On a monodromy theorem for sheaves of local fields and applications

TL;DR

This paper establishes a monodromy theorem for sheaves of local fields with unique continuation, enabling global extension of local fields on simply connected manifolds, generalizing previous results in geometric structures.

Contribution

It introduces a monodromy theorem for sheaves of local vector fields satisfying unique continuation, extending prior work to broader geometric contexts.

Findings

Global extension of local fields on simply connected manifolds

Applicability to Finsler and pseudo-Finsler Killing fields

Generalization of classical results in geometric structures

Abstract

We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation property. In particular, in the case of admissible regular sheaves of local fields defined on a simply connected manifold, we obtain a global extension result for every local field of the sheaf. This generalizes previous works of Nomizu for semi-Riemannian Killing fields, of Ledger--Obata for conformal fields, and of Amores for fields preserving a $G$-structure of finite type. The result applies to Finsler or pseudo-Finsler Killing fields and, more generally, to affine fields of a spray. Some applications are discussed.