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Harmonic morphisms between spaces of constant curvature. (English) Zbl 0790.58012

Harmonic morphisms are maps between Riemannian manifolds \(\pi: (M,g) \to (N,h)\) which pull back germs of harmonic functions on \(N\) to germs of harmonic functions on \(M\). In this paper \((M,g)\) and \((N,h)\) are assumed to be simply connected space forms and \(\pi: U\to N\) a horizontally homothetic harmonic morphism from an open and connected subset \(U\) of \(M\). It is shown that if \(\pi\) 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. Furthermore it is proved that horizontally homothetic harmonic morphisms of codimension one are either Riemannian submersions modulo a constant, or up to isometries of \(M\) and \(N\) one of six well known examples.

MSC:

58E20 Harmonic maps, etc.
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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