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

Rovenski and Wolak introduced weak metric structures on a differentiable manifold that generalize the Yano $f$-structure and almost contact metric structure, and offer a new perspective on the traditional theory and its applications. In this paper, we study curvature-related properties of weak almost $\mathcal{S}$-manifolds (w.a.$\mathcal{S}$-manifolds) with additional conditions trivial for almost $\mathcal{S}$-manifolds, and with the $f$-$(\kappa ,\mu )$-nullity condition, and prove theorems that generalize known results. Using the partial Ricci flow, we characterize $\mathcal{S}$-manifolds as limits of w.a.$\mathcal{S}$-manifolds satisfying the key properties of $\mathcal{S}$-manifolds, or the $f$-$(1,\mu )$-nullity condition, which agrees with results of Cappelletti Montano and Di Terlizzi. For w.a.$\mathcal{S}$-manifolds with $\kappa =\mu =0$, we prove a splitting theorem in which one of the factors is flat. Our main results consequences on the dynamical characteristics of Sasakian manifolds and the splitting of weak metric contact manifolds.