The geometry of 3-quasi-Sasakian manifolds

3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which reveals to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that they are multiply foliated by four distinct fundamental foliations. The study of the transversal geometries with respect to these foliations allows us to link the 3-quasi-Sasakian manifolds to the more famous hyper-Kaehler and quaternionic-Kaehler geometries. Furthermore, we strongly improve the splitting results previously found; we prove that any 3-quasi-Sasakian manifold of rank 4l+1 is 3-cosymplectic and any 3-quasi-Sasakian manifold of maximal rank is 3-alpha-Sasakian.