Harmonic morphisms with one-dimensional fibres

Abstract

AbstractThis chapter shows that a harmonic morphism from a manifold of dimension n+1 to a manifold of dimension n is, locally or globally, a principal bundle with a certain metric. When n = 3, in a neighbourhood of a critical point, a harmonic morphism behaves like the Hopf polynomial map; when n > 3, there can be no critical points. A factorization theorem and a circle action are obtained in all cases, leading to topological restrictions. Given a nowhere-zero Killing field V, it is shown how to find harmonic morphisms with fibres tangent to V. Harmonic morphisms of warped product type are discussed; these are related to isoparametric functions. These two types are the only types that can occur on a space form or on an Einstein manifold when n > 3. When n = 3, a third type of harmonic morphism is found related to the Beltrami fields equation of hydrodynamics.