CR Submanifolds of the Nearly Kähler S3×S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^3\times \mathbb {S}^3$$\end{document} Characterised by Properties of the Almost Product Structure

In a previous paper (Antić et al., three-dimensional CR submanifolds of the nearly Kähler S3×S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{3}\times \mathbb {S}^{3}$$\end{document}, 2017), the authors together with L. Vrancken initiated the study of 3-dimensional CR submanifolds of the nearly Kähler homogeneous S3×S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^3\times \mathbb {S}^3$$\end{document}. 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 S3×S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^3\times \mathbb {S}^3$$\end{document} can be naturally split as the orthogonal sum of three 2-dimensional vector bundles D1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}_1$$\end{document}, D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}_2$$\end{document} and D3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {D}_3$$\end{document}. We study the CR submanifolds in relation with the behavior of the almost product structure on these vector bundles.