The Critical Point Equation And Contact Geometry

In this paper, we consider the CPE conjecture in the frame-work of $K$-contact and $(\kappa, \mu)$-contact manifolds. First, we prove that if a complete $K$-contact metric satisfies the CPE is Einstein and is isometric to a unit sphere $S^{2n+1}$. Next, we prove that if a non-Sasakian $(\kappa, \mu)$-contact metric satisfies the CPE, then $M^{3}$ is flat and for $n > 1$, $M^{2n+1}$ is locally isometric to $E^{n+1}\times S^{n}(4)$.