Let be a contact metric manifold with a Killing structure vector field (called a -contact manifold) and its Weyl conformal tensor. Then , decomposes into , where is a 1-dimensional linear subspace of generated by . It is natural to study the following particular cases:
It was shown by the last and first author that in case (i) is locally isometric to the unit sphere; in case (ii) is an -Einstein Sasakian manifold. This paper shows that in case (iii) if is compact and (i.e., is -conformally flat), then is a principal -bundle over an almost Kähler space of constant holomorphic sectional curvature.