Isotropic Lagrangian submanifolds in the homogeneous nearly Khler S^3× S^3

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kahler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly KahlerS3 × S3 is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function A, such that g（（△↓h）（v, v, v）, Jr） = λ holds for all unit tangent vector v.

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Khler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Khler S^3× S^3 is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((▽h)(v, v, v), J v) = λ holds for all unit tangent vector v.