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

TL;DR

This paper introduces weak $eta$-Kenmotsu $f$-structures as a generalization of classical structures, proving that certain manifolds with these structures and $eta$-constant are $ ext{ exteta}$-Einstein with constant scalar curvature.

Contribution

It defines weak $eta$-Kenmotsu $f$-structures and demonstrates their geometric properties, including local twisted product structure and conditions for $ ext{ exteta}$-Einstein metrics.

Findings

Weak $eta$-Kenmotsu $f$-manifolds are locally twisted products of Euclidean space and weak Kähler manifolds.

Manifolds with $eta=const$ and $ ext{ exteta}$-Ricci soliton structure are $ ext{ exteta}$-Einstein.

Such manifolds have constant scalar curvature.

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.