This paper investigates the conditions under which locally Lipschitz maps between Finsler manifolds are globally invertible, introducing a pseudo-Jacobian concept and connecting metric regularity with covering properties.

Contribution

It introduces a pseudo-Jacobian framework for Finsler manifolds and establishes new global invertibility criteria based on metric regularity and covering properties.

Findings

Established a pseudo-Jacobian notion for Finsler manifolds

Derived conditions for global invertibility of Lipschitz maps

Extended Hadamard integral condition to Finsler manifold setting

Abstract

Our aim in this paper is to study the global invertibility of a locally Lipschitz map $f:X \to Y$ between (possibly infinite-dimensional) Finsler manifolds, stressing the connections with covering properties and metric regularity of $f$. To this end, we introduce a natural notion of pseudo-Jacobian $Jf$ in this setting, as is a kind of set-valued differential object associated to $f$. By means of a suitable index, we study the relations between properties of pseudo-Jacobian $Jf$ and local metric properties of the map $f$, which lead to conditions for $f$ to be a covering map, and for $f$ to be globally invertible. In particular, we obtain a version of Hadamard integral condition in this context.