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 (2022), allowed a new look at the theory of contact manifolds. This paper studies the curvature and topology of new structures of this type, called the weak nearly cosymplectic structure and weak nearly K\"{a}hler structure. We found conditions under which weak nearly cosymplectic manifolds become Riemannian products of two kinds, and characterized 5-dimensional weak nearly cosymplectic manifolds. Our theorems generalize results by H. Endo (2005) and A. Nicola--G. Dileo--I. Yudin (2018) on curvature and splitting of nearly cosymplectic manifolds.PDF: On the splitting of weak nearly cosymplectic manifolds.pdf
