Contact twistor spaces and almost contact metric structures

The notions of a twistor space of a contact manifold and a contact connection on such a manifold have been introduced by L. Vezzoni as extensions of the corresponding notions in the case of a symplectic manifold. Given a contact connection on a contact manifold one can define an almost $CR$-structure on its twistor space and Vezzoni has found the integrability condition for this structure. In the present paper it is observed that the $CR$-structure is induced by an almost contact metric structure. The main goal of the paper is to obtain necessary and sufficient conditions for normality of this structure in terms of the curvature of the given contact connection. Illustrating examples are discussed at the end of the paper.