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

TL;DR

This paper investigates 3-dimensional CR submanifolds within the nearly Kähler manifold ^3 ^3, focusing on how the almost product structure influences their geometric properties and classification.

Contribution

It characterizes CR submanifolds based on the behavior of the almost product structure on associated vector bundles, extending previous work on nearly Kähler manifolds.

Findings

Classification of CR submanifolds according to the almost product structure

Analysis of the orthogonal decomposition of tangent spaces

Insights into the geometric behavior of submanifolds in nearly Kähler ^3 ^3

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 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 $\mathbb S^3\times\mathbb S^3$ 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.