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Harmonic morphisms between spaces of constant curvature

Published online by Cambridge University Press:  20 January 2009

Sigmundur Gudmundsson
Affiliation:
Department of Pure MathematicsUniversity of LeedsLeeds LS2 9JT, England Department of MathematicsScience InsituteUniversity of IcelandDunhaga 3107 Reykjavik, Iceland
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Abstract

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Let M and N be simply connected space forms, and U an open and connected subset of M. Further let π: UN be a horizontally homothetic harmonic morphism. In this paper we show that if π has totally geodesic fibres and integrable horizontal distribution, then the horizontal foliation of U is totally umbilic and isoparametric. This leads to a classification of such maps. We also show that horizontally homothetic harmonic morphisms of codimension one are either Riemannian submersions modulo a constant, or up to isometrics of M and N one of six well known examples.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

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