Johann Davidov

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. Bobieński 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.

Every Walker [Formula: see text]-manifold [Formula: see text], endowed with a canonical neutral metric [Formula: see text], admits a specific almost complex structure called proper. In this paper, we find the conditions under which a proper almost complex structure is a harmonic section or a harmonic map from [Formula: see text] to its hyperbolic twistor space.