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 ,n≥2, except the totally geodesic ones. In this paper we introduce the notion of LagrangianH-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-sphereS ³ (respectively, in a 3-dimensional antide Sitter space-timeH ₁ ³ ), there associates a LagrangianH-umbilical submanifold in ℂP ⁿ (respectively, in ℂH ⁿ ) via warped products. The main part of this paper is devoted to the classification of LagrangianH-umbilical submanifolds in ℂP ⁿ and in ℂH ⁿ . Our classification theorems imply in particular that “except some exceptional classes”, LagrangianH-umbilical submanifolds of ℂP ⁿ and of ℂH ⁿ are obtained from Legendre curves inS ³ or inH ₁ ³ via warped products. This provides us an interesting interaction of Legendre curves and LagrangianH-umbilical submanifolds in non-flat complex-space-forms. As an immediate by-product, our results provide us many concrete examples of LagrangianH-umbilical isometric immersions of real-space-forms into non-flat complex-space-forms.