Abstract: Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic function on the codomain is a harmonic function on the domain; it thus ‘pulls back’ germs of harmonic functions to germs of harmonic functions. Harmonic morphisms can be characterized as harmonic maps satisfying a condition dual to weak conformality called ‘horizontal weak conformality’ or ‘semiconformality’. Examples include harmonic functions, conformal mappings in the plane, holomorphic mappings with values in a Riemann surface, and certain submersions arising from Killing fields and geodesic fields. The study of harmonic morphisms involves many different branches of mathematics: the book includes discussion on aspects of the theory of foliations, polynomials induced by Clifford systems and orthogonal multiplications, twistor and mini-twistor spaces, and Hermitian structures. Relations with topology are discussed, including Seifert fibre spaces and circle actions, also relations with isoparametric functions and the Beltrami fields equation of hydrodynamics.

Keywords: harmonic map, Laplace's equation, conformal, Killing field, twistor, foliation