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

Abstract

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 ${\cal S}$-manifolds (w.a.$\,\cal S$-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 ${\cal S}$-manifolds. First, using the partial Ricci flow, we obtain dynamical characterizations of $\cal S$-manifolds: starting from a w.a.$\,\cal S$-structure satisfying the curvature condition of $\cal S$-manifolds or the $f$-$(1,\mu)$-nullity condition, the flow evolves the structure exponentially fast toward an $\cal S$-structure. This extends results of Cappelletti Montano and Di Terlizzi to the weak metric setting. Next, we identify conditions under which a w.a.$\,{\cal S}$-manifold admits a bi-Legendrian structure with totally geodesic foliations. Finally, for w.a.$\,\cal S$-manifolds with $\kappa=\mu=0$, we prove a splitting theorem in which one factor is flat, generalizing classical results for almost $\cal S$-geometry. These findings have consequences for the theory of Sasakian and $\cal S$- manifolds, the geometry of bi-Legendrian structures, and the behavior of weak metric contact manifolds under curvature constraints.