Characterizations of weak almost ${\mathcal S}$-manifolds with curvature properties

The interest of geometers in $f$-structures is motivated by the study of the dynamics of contact foliations, as well as their applications in physics. A weak $f$-structure on a smooth manifold, introduced by V. Rovenski and R. Wolak (2022), generalizes K. Yano's (1961) $f$-structure. This generalization allows us to revisit classical theory and discover new applications related to Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. In this paper, we investigate some fundamental curvature properties of weak almost $\mathcal{S}$-manifolds and examine those satisfying the condition ``the curvature tensor in the directions of the Reeb vector fields is zero", as well as its generalization, the $(\kappa, \mu)$-nullity condition. We~find when a weak almost ${\mathcal S}$-manifold satisfying this curvature tensor condition admits two complementary orthogo\-nal foliations, both of which are totally geodesic, with one being flat (in the (2+s)-dimensional case, the manifold is flat). We also characterize weak almost ${\mathcal S}$-manifolds, which in the case of the $f$-$(1,\mu)$-nullity condition become ${\cal S}$-manifolds; this agrees with the results of B. Cappelletti Montano and L. Di Terlizzi (2007) on $f$-manifolds.