Harmonic morphisms between semi-riemannian manifolds

A smooth map f: M --> N between semi-riemannian manifolds is called a harmonic morphism if f pulls back harmonic functions (i.e., local solutions of the Laplace-Beltrami equation) on N into harmonic functions on M. It is shown that a harmonic morphism is the same as a harmonic map which is moreover horizontally weakly conformal, these two notions being likewise carried over from the riemannian case. Additional characterizations of harmonic morphisms are given. The case where M and N have the same dimension n is studied in detail. When n = 2 and the metrics on M and N are indefinite, the harmonic morphisms are characterized essentially by preserving characteristics.