On isotropic Lagrangian submanifolds in the homogeneous nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$

TL;DR

This paper proves that isotropic Lagrangian submanifolds in 6-dimensional strict nearly Kähler manifolds are totally geodesic and provides a classification of J-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S^3×S^3.

Contribution

It establishes the total geodesicity of isotropic Lagrangian submanifolds and classifies J-isotropic Lagrangian submanifolds in a specific nearly Kähler manifold.

Findings

Isotropic Lagrangian submanifolds are totally geodesic.

Classification of J-isotropic Lagrangian submanifolds in S^3×S^3.

Conditions under which classification holds.

Abstract

In this paper, we show that isotropic Lagrangian submanifolds in a $6$-dimensional strict nearly K\"ahler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the $J$-isotropic Lagrangian submanifolds in the homogeneous nearly K\"ahler $\mathbb{S}^3\times \mathbb{S}^3$ is also obtained. Here, a Lagrangian submanifold is called $J$-isotropic, if there exists a function $\lambda$, such that $g((\nabla h)(v,v,v),Jv)=\lambda$ holds for all unit tangent vector $v$.