$\eta$-Ricci solitons and $\eta$-Einstein metrics on weak $\beta$-Kenmotsu $f$-manifolds

Abstract

Recent interest among geometers in $f$-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by V. Rovenski and R. Wolak as a generalization of Hermitian and K\"{a}hler structures, as well as $f$-structures, allow a fresh look at the classical theory. In this paper, we study a new $f$-structure of this kind, called the weak $\beta$-Kenmotsu $f$-structure, as a generalization of K. Kenmotsu's concept. We prove that a weak $\beta$-Kenmotsu $f$-manifold is locally a twisted product of the Euclidean space and a weak K\"{a}hler manifold. Our main results show that such manifolds with $\beta=const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature.