$η$-Ricci solitons and $η$-Einstein metrics on weak $β$-Kenmotsu $f$-manifolds
Vladimir Rovenski
TL;DR
This work extends η-Ricci soliton and η-Einstein geometry to weak $\beta f$-Kenmotsu ($\beta f$-KM) manifolds, showing that these structures are locally twisted products $\mathbb{R}^s \times_\sigma \bar{M}$ with $\bar{M}$ weak Kähler. For constant $\beta$, the authors derive explicit curvature formulas, establish that $\mathcal{D}^\perp$ induces flat leaves and that the horizontal distribution is totally umbilical with mean curvature $-\beta\bar{\xi}$, and prove that $\nabla_{\bar{\xi}}\operatorname{Ric}^\sharp=0$ implies the manifold is $\eta$-Einstein with constant scalar curvature $r=-2sn(2n+1)\beta^2$. In the setting of $\eta$-Ricci solitons, they show that the soliton relations force $\lambda+\mu=-2n\beta^2$ and, when the potential field is either a contact vector field or collinear to $\bar{\xi}$, yield explicit $\eta$-Einstein structures with constant $r$, extending known $s=1$ results. Overall, the paper connects soliton theory with twisted/product geometry in weak $f$-structures and provides concrete rigidity results under η-Ricci soliton scenarios.
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 Kähler 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 $β$-Kenmotsu $f$-structure, as a generalization of K. Kenmotsu's concept. We prove that a weak $β$-Kenmotsu $f$-manifold is locally a twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $β=const$ and equipped with an $η$-Ricci soliton structure whose potential vector field satisfies certain conditions are $η$-Einstein manifolds of constant scalar curvature.