Lagrangian H-umbilical submanifolds in complex space forms and pseudo-parallel cubic form

Abstract

Lagrangian H-umbilical submanifolds in complex space forms, as the “simplest” Lagrangian submanifolds next to the geodesic ones, were introduced and determined by B.-Y. Chen. Many interesting examples belong to this class, such as the Whitney spheres, isotropic non-minimal immersions, and special Calabi product immersions. In this paper, such submanifolds are proved to be of a conformally flat, quasi-Einstein metric and the pseudo-parallel cubic form. As the main results, we find a geometric characterization of those submanifolds as not being of constant sectional curvature. Meanwhile, for Lagrangian submanifolds in complex space forms with pseudo-parallel cubic form, we completely determine the three dimensional case, and all dimensions for the conformally flat case.