Harmonic morphism

rdf:langString Harmonic morphism

rdf:langString In mathematics, a harmonic morphism is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps i.e. those that are horizontally (weakly) conformal. In local coordinates, on and on , the harmonicity of is expressed by the non-linear system where and are the Christoffel symbols on and , respectively. The horizontal conformality is given by where the conformal factor is a continuous function called the dilation. Harmonic morphisms are therefore solutions to non-linear over-determined systems of partial differential equations, determined by the geometric data of the manifolds involved. For this reason, they are difficult to find and have no general existence theory, not even locally.