From the Eisenhart Problem to Ricci Solitons in f-Kenmotsu Manifolds

The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular f-Kenmotsu framework. In 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.