Harmonic morphisms between riemannian manifolds

A harmonic morphism f : M → N between Riemannian manifolds M and N is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim M ≥ dim N , since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where d f vanishes. Every non-constant harmonic morphism is shown to be an open mapping.