On f-harmonic morphisms between Riemannian manifolds

TL;DR

This paper investigates f-harmonic morphisms between Riemannian manifolds, characterizing them as horizontally weakly conformal f-harmonic maps, and explores their properties, examples, and non-existence results.

Contribution

It generalizes the Fuglede-Ishihara characterization for harmonic morphisms to the f-harmonic case and studies properties and examples of these morphisms.

Findings

Characterization of f-harmonic morphisms as horizontally weakly conformal f-harmonic maps

Provision of examples and non-existence results for f-harmonic morphisms

Analysis of f-harmonicity in conformal immersions

Abstract

f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map. This generalizes the well-known Fuglede-Ishihara characterization for harmonic morphisms. Some properties and many examples as well as some non-existence of f-harmonic morphisms are given. We also study the f-harmonicity of conformal immersions.