Principal torus bundles of Lorentzian S-manifolds and the {\phi}-null   Osserman condition
  The main result we give in this brief note relates, under suitable hypotheses, the {\phi}-null Osserman, the null Osserman and the classical Osserman conditions to each other, via semi-Riemannian submersions as projection maps of principal torus bundles arising from a Lorentzian S-manifold. 
Letizia Brunetti and Angelo V. Caldarella
math.DG