Isometric Actions And Harmonic Morphisms

We give the necessary and sufficient condition for a Riemannian foliation, of arbitrary dimension, locally generated by Killing fields to produce harmonic morphisms. Natural constructions of harmonic maps and morphisms are thus obtained. Introduction It is well-known that a Riemannian foliation with minimal leaves has the property that it produces harmonic morphisms i.e. its leaves are locally fibres of submersive harmonic morphisms. This is an immediate consequence of the fact that Riemannian submersions with minimal fibres are harmonic morphisms. More generally, a Riemannian foliation (of codimension not equal to two) produces harmonic morphisms if and only if the vector field determined by the mean curvatures of the leaves is locally a gradient vector field. This is a consequence of the fundamental equation of P. Baird and J. Eells [1] (see Proposition 1.2 below). Although this condition is quite simple, there were not known many examples of such Riemannian foliations; our work wil..