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 ${\cal S}$-manifolds (w.a.$\cal S$-manifolds) with additional conditions trivial for almost ${\cal S}$-manifolds, and with the $f$-$(\kappa,\mu)$-nullity condition, and prove theorems that generalize known results.

Using the partial Ricci flow, we characterize $\cal S$-manifolds as limits of w.a.$\cal S$-manifolds satisfying the key properties of $\cal S$-manifolds, or the $f$-$(1,\mu)$-nullity condition, which agrees with results of Cappelletti Montano and Di Terlizzi.

For w.a.$\cal S$-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.