Abstract
Lagrangian H-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian H-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean n-space and each unit speed curve Fin the complex plane, we introduce the notion of the complex extensor of M in the complex Euclidean n-space via F. The main purpose of this paper is to classify Lagrangian H-umbilical submanifolds of the complex Euclidean n-space by utilizing complex extensors. We prove that, except the flat ones, Lagrangian H-umbilical submanifolds of complex Euclidean n-space with n greater than 2 are Lagrangian pseudo-spheres and complex extensors of the unit hypersphere of the Euclidean n-space. For completeness we also include in the last section the classification of flat Lagrangian H-umbilical submanifolds of complex Euclidean spaces.
