Contact twistor spaces and almost contact metric structures

TL;DR

This paper explores the relationship between contact twistor spaces and almost contact metric structures, establishing conditions for their normality based on curvature, and providing illustrative examples.

Contribution

It demonstrates that the CR-structure on the twistor space is induced by an almost contact metric structure and derives curvature-based criteria for its normality.

Findings

CR-structure is induced by an almost contact metric structure

Normality conditions are characterized by curvature criteria

Examples illustrate the theoretical results

Abstract

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.