Interaction of Legendre curves and Lagrangian submanifolds
type: Item
instance of: scholarly article
title: Interaction of Legendre curves and Lagrangian submanifolds
zbMATH Open document ID: 0884.53014
DOI: 10.1007/BF02760677
author: Bang-Yen Chen
published in: Israel Journal of Mathematics
publication date: 1997-09-17T00:00:00Z
review text: Let \(\widetilde M^n(4c)\) be a complex space form of holomorphic sectional curvature \(4c\neq 0\) and \(\pi:S^{2n+1} (c)\to \widetilde M^n(4c)\) the Hopf fibration from the sphere of radius \(1/ \sqrt c\) for \(c>0\) resp. from the anti-de Sitter space of curvature \(c<0\). The totally real submanifolds \(M\subset \widetilde M^n(4c)\) of (real) dimension \(n\) are called Lagrangian submanifolds. Locally there exists a 1:1-correspondence to the \(n\)-dimensional horizontal submanifolds in \(S^{2n+1}(c)\) with respect to \(\pi\). For \(n=1\) the latter submanifolds are called Legendre curves. Besides the totally geodesic submanifolds, the simplest Lagrangian submanifolds are the \(H\)-umbilical ones, whose shape operator has a special, simple structure. In this article a classification of the latter class is obtained. It is shown that many of the examples can be constructed by starting from a Legendre curve, deriving a horizontal warped product in \(S^{2n+1} (c)\) from it and then projecting this submanifold into \(\widetilde M^n(4c)\).
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1062226
zbMATH Keywords: submanifolds in complex space forms
zbMATH Keywords: Legendrian curves
zbMATH Keywords: Lagrangian submanifolds
reviewed by: Helmut Reckziegel
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02760677
OpenAlex ID: W2025822592
description: scientific article; zbMATH DE number 1062226