We give a generalization of the result obtained by C. Currais-Bosch. We
consider the -operator associated to a transverse Killing field v on a
complete foliated Riemannian manifold . Under a certain assumption,
we prove that, for each belongs to the Lie algebra of the linear
holonomy group . A special case of our result, the version of the foliation
by points, implies the results given by B. Kostant (compact case) and
C. Currfis-Bosch (non-compact case).