ON HARMONIC MORPHISMS PROJECTING HARMONIC FUNCTIONS TO HARMONIC FUNCTIONS

For Riemannian manifolds and , admitting a submersion with compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians on and , we determine conditions under which a harmonic function on projects down, via its horizontal component, to a harmonic function on .


Introduction and Preliminaries
Harmonic morphisms are the maps between Riemannian manifolds which preserve germs of harmonic functions i.e. these (locally) pull back harmonic functions to harmonic functions. The aim of this note is to analyse the converse situation and to investigate the class of harmonic morphisms that (locally) projects or pushes forward harmonic functions to harmonic functions, in the sense of Definition 2.4. If such a class exists, another interesting question arises "to what extent does the pull back of the projected function preserve the original function".
The formal theory of harmonic morphisms between Riemannian manifolds began with the work of Fuglede [6] and Ishihara [10]. For a smooth map φ : M m → N n , let C φ = {x ∈ M |rankdφ x < n} be its critical set. The points of the set M \ C φ are called regular points.
conformal if dφ = 0 on C φ and the restriction of φ to M \C φ is a conformal submersion, that is, for each x ∈ M \ C φ , the differential dφ x : T H x M → T φ(x) N is conformal and surjective. This means that there exists a function λ : By setting λ = 0 on C φ , we can extend λ : M → R + 0 to a continuous function on M such that λ 2 is smooth. The extended function λ : M → R + 0 is called the dilation of the map.
Recall that a map φ : M m → N n is said to be harmonic if it extremizes the It is well-known that a map φ is harmonic if and only if its tension field vanishes.
Harmonic morphisms can be viewed as a subclass of harmonic maps in the light of the following characterization, obtained in [6,10].   For the fundamental results and properties of harmonic morphisms, the reader is referred to [1,4,6,11] and for an updated online bibliography to [9]. Precisely, for any V ⊂ N and integrable function f :

Harmonic morphisms projecting harmonic functions
The vertical component of f is given by (1) If f is horizontally homothetic at x then Hf is also horizontally homothetic at x.
(2) If Hf is horizontally homothetic at x and either X i (f ) ≥ 0 or X i (f ) ≤ 0 (for all i) on the fibre through x then f is horizontally homothetic, where is a local orthonormal frame for the horizontal distribution.
(3) If f is constant along the fibre through x then Vf = 0.
Proof. The proof can be completed by following the calculations in Proposition 2.5 (below).  Then for any V ⊂ N and f : Proof. First notice from Theorem 1.3 that λ is horizontally homothetic; a fact which will be used repeatedly in the proof.
Choose a local orthonormal frame {X i } n i=1 for T N . If X i denotes the horizontal lift of X i for i = 1, . . . , n then {λX i } n i=1 is a local orthonormal frame for the horizontal distribution. Let {X α } m α=n+1 be a local orthonormal frame for the vertical distribution. Then we can write the Laplacian ∆ M on M as Now the Laplacian of the fibre φ −1 (y) is Therefore, from Equation 2.1 we obtain where H is (m − n) times the mean curvature vector field of the fibres and H, V denote the orthogonal projections on the horizontal and vertical subbundles of T M , respectively.
Since X i is horizontal lift of X i for i = 1, . . . , n, therefore for the functionf we have The volume of the fibres does not vary in the horizontal direction because of the relation X i (vol(φ −1 (y))) = − φ −1 (y) g(H, X i )v φ −1 (y) and the fact that the fibres are minimal.
Similarly, we obtain The horizontal homothety of the dilation implies that H∇ M X i X i is the horizontal lift of ∇ N X i X i , cf. [3, Lemma 3.1], therefore, we have Proof. Since the fibres are compact and dilation is constant, the harmonicity off follows from Proposition 2.5 because Rest of the proof follows from the construction off .
As an application, we give a description of harmonic functions on manifolds admitting harmonic Riemannian submersions with compact fibres.  construction φ : G/K → G/H with K ⊂ H ⊂ G and K, H compact; see [5] for the details of the metrics for which φ is a harmonic morphism. Another reference for such examples is [7,Ch. 6].