On a class of almost Kenmotsu manifolds admitting an Einstein like structure

Abstract

In the present paper, we introduce the notion of $$*$$ -gradient $$\rho$$ -Einstein soliton on a class of almost Kenmotsu manifolds. It is shown that if a $$(2n+1)$$ -dimensional $$(k,\mu )'$$ -almost Kenmotsu manifold M admits $$*$$ -gradient $$\rho$$ -Einstein soliton with Einstein potential f, then (1) the manifold M is locally isometric to $$\mathbb {H}^{n+1}(-4)$$ $$\times$$ $$\mathbb {R}^n$$ , (2) the manifold M is $$*$$ -Ricci flat and (3) the Einstein potential f is harmonic or satisfies a physical Poisson’s equation. Finally, an illustrative example is presented.