Killing vector fields and Lagrangian submanifolds of the nearly Kaehler \(S^6\)
type: Item
instance of: scholarly article
title: Killing vector fields and Lagrangian submanifolds of the nearly Kaehler \(S^6\)
zbMATH Open document ID: 0907.53015
DOI: 10.1016/S0021-7824(98)80001-9
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1999-03-04T00:00:00Z
review text: It is well known that the algebraic structure of the Cayley numbers can be used to define an almost complex structure \(J\) on the 6-dimensional unit sphere \(S^6\) turning \(S^6\) into a nearly Kähler manifold. A submanifold \(M\) of \(S^6\) is said to be totally real if \(J\) maps the tangent bundle \(TM\) of \(M\) into the normal bundle of \(M\), and it said to be almost complex if \(J\) leaves \(TM\) invariant. The author gives a local classification of all 3-dimensional totally real submanifolds \(M\) of \(S^6\) admitting a unit Killing vector field whose integral curves are great circles. Roughly, they can be constructed from either superminimal almost complex curves in \(S^6\) or via Hopf lifts from holomorphic curves in complex projective plane \(\mathbb{C} P^2\).
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 1207281
zbMATH Keywords: nearly Kähler 6-sphere
zbMATH Keywords: totally real submanifolds
zbMATH Keywords: almost complex submanifolds
zbMATH Keywords: superminimal immersions
zbMATH Keywords: Killing vector fields
author: Luc Vrancken
reviewed by: Jürgen Berndt
MaRDI profile type: Publication
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Wikidata QID: Q126183185
full work available at URL: s0021-7824(98)80001-9
OpenAlex ID: W2010590063
description: scientific article; zbMATH DE number 1207281