From the Eisenhart problem to Ricci solitons in $f$-Kenmotsu manifolds

TL;DR

This paper solves the Eisenhart problem for symmetric tensors in $f$-Kenmotsu manifolds, linking it to Ricci solitons and Einstein manifolds, and introduces new classes of Einstein-Kenmotsu manifolds.

Contribution

It provides the first solution to the Eisenhart problem in the $f$-Kenmotsu setting and connects it to Ricci solitons and Einstein manifolds.

Findings

Recovered the Olszack-Rosca Einstein manifold example within $f$-Kenmotsu manifolds.

Identified cases of Killing vector fields in this framework.

Presented new classes of Einstein-Kenmotsu manifolds.

Abstract

The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular $f$-Kenmotsu framework. On this way, the Olszack-Rosca example of Einstein manifolds provided by $f$-Kenmotsu manifolds via locally symmetric Ricci tensors is recovered as well as a case of Killing vector fields. Some other classes of Einstein-Kenmotsu manifolds are presented. Our result is interpreted in terms of Ricci solitons and special quadratic first integrals.