Critical point equation within the framework of various contact metric manifolds

The aim of the present paper is to study the critical point equation (shortly CPE) conjecture within the framework of various contact metric manifolds. First we establish that Kenmotsu manifold satisfying the CPE either becomes an Einstein manifold or the derivative of potential function along characteristic vector field satisfy a certain relation on the distribution of η. Next we study CPE on (κ, μ)'-almost Kenmotsu manifold and obtain that the manifold is Einstein. Later in case of 3-dimensional trans-Sasakian manifold, we get that either the manifold becomes α-Sasakian or it becomes Einstein. Finally we give examples of 3-dimensional trans-Sasakian manifold and (κ ,μ)'-almost Kenmotsu manifold to verify our outcomes.