From the Eisenhart problem to Ricci solitons in \(f\)-Kenmotsu manifolds (Q708452)

The authors consider \(f\)-Kenmotsu manifolds, that is, Riemannian manifolds equipped with almost contact metric structures satisfying a special condition, expressed in terms of the Levi-Civita connection \(\nabla\). They show that (1) any parallel symmetric \((0,2)\)-tensor field on such a manifold is a constant multiple of the metric tensor; (2) locally Ricci symmetric (\(\nabla S = 0\), \(S\) being the Ricci tensor) \(f\)-Kenmotsu manifolds are Einstein. They provide conditions under which Ricci solitons on such manifolds are expanding.