Complex extensors and lagrangian submanifolds in indefinite complex Euclidean spaces

摘要

A general method to construct SO(n)-invariant Lagrangian submanifolds in complex Euclidean n-space was introduced in [6]. In this paper we extend the method to construct SO(k,n-k)-invariant Lagrangian submanifolds in an indefinite complex Euclidean spaces C_k~n. To do so, we introduce the notion of complex extensors in C_k~n. We show that a complex extensor in C_k~n is a Lagrangian H-umbilical submanifold. Conversely, we prove that, except the flat cases, Lagrangian H-umbilical submanifolds in C_k~n are Lagrangian pseudo-Riemannian spheres, Lagrangian pseudo-hyperbolic spaces, complex extensors of a unit pseudo-Riemannian sphere, or complex extensors of a unit pseudo-hyperbolic space.