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

Weak metric structures, introduced by Rovenski and Wolak in 2022, extend Yano's $f$-structure and almost contact metric structure. In this paper, we investigate curvature phenomena of weak almost $\mathcal{S}$-manifolds (w.a.$\mathcal{S}$-manifolds) focusing on the $f$-$(\kappa ,\mu )$-nullity condition and its special case $R_{X,Y}\xi =0$. We establish several results that generalize known rigidity theorems for almost $\mathcal{S}$-manifolds. First, using the partial Ricci flow, we obtain dynamical characterizations of $\mathcal{S}$-manifolds: starting from a w.a.$\mathcal{S}$-structure satisfying the curvature condition of $\mathcal{S}$-manifolds or the $f$-$(1,\mu )$-nullity condition, the flow evolves the structure exponentially fast toward an $\mathcal{S}$-structure. This extends results of Cappelletti Montano and Di Terlizzi to the weak metric setting. Next, we identify conditions under which a w.a.$\mathcal{S}$-manifold admits a bi-Legendrian structure with totally geodesic foliations. Finally, for w.a.$\mathcal{S}$-manifolds with $\kappa =\mu =0$, we prove a splitting theorem in which one factor is flat, generalizing classical results for almost $\mathcal{S}$-geometry. These findings have consequences for the theory of Sasakian and $\mathcal{S}$- manifolds, the geometry of bi-Legendrian structures, and the behavior of weak metric contact manifolds under curvature constraints.