Gradient $$\rho $$-Einstein soliton on almost Kenmotsu manifolds

Abstract

In this paper, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation admits a gradient \(\rho \)-Einstein soliton, then either \(M^{2n+1}\) is Einstein or the potential function is pointwise collinear with the Reeb vector field \(\xi \) on an open set \({\mathcal {O}}\) of \(M^{2n+1}\). Moreover, we prove that if the metric of a \((\kappa ,-2)'\)-almost Kenmotsu manifold with \(h'\ne 0\) admits a gradient \(\rho \)-Einstein soliton, then the manifold is locally isometric to the Riemannian product \({\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n\) and potential vector field is tangential to the Euclidean factor \({\mathbb {R}}^n\). We show that there does not exist gradient \(\rho \)-Einstein soliton on generalized \((\kappa ,\mu )\)-almost Kenmotsu manifold of constant scalar curvature. Finally, we construct an example for gradient \(\rho \)-Einstein soliton.