CR submanifolds of the nearly K\" ahler ${\mathbb{S}}^3\times {\mathbb{S}}^3$ characterised by properties of the almost product structure

In a previous paper, the authors together with L. Vrancken initiated the study of $3$-dimensional CR submanifolds of the nearly K" ahler homogeneous ${\mathbb{S}}^3\times {\mathbb{S}}^3$. As is shown by Butruille this is one of only four homogeneous $6$-dimensional nearly K"ahler manifolds. Besides its almost complex structure $J$ it also admits a canonical almost product structure $P$. Along a $3$-dimensional CR submanifold the tangent space of $\mathbb S^3\times\mathbb S^3$canbenaturallysplitastheorthogonalsumofthree$2$-dimensionalvectorbundles$\mathcal D_1$,$\mathcal D_2$and$\mathcal D_3$. We study the CR submanifolds in relation to the behavior of the almost product structure on these vector bundles.