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

Abstract

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 Kähler 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 ${\mathbb{S}}^3\times {\mathbb{S}}^3$ 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 to the behavior of the almost product structure on these vector bundles.

References in corpus (1)

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

Cited by in corpus (1)

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