Complex extensors and Lagrangian submanifolds in complex Euclidean spaces

Abstract

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