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

In the present paper, we introduce the notion of ∗ -gradient ρ -Einstein soliton on a class of almost Kenmotsu manifolds. It is shown that if a ( 2 n + 1 ) -dimensional ( k , μ ) ′ -almost Kenmotsu manifold M admits ∗ -gradient ρ -Einstein soliton with Einstein potential f , then (1) the manifold M is locally isometric to H n + 1 ( - 4 ) × 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.