  Weak contact metric 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 a new look at the theory of contact
manifolds. In this paper we study the new structure of this type, called the
weak nearly Sasakian structure. We find conditions that are automatically
satisfied by almost contact manifolds and under which the contact distribution
is curvature invariant and the weak nearly Sasakian structure foliates into two
types of totally geodesic foliations. Our main result generalizes the theorem
by B. Cappelletti-Montano and G. Dileo (2016) about the foliated structure of
nearly Sasakian manifolds.
