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Harmonic Morphisms Between Riemannian Manifolds$
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Paul Baird and John C. Wood

Print publication date: 2003

Print ISBN-13: 9780198503620

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198503620.001.0001

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Mini-twistor theory on three-dimensional space forms

Mini-twistor theory on three-dimensional space forms

(p.175) 6 Mini-twistor theory on three-dimensional space forms
Harmonic Morphisms Between Riemannian Manifolds

Paul Baird

John C. Wood

Oxford University Press

This chapter discusses harmonic morphisms from 3-manifolds to surfaces, especially those of constant curvature. First, the local behaviour is considered, showing that the foliation given by its fibres is a smooth conformal foliation by geodesics, even at critical points. This leads to a local factorization of a harmonic morphism as a submersion followed by a weakly conformal map of surfaces and a local normal form. The mini-twistor spaces of geodesics in the three complete simply connected space-forms are described: that these are complex surfaces leading to Weierstrass-type representations of submersive harmonic morphisms on domains of these spaces. This, together with a stronger version of the factorization theorem, determines all harmonic morphisms on such domains leading to classifications of the harmonic morphisms from these spaces to a surface. Finally, the constructions are generalized to give a class of harmonic morphisms from higher-dimensional space forms.

Keywords:   3-manifold, constant curvature, space form, normal form, Weierstrass representation

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