Nearly Sasakian manifolds are a special class of almost contact metric manifolds, providing a natural odd dimensional counterpart of nearly Kähler manifolds. In the present talk I will describe the main properties of nearly Sasakian manifolds, referring also to geometric structures which are strictly related to nearly Sasakian structures (Sasakian, coKähler, nearly cosymplectic, Kähler and nearly Kähler structures). One of the main features of nearly Sasakian manifolds is the existence of some canonical foliations, showing that the 5-dimensional case is particularly interesting. Focusing on this case, I will provide an equivalent notion of nearly Sasakian structures in terms of SU(2)-structures. In the last part of the talk I will introduce a class of canonical connections for nearly Sasakian manifolds, which play a role similar to the Gray connection in the context of nearly Kähler geometry. The results are obtained jointly with B. Cappelletti-Montano.
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