Lagrangian submanifolds in the homogeneous nearly Kähler \({\mathbb {S}}^3 \times {\mathbb {S}}^3\)
type: Item
instance of: scholarly article
title: Lagrangian submanifolds in the homogeneous nearly Kähler \({\mathbb {S}}^3 \times {\mathbb {S}}^3\)
zbMATH Open document ID: 1403.53050
author: B. Dioos
author: Luc Vrancken
author: Xianfeng Wang
published in: Annals of Global Analysis and Geometry
publication date: 2018-01-25T00:00:00Z
review text: In this paper, the authors investigate Lagrangian submanifolds \(M\) in the nearly Kähler manifold \(\mathbf{S}^{3}\times \mathbf{S}^{3}\). Since \(M\) is Lagrangian, we can consider two endomorphisms \(A,B:TM\rightarrow TM\) such that the restriction of the almost product structure \(P\) to \(M\) is \(A+JB\). At each point \(p\) in \(M\), there is an orthonormal basis \(e_{1},e_{2},e_{3}\in T_{p}M\) such that \[ Pe_{i}=\cos 2\theta_{i}e_{i}+\sin2\theta_{i}Je_{i}, i=1,2,3 \] According to several properties of the angle functions \(\theta_{i}\), the following results are provided. Corollary 3.6: Let \(M\) be a Lagrangian submanifold of the nearly Kähler \(\mathbf{S}^{3}\times \mathbf{S}^{3}\). If \(M\) is totally geodesic, then the angles \(\theta_{1},\theta_{2},\theta_{3}\) are constant. Conversely, if the angles are constant and \(h_{12}^{3}=0\), then \(M\) is totally geodesic. \newline Lemma 3.9 states that for a totally geodesic Lagrangian submanifolds in \(\mathbf{S}^{3}\times \mathbf{S}^{3}\), there exist 6 possible cases for angle functions after a possible permutation of the angles. According to these 6 possible angle functions, 6 examples of totally geodesic submanifolds were obtained and there are two more non totally geodesic Lagrangian submanifolds are provided. In the last section, it is proved that these are congruent to one of the examples of constant sectional curvature. As a result, the radius of a round Lagrangian sphere in \(\mathbf{S}^{3}\times \mathbf{S}^{3}\) can be either \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).
reviewed by: Yun Myung Oh
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 6829770
zbMATH Keywords: Lagrangian submanifolds
zbMATH Keywords: Lagrangian torus
zbMATH Keywords: nearly Kähler manifold
zbMATH Keywords: Lagrangian sphere
zbMATH Keywords: constant sectional curvature
MaRDI profile type: Publication
full work available at URL: s10455-017-9567-z
OpenAlex ID: W2733126359
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DOI: 10.1007/S10455-017-9567-Z
description: scientific article; zbMATH DE number 6829770