  Harmonic morphisms are maps between Riemannian manifolds that pull back
harmonic functions to harmonic functions. These maps are characterized as
horizontally weakly conformal harmonic maps and they have many interesting
links and applications to several areas in mathematics (see the book by Baird
and Wood for details). In this paper, we study generalized harmonic morphisms
which are defined to be maps between Riemannian manifolds that pull back
harmonic functions to biharmonic functions. We obtain some characterizations of
generalized harmonic morphisms into a Euclidean space and give two methods of
constructions that can be used to produce many examples of generalized harmonic
morphisms which are not harmonic morphisms. We also give a complete
classification of generalized harmonic morphisms among the projections of a
warped product space, which provides infinitely many examples of proper
biharmonic Riemannian submersions and conformal submersions from a warped
product manifold.
