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We carry on a systematic study of nearly Sasakian manifolds. We prove thatany nearly Sasakian manifold admits two types of integrable distributions withtotally geodesic leaves which are, respectively, Sasakian or $5$-dimensionalnearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is acontact manifold. Focusing on the $5$-dimensional case, we prove that thereexists a one-to-one correspondence between nearly Sasakian structures and aspecial class of nearly hypo $SU(2)$-structures. By deforming such a$SU(2)$-structure one obtains in fact a Sasaki-Einstein structure. Further weprove that both nearly Sasakian and Sasaki-Einstein $5$-manifolds are endowedwith supplementary nearly cosymplectic structures. We show that there is aone-to-one correspondence between nearly cosymplectic structures and a specialclass of hypo $SU(2)$-structures which is again strictly related toSasaki-Einstein structures. Furthermore, we study the orientable hypersurfacesof a nearly K\"{a}hler 6-manifold and, in the last part of the paper, we definecanonical connections for nearly Sasakian manifolds, which play a role similarto the Gray connection in the context of nearly K\"{a}hler geometry. Indimension $5$ we determine a connection which parallelizes all the nearlySasakian $SU(2)$-structure as well as the torsion tensor field. An analogousresult holds also for Sasaki-Einstein structures.

Nearly Sasakian geometry and $SU(2)$-structures