Harmonic morphisms between spaces of constant curvature

Let M and N be simply connected space forms, and U an open and connected subset of M. Further let
n: U-*N be a horizontally homothetic harmonic morphism. In this paper we show that if n 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
isometries of M and N one of six well known examples.