On a monodromy theorem for sheaves of local fields and applications

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.