Foliated structure of weak nearly Sasakian manifolds

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of almost contact metric manifolds. In this paper we study the new structure of this type, called the weak nearly Sasakian structure. We find conditions that are satisfied by almost contact manifolds and under which the contact distribution is curvature invariant and weak nearly Sasakian manifolds admit two types of totally geodesic foliations. Our main result generalizes the theorem by Cappelletti-Montano and Dileo (Ann Matem Pura Appl 195:897-922, 2016) to the context of weak almost contact geometry.