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Interaction of Legendre curves and Lagrangian submanifolds

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Abstract

It is proved in [8] that there exist no totally umbilical Lagrangian submanifolds in a complex-space-form\(\tilde M^n (4c)\),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 31 ), 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 31 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.

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  1. Department of Mathematics, Michigan State University, 48824-1027, East Lansing, Michigan, USA

    Bang-Yen Chen

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  1. Bang-Yen Chen
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Chen, BY. Interaction of Legendre curves and Lagrangian submanifolds. Isr. J. Math. 99, 69–108 (1997). https://doi.org/10.1007/BF02760677

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