Interaction of Legendre curves and Lagrangian submanifolds

It is proved in [8] that there exist no totally umbilical Lagrangian submanifolds in a complex-space-form (M) over tilde(n)(4c), n greater than or equal to 2, except the totally geodesic ones. In this paper we introduce the notion of Lagrangian H-umbilical 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 S-dimensional anti-de Sitter space-time H-1(3)), there associates a Lagrangian H-umbilical submanifold in CPn (respectively, in CHn) via warped products. The main part of this paper is devoted to the classification of Lagrangian H-umbilical submanifolds in CPn and in CHn. Our classification theorems imply in particular that ''except some exceptional classes'', Lagrangian H-umbilical submanifolds of CPn and of CHn are obtained from Legendre curves in S-3 or in H-1(3) via warped products. This provides us an interesting interaction of Legendre curves and Lagrangian H-umbilical submanifolds in non-flat complex-space-forms. As an immediate by-product, cur results provide us many concrete examples of Lagrangian H-umbilical isometric immersions of real-space-forms into non-flat complex-space-forms.