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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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Curvature considerations

Curvature considerations

Chapter:
(p.319) 11 Curvature considerations
Source:
Harmonic Morphisms Between Riemannian Manifolds
Author(s):

Paul Baird

John C. Wood

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198503620.003.0011

The curvature of the domain on various combinations of horizontal and vertical vectors is calculated for a horizontally conformal submersion and for a (submersive) harmonic morphism. It is related to the dilation and the curvature of the codomain, and local and global non-existence results are deduced for maps with or without critical points. It is shown that any harmonic morphism with totally geodesic fibres — defined on a Euclidean space with values in a manifold of dimension not equal to two — is orthogonal projection to a subspace followed by a surjective homothetic covering. Together with Theorem 6.7.3, this gives a complete picture of the globally defined harmonic morphisms with totally geodesic fibres on Euclidean space.

Keywords:   curvature conditions, nonexistence, codomain, entire harmonic morphisms

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