CR Submanifolds of the Nearly Kähler $\mathbb {S}^3\times \mathbb {S}^3$ Characterised by Properties of the Almost Product Structure

Abstract

In a previous paper (Antić et al., three-dimensional CR submanifolds of the nearly Kähler $\mathbb {S}^{3}\times \mathbb {S}^{3}$, 2017), the authors together with L. Vrancken initiated the study of 3-dimensional CR submanifolds of the nearly Kähler homogeneous $\mathbb {S}^3\times \mathbb {S}^3$. As is shown by Butruille, this is one of only four homogeneous 6-dimensional nearly Kähler manifolds. Besides its almost complex structure J, it also admits a canonical almost product structure P, see (Moruz and Vrancken, Publ Inst Math 2018) and (Bolton et al., Tôhoku Math J 67:1–17, 2015). Along a proper 3-dimensional CR submanifold, the tangent space of $\mathbb {S}^3\times \mathbb {S}^3$ can be naturally split as the orthogonal sum of three 2-dimensional vector bundles $\mathcal {D}_1$, $\mathcal {D}_2$ and $\mathcal {D}_3$. We study the CR submanifolds in relation with the behavior of the almost product structure on these vector bundles.