HARMONIC MORPHISMS WITH ONE-DIMENSIONAL FIBRES

Abstract

We study harmonic morphisms by placing them into the context of conformal foliations. Most of the results we obtain hold for fibres of dimension one and codomains of dimension not equal to two. We consider foliations which produce harmonic morphisms on both compact and noncompact Riemannian manifolds. By using integral formulae, we prove an extension to one-dimensional foliations which produce harmonic morphisms of the well-known result of S. Bochner concerning Killing fields on compact Riemannian manifolds with nonpositive Ricci curvature. From the noncompact case, we improve a result of R. L. Bryant[9] regarding harmonic morphisms with one-dimensional fibres defined on Riemannian manifolds of dimension at least four with constant sectional curvature. Our method gives an entirely new and geometrical proof of Bryant's result. The concept of homothetic foliation (or, more generally, homothetic distribution) which we introduce, appears as a useful tool both in proofs and in providing new examples of harmonic morphisms, with fibres of any dimension.