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

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$isalsoobtained\mathrm{.}Here,aLagrangiansubmanifoldiscalled$J$-isotropic,ifthereexistsafunction$\lambda$,suchthat$g((\nabla h)(v,v,v),Jv)=\lambda$holdsforallunittangentvector$v$.