A twistor description of harmonic maps of a 2-sphere into a Grassmannian
type: Item
instance of: scholarly article
title: A twistor description of harmonic maps of a 2-sphere into a Grassmannian
zbMATH Open document ID: 0575.58018
DOI: 10.1007/BF01458017
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164087
review text: A twistor fibration \(\pi\) : \(Z\to N\) over a Riemannian manifold N is a Riemannian submersion of an almost Hermitian manifold Z onto N with the following property: If M is an almost Hermitian manifold with co-closed Kähler form and \(\psi\) : \(M\to Z\) is a holomorphic map then \(\pi_ 0\psi: M\to N\) is a harmonic map. In this article, certain natural twistor fibrations of complex flag manifolds over complex Grassmannians are exhibited. It is then shown that all harmonic maps of a Riemann sphere into a Grassmannian arise as projections of holomorphic maps into a flag manifold. Further, it is demonstrated that from every harmonic map of \(S^ 2\) into a Grassmannian, there may be constructed a family of transformed maps of \(S^ 2\) into other Grassmannians which are also harmonic.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3919405
zbMATH Keywords: twistor fibration
zbMATH Keywords: Hermitian manifold
zbMATH Keywords: Kähler form
zbMATH Keywords: harmonic map
zbMATH Keywords: complex flag manifolds over complex Grassmannians
zbMATH Keywords: holomorphic maps into a flag manifold
author: Francis E. Burstall
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3919405