The structure of some classes of K-contact manifolds

Abstract

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ -projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and ϕ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, ϕ-projectively flat and locally isometric to the unit sphere S 2n+1 (1) are equivalent. Finally, we prove that a compact ϕ-projectively flat K-contact manifold with regular contact vector field is a principal S 1 -bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.