Normality of the twistor space of a $5$-manifold with a $SO(3)$-structure

TL;DR

This paper investigates the conditions under which certain geometric structures called $CR$-structures on a twistor space of a 5-manifold with an $SO(3)$-structure are normal, providing necessary and sufficient criteria and examples.

Contribution

It establishes a geometric criterion for the normality of $CR$-structures induced by almost contact metric structures on the twistor space of such manifolds.

Findings

Necessary and sufficient condition for normality of $CR$-structures

Examples illustrating the normality condition

Connection between $SO(3)$-structures and $CR$-structures

Abstract

A manifold with an irreducible $SO(3)$-structure is a $5$-manifold $M$ whose structure group can be reduced to the group $SO(3)$, non-standardly imbedded in $SO(5)$. The study of such manifolds has been initiated by M. Bobie\'nski and P. Nurowski who, in particular, have shown that one can define four $CR$-structures on a twistor-like $7$-dimensional space associated to $M$. In the present paper it is observed that these $CR$-structures are induced by almost contact metric structures. The purpose of the paper is to study the problem of normality of these structures. The main result gives necessary and sufficient condition for normality in geometric terms of the base manifold $M$. Examples illustrating this result are presented at the end of the paper.