Harmonic maps of nonorientable surfaces to four-dimensional manifolds
type: Item
instance of: scholarly article
title: Harmonic maps of nonorientable surfaces to four-dimensional manifolds
zbMATH Open document ID: 0770.58006
author: Toru Ishihara
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1993-06-17T00:00:00Z
review text: In the present paper, it will be tried to construct harmonic maps from nonorientable surfaces into 4-dimensional Riemannian manifolds. Nonorientable surfaces \(M\) which are considered are quotients of Riemann surfaces \(\widehat M\) by the equivalence relation \(z\sim I(z)\), where \(I\) is an anti-holomorphic involution of \(\widehat M\) without fixed points. Especially, a projective plane and hyperelliptic Riemann surface are treated. Let \(\pi: \widehat M \to M\) be the natural projection of \(\widehat M\) to the quotient space. A map \(\widehat h\) of \(\widehat M\) into a Riemannian manifold \(N\) is factored as \(\widehat h = h \cdot \pi\), where \(h\) is a map of \(M\) into \(N\), if and only if \(\widehat hI = \widehat h\). Hence harmonic maps \(\widehat h: \widehat M \to N\) with \(\widehat h \cdot I = \widehat h\) are investigated. Let \(N\) be a 4-dimensional oriented Riemannian manifold and \(S\) its twistor space with almost complex structures \(J_ 1\) and \(J_ 2\). By introducing a natural involution \(I_ S\) of \(S\) which is anti- holomorphic with respect to \(J_ 1\) and \(J_ 2\) and using the twistor lifts \(\widetilde h: M \to S\) of Eells and Salamon for harmonic maps \(\widehat h: \widehat M \to N\), it is shown: The assignment \(\widehat h \mapsto \widetilde h\) is a bijective correspondence between the set of nonconstant conformal harmonic maps \(\widehat h: \widehat M \to N\) with \(\widehat h\cdot I = \widehat h\) and the set of nonvertical \(J_ 2\)- holomorphic curves \(\widetilde h: M\to S\) with \(\widetilde h\cdot I = I_ S \cdot \widetilde h\). Now, let \(N\) be the unit 4-sphere \(S^ 4\). Then its twistor space is the complex projective 3-space \(CP^ 3\). Bryant gives the correspondence between the set of meromorphic functions \(f\) and \(g\) on \(M\) with \(g\) nonconstant on \(\widehat M\) and the set of horizontal and holomorphic maps of \(\widehat M\) to \(CP^ 3\). It is shown: A conformal isotropic harmonic map \(\widehat h: \wideh…
reviewed by: Toru Ishihara
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 203567
zbMATH Keywords: twister space
zbMATH Keywords: twistor space
zbMATH Keywords: harmonic maps
zbMATH Keywords: nonorientable surfaces
zbMATH Keywords: 4- dimensional Riemannian manifolds
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225951
OpenAlex ID: W2086007951
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DOI: 10.2748/TMJ/1178225951
description: scientific article; zbMATH DE number 203567