$\ast$-$η$-Ricci solitons on weak Kenmotsu $f$-manifolds
Vladimir Rovenski
TL;DR
The paper develops the ∗-Ricci tensor for weak metric f-manifolds and studies how ∗-η-Ricci solitons interact with weak βf-Kenmotsu geometry. By formulating ∗-η-RS and deriving explicit curvature relations, the authors obtain η-Einstein characterizations under several natural conditions, including when the soliton potential is a contact vector field, when the vector field lies in ker f, or in gradient almost soliton settings. The results generalize known theorems in related geometries and provide new Weinstein-type Einstein-type metrics on a broad class of higher-dimensional, non-Einstein manifolds. These findings broaden the scope of Ricci-type soliton theory to weak f-structures and suggest further exploration of almost gradient cases and other weak metric f-manifolds with potential applications to contact foliations and geometric flows.
Abstract
Recent interest among geometers in $f$-structures of K. Yano is due to the study of topology and dynamics of contact foliations and generalized A. Weinstein conjectures. Weak metric $f$-structures, introduced by the author and R. Wolak as a generalization of Hermitian structure, as well as $f$-structure allow for a fresh perspective on the classical theory. An important case of such manifolds, which is locally a twisted product, is a weak $βf$-Kenmotsu manifold defined as a generalization of K. Kenmotsu's concept. In this paper, the concept of the $\ast$-Ricci tensor of S. Tashibana is adapted to weak metric $f$-manifolds, the interaction of $\ast$-$η$-Ricci soliton with the weak $βf$-Kenmotsu structure is studied and new characteristics of $η$-Einstein metrics are obtained.