Interaction of Legendre curves and Lagrangian submanifolds

Abstract

It is proved in [8] that there exist no totally umbilical Lagrangian submanifolds in a complex-space-form Mn(4c), n >_ 2, except the totally geodesic ones. In this paper we introduce the notion of Lagrangian Humbilical submanifolds which are the "simplest" Lagrangian submanifolds next to the totally geodesic ones in complex-space-forms. We show that for each Legendre curve in a 3-sphere S 3 (respectively, in a 3-dimensional antide Sitter space-time H3), there associates a Lagrangian H-umbilical submanifold in CP n (respectively, in CH n ) via warped products. The main part of this paper is devoted to the classification of Lagrangian H-umbi|ical submanifolds in CP n and in C/'/n . Our classification theorems imply in particular that "except some exceptional classes", Lagrangian H-umbilical submanlfolds of CP n and of CH n axe obtained from Legendre curves in S 3 or in/-13 via warped products. This provides us an interesting interruption of Legendre curves and Lagrangian H-umbilicM submanifolds in non-fiat complex-space-forms. As an immediate by-product, our results provide us many concrete examples of Lagrangian H-umbilical isometric immersions of real-space-forms into non-fiat complex-space-forms.