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

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 $v$.