Av-operator on complete foliated Riemannian manifolds

We give a generalization of the result obtained by C. Currás-Bosch. We consider the Av-operator associated to a transverse Killing fieldν on a complete foliated Riemannian manifold (M, ℱ, g). Under a certain assumption, we prove that, for eachx ∈M, (Av) _x belongs to the Lie algebra of the linear holonomy group ψv(x). A special case of our result, the version of the foliation by points, implies the results given by B. Kostant (compact case) and C. Currás-Bosch (non-compact case).