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

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.