On f-harmonic morphisms between Riemannian manifolds

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f-Harmonic maps were first introduced and studied by Lichnerowicz in Li (see also Section 10.20 in Eells-Lemaire's report 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.