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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

(p.319) 11 Curvature considerations
Harmonic Morphisms Between Riemannian Manifolds

Paul Baird

John C. Wood

Oxford University Press

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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