Einstein-Type Metrics on Almost Kenmotsu Manifolds

Abstract

This article aims to classify the Einstein-type metrics on Kenmotsu and almost Kenmotsu manifolds. In Kenmotsu case, we find that it is T-Einstein. Also, if the manifold is complete and the scalar curvature remains invariant along the Reeb vector field, then either, it is isometric to the hyperbolic space $\mathbb {H}^{2n+1}(1)$ or, the warped product $\widetilde{M}\times _\gamma \mathbb {R}$, provided $\zeta \psi \ne \psi$. Next, we investigate non-Kenmotsu $(\kappa ,\mu )'$-almost Kenmotsu manifolds obeying the Einstein-type metrics and give some classification. Finally, we establish that if $(\psi ,g)$ is a non-trivial solution of Einstein-type metrics with smooth function $\psi$ which is constant along the Reeb vector field on almost Kenmotsu 3-H-manifold, then either, it is locally isometric to the hyperbolic space $\mathbb {H}^3(1)$ or, the Riemannian product $\mathbb {H}^2(4)\times \mathbb {R}$. Finally, we construct several non-trivial examples to verify our main results.