Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.
In this paper,we study f-harmonicity of some special maps from or into a doubly warped product manifold.First we recall some properties of doubly twisted product manifolds.After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M ×(μ,λ) N can not be proper f-harmonic maps,we use projection maps and product maps to construct nontrivial f-harmonic maps.Thus we obtain some similar results given in [21],such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps.Furthermore,we investigate non-trivial f-harmonicity of the product of two harmonic maps.
f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970.In this paper,the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions.The author proves 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 characterization for harmonic morphisms.Some properties and many examples as well as some non-existence of f-harmonic morphisms are given.The author also studies the f-harmonicity of conformal immersions.
