Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps

TL;DR

This paper extends the concept of harmonic morphisms to generalized harmonic morphisms that pull back harmonic functions to biharmonic functions, providing characterizations, constructions, and classifications of such maps.

Contribution

It introduces the notion of generalized harmonic morphisms, offers new characterization methods, and classifies them among warped product space projections.

Findings

Characterization of generalized harmonic morphisms into Euclidean space

Two new construction methods for non-harmonic generalized harmonic morphisms

Complete classification of such morphisms among warped product space projections

Abstract

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.