The Critical Point Equation And Contact Geometry

Abstract

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 $E^{n+1}\times S^n(4)$.

In the published version there was a sign error in Eq. (1.3). We have fixed it here