Nearly Sasakian geometry and SU(2)-structures

We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian and 5-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the 5-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo SU(2)-structures. By deforming such an SU(2)-structure, one obtains in fact a Sasaki–Einstein structure. Further we prove that both nearly Sasakian and Sasaki–Einstein 5-manifolds are endowed with supplementary nearly cosymplectic structures. We show that there is a one-to-one correspondence between nearly cosymplectic structures and a special class of hypo SU(2)-structures which is again strictly related to Sasaki–Einstein structures. Furthermore, we study the orientable hypersurfaces of a nearly Kähler 6-manifold, and in the last part of the paper, we define canonical connections for nearly Sasakian manifolds, which play a role similar to the Gray connection in the context of nearly Kähler geometry. In dimension 5, we determine a connection which parallelizes all the nearly Sasakian SU(2)-structure as well as the torsion tensor field. An analogous result holds also for Sasaki–Einstein structures.