Lagrangian submanifolds in the 6-dimensional nearly Kähler manifolds with parallel second fundamental form

In this paper, we study the Lagrangian submanifolds in the homogeneous nearly Kähler S3×S3S3×S3 with parallel second fundamental form. We first prove that every Lagrangian submanifold with parallel second fundamental form in any 6-dimensional strict nearly Kähler manifold is totally geodesic. Then we give a complete classification of the totally geodesic Lagrangian submanifolds in the homogeneous nearly Kähler S3×S3S3×S3.