Weak Nearly $\mathcal S$- and Weak Nearly $\mathcal C$- Manifolds

The recent interest of geometers in the $f$-structures of K. Yano is motivated by the study of the dynamics of contact foliations, as well as their applications in theoretical physics. Weak metric $f$-structures on a smooth manifold, recently introduced by the author and R. Wolak, open a new perspective on the theory of classical structures. In the paper, we define structures of this kind, called weak nearly ${\cal S}$- and weak nearly ${\cal C}$- structures, study their geometry, e.g. their relations to Killing vector fields, and characterize weak nearly ${\cal S}$- and weak nearly ${\cal S}$- submanifolds in a weak nearly K\"{a}hler manifold.