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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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Harmonic morphisms from compact 3-manifolds

Harmonic morphisms from compact 3-manifolds

(p.295) 10 Harmonic morphisms from compact 3-manifolds
Harmonic Morphisms Between Riemannian Manifolds

Paul Baird

John C. Wood

Oxford University Press

This chapter shows that a non-constant harmonic morphism with compact fibres from a 3-manifold endows it with the structure of a Seifert fibre space — a 3-manifold with a certain type of one-dimensional foliation with leaf space an orbifold. Conversely, any Seifert fibre space can be obtained from a harmonic morphism by smoothing the orbifold leaf space. A global version of the factorization theorem is provided, then the metrics on a 3-manifold which support a harmonic morphism to a surface are described locally and globally. It is shown how fundamental invariants of a one-dimensional foliation propagate along its leaves. The necessary curvature conditions where a Riemannian 3-manifold supports a non-constant harmonic morphism to a surface are given. These show that there are never more than two such harmonic morphisms up to equivalence, even locally, unless it is a space form.

Keywords:   Seifert fibre space, orbifold, factorization theorem, curvature conditions

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