# Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

Authors: M. T. Mustafa
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

For Riemannian manifoldsM and N, admitting a submersion ϕ with compact fibres, we introduce the projection of a function via its decomposition into
horizontal and vertical components. By comparing the Laplacians on M and N, we determine conditions under which a harmonic function on U=ϕ−1(V)⊂M projects down, via its horizontal component, to a harmonic function on V⊂N.


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## Body

## 1. Introduction and Preliminaries

Harmonic morphisms are the maps between Riemannian manifolds which preserve germs of harmonic functions, that is, these (locally) pull back harmonic functions to harmonic functions. The aim of this paper 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 Definition2.4. 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 [1] and Ishihara [2].Definition 1.1.
A smooth mapϕ:Mm→Nn between Riemannian manifolds is called a harmonic morphism if, for every real-valued function f which is harmonic on an open subset V of N with ϕ-1(V) nonempty, f∘ϕ is a harmonic function on ϕ-1(V).These maps are related to horizontally (weakly) conformal maps which are a natural generalization of Riemannian submersions.For a smooth mapϕ:Mm→Nn, let Cϕ={x∈M|rankdϕx<n} be its critical set. The points of the set M∖Cϕ are called regular points. For each x∈M∖Cϕ, the vertical space at x is defined by TxVM=Kerdϕx. The horizontal spaceTxHM at x is given by the orthogonal complement of TxVM in TxM.Definition 1.2 (see [3, Section 2.4]).
A smooth mapϕ:(Mm,g)→(Nn,h) is called horizontally (weakly) conformal if dϕ=0 on Cϕ and the restriction of ϕ to M∖Cϕ is a conformal submersion, that is, for each x∈M∖Cϕ, the differential dϕx:TxHM→Tϕ(x)N is conformal and surjective. This means that there exists a function λ:M∖Cϕ→ℝ+ such that
(1.1)h(dϕ(X),dϕ(Y))=λ2g(X,Y),∀X,Y∈TxHM.By settingλ=0 on Cϕ, we can extend λ:M→ℝ0+ to a continuous function on M such that λ2 is smooth. The extended function λ:M→ℝ0+ is called the dilation of the map.Forx∈M∖Cϕ, the assignments x↦TxHM and x↦TxVM define smooth distributions THM and  TVM on M∖Cϕ or subbundles of TM|M∖Cϕ, the tangent bundle of M∖Cϕ. The distributions THM and TVM are, respectively, called horizontal distribution (or horizontal subbundle) and vertical distribution (or vertical subbundle) defined by ϕ.Recall that a mapϕ:Mm→Nn is said to be harmonic if it extremizes the associated energy integral  E(ϕ)=(1/2)∫Ω∥ϕ*∥2dυM for every compact domain Ω⊂M. 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 [1, 2].A smooth map is a harmonic morphism if and only if it is harmonic and horizontally (weakly) conformal.What is special about this characterization of harmonic morphism is that it equips them with geometric as well as analytic features. For instance, the following result of Baird and Eells [4, Riemannian case] and Gudmundsson [5, semi-Riemannian case] reflects such properties of harmonic morphisms.Theorem 1.3.
Letϕ:Mm→Nn be a horizontally conformal submersion with dilation λ. If(1)
n=2, then ϕ is a harmonic map if and only if it has minimal fibres;(2)
n≥3, then two of the following imply the other:
(a)
ϕ is a harmonic map,(b)
ϕ has minimal fibres,(c)
gradHλ2=0 where gradHλ2 denotes the projection of gradλ2 on the horizontal subbundle of TM, obtained through the unique orthogonal decomposition into vertical and horizontal parts.For the fundamental results and properties of harmonic morphisms, the reader is referred to [1, 3, 6, 7] and for an updated online bibliography to [8].


## 2.  The Projection of a Function via a Submersion

Given a smooth mapϕ:Mm→Nn with compact fibres ϕ-1(ϕ(x)) for x∈M∖Cϕ, we can use fibre integration to define the horizontal and vertical components of every integrable function f on U⊂M at regular points.Definition 2.1.
Letϕ:Mm→Nn be a smooth map between Riemannian manifolds with compact fibres. Define the horizontal component of an integrable function f, on M, at a regular point x as the average of f taken over the fibre ϕ-1(ϕ(x)). Precisely, for any V⊂N and integrable function f:U=ϕ-1(V)⊂M→ℝ, the horizontal component of f at a regular point x is defined as
(2.1)(Hf)(x)=1vol(ϕ-1(y))(∫ϕ-1(y)fdvϕ-1(y))(ϕ(x)),
where x∈U, ϕ(x)=y,  dvϕ-1(y) is the volume element of the fibre ϕ-1(y), vol(ϕ-1(y)) is the volume of the fibre ϕ-1(y), and (∫ϕ-1(y)fdvϕ-1(y))(ϕ(x)) denotes the integral of f|ϕ-1(ϕ(x)).
Thevertical component of f is given by
(2.2)(Vf)(x)=(f-Hf)(x).
Note that the horizontal component of a function depends only on the fibreϕ-1(y) and not the choice of x∈ϕ-1(y).Definition 2.2.
Letϕ:Mm→Nn be a submersion. A function f:U⊂M→ℝ is called horizontally homothetic if the vector field  grad(f) is vertical, that is, at each point grad(f) is tangent to the fibre.The componentsℋf and 𝒱f have the following basic properties for submersions.Lemma 2.3.
Letϕ:Mm→Nn be a submersion with compact fibres. Suppose that the fibres ϕ-1(y), y∈N are minimal submanifolds of M. Consider x∈U and a function f:U⊂M→ℝ.(1)
Iff is horizontally homothetic at x, then ℋf is also horizontally homothetic at x.(2)
Ifℋf is horizontally homothetic at x and either Xi(f)≥0 or Xi(f)≤0 (for all i) on the fibre through x, then f is horizontally homothetic, where {Xi}i=1n is a local orthonormal frame for the horizontal distribution.(3)
Iff is constant along the fibre through x then 𝒱f=0.Proof.
The proof can be completed by following the calculations in Proposition3.1.Definition 2.4.
Letϕ:Mm→Nn be a submersion with compact fibres, and let f:U=ϕ-1(V)⊂M→ℝ be an integrable function. The horizontal component of f defines a function f̃:V⊂N→ℝ as
(2.3)f̃(y)=(Hf)(x),
where x∈U and y=ϕ(x). The function f̃ is called the projection of f on N, via the mapϕ.We next focus on projection of harmonic functions to harmonic functions via harmonic morphisms.


## 3.  Harmonic Morphisms Projecting Harmonic Functions

The conditions under which harmonic morphisms project harmonic functions to harmonic functions can be obtained by employing an identity relating the Laplacian on the fibre with the Laplacians on the domain and target manifolds.Recalling that for a submersionϕ:Mm→Nn, the vector fields X on M and Y on N are said to be ϕ-related if dϕ(Xx)=Yϕ(x) for every x∈M. A horizontal vector field X on M is called basic if it is ϕ-related to some vector field  X′ on N, and  X is called horizontal lift of  X′. It is well known that for a given vector field X′ on N, there exists a unique horizontal lift X of X′ such that X and X′ are ϕ-related.Proposition 3.1.
Letϕ:(Mm,g)→(Nn,h)(n>2) be a nonconstant submersive harmonic morphism with dilation λ, having compact, connected, and minimal fibres. Then for any V⊂N and f:U=ϕ-1(V)⊂M→ℝ,
(3.1)ΔNf̃=1vol(ϕ-1(y))∫ϕ-1(y)1λ2(ΔMf-Δϕ-1(y)f)dvϕ-1(y)+1vol(ϕ-1(y))∑i=1n∫ϕ-1(y)(∇XiMXi)Vfdvϕ-1(y),
where x∈U,ϕ(x)=y,f̃ is as defined in Definition 2.4 and ΔM, ΔN, Δϕ-1(y) are the Laplacians on M, N, ϕ-1(y), respectively, ∇M is the Levi-Civita connection on M, (∇XiMXi)V denotes the vertical component of ∇XiMXi, and {Xi}i=1n denote the horizontal lift of a local orthonormal frame {Xi′}i=1n for TN.Proof.
First notice from Theorem1.3 that λ is horizontally homothetic, a fact which will be used repeatedly in the proof.
Choose a local orthonormal frame{Xi′}i=1n for TN. If Xi denotes the horizontal lift of Xi′ for i=1,…,n, then {λXi}i=1n is a local orthonormal frame for the horizontal distribution. Let {Xα}α=n+1m be a local orthonormal frame for the vertical distribution. Then we can write the Laplacian ΔM on M as
(3.2)ΔM=∑i=1n{λXi∘λXi-∇λXiMλXi}+∑α=n+1m{Xα∘Xα-∇XαMXα}=λ2∑i=1n{Xi∘Xi-∇XiMXi}+∑α=n+1m{Xα∘Xα-∇XαMXα}.
Now the Laplacian of the fibre ϕ-1(y) is
(3.3)Δϕ-1(y)=∑α=n+1m{Xα∘Xα-∇Xαϕ-1(y)Xα}.
If B is the second fundamental form of the fibre ϕ-1(y) as a submanifold in M, we can write ∇XαMXα as
(3.4)∇XαMXα=∇Xαϕ-1(y)Xα+B(Xα,Xα).
Let μ denote the mean curvature vector of ϕ-1(y) given by
(3.5)μ=1m-n∑α=n+1mB(Xα,Xα).
Setting H=(m-n)μ, we obtain from (3.2)
(3.6)ΔM=λ2∑i=1n{Xi∘Xi-(∇XiMXi)H}+Δϕ-1(y)-H-λ2∑i=1n(∇XiMXi)V=λ2∑i=1n{Xi∘Xi-(∇XiMXi)H}+Δϕ-1(y)-λ2∑i=1n(∇XiMXi)V,
where XH, XV denote the orthogonal projections of a vector field X on the horizontal and vertical subbundles of TM, respectively.
SinceXi is the horizontal lift of  Xi′(i=1,…,n), we have
(3.7)Xi′(f̃)=1vol(ϕ-1(y)){∫ϕ-1(y)Xi(f)dvϕ-1(y)+∫ϕ-1(y)fLXi(dvϕ-1(y))}=1vol(ϕ-1(y)){∫ϕ-1(y)Xi(f)dvϕ-1(y)+∑α=n+1m∫ϕ-1(y)fg(∇XαMXi,Xα)dvϕ-1(y)}=1vol(ϕ-1(y)){∫ϕ-1(y)Xi(f)dvϕ-1(y)-∫ϕ-1(y)fg(H,Xi)dvϕ-1(y)},
where ℒXi denotes the Lie derivative along Xi. The volume of the fibres does not vary in the horizontal direction because of the relation Xi′(vol(ϕ-1(y)))=-∫ϕ-1(y)g(H,Xi)dvϕ-1(y) and the fact that the fibres are minimal.
Similarly, we obtain(3.8)Xi′∘Xi′(f̃)=1vol(ϕ-1(y)){∫ϕ-1(y)Xi∘Xi(f)dvϕ-1(y)-∫ϕ-1(y)Xi(f)⋅g(H,Xi)dvϕ-1(y)}-1vol(ϕ-1(y)){∫ϕ-1(y)Xi(fg(H,Xi))dvϕ-1(y)-∫ϕ-1(y)f(g(H,Xi))2dvϕ-1(y)}.
The horizontal homothety of the dilation implies that (∇XiMXi)H is the horizontal lift of ∇Xi′NXi′, cf. [9, Lemma 3.1]; therefore, we have
(3.9)∇Xi′NXi′(f̃)=1vol(ϕ-1(y))×{∫ϕ-1(y)(∇XiMXi)H(f)dvϕ-1(y)-∫ϕ-1(y)f⋅g(H,(∇XiMXi)H)dvϕ-1(y)}.
Now using (3.7), (3.8), (3.9), along with the condition that the fibres are minimal, in (3.6) completes the proof.From the above proposition, we see that it suffices to takeλ constant to have both f and f̃ harmonic on M and N, respectively. In this case, by a homothety of M we may suppose that λ≡1 and ϕ is a harmonic Riemannian submersion. We then have the following consequence.Theorem 3.2.
Letϕ:(Mm,g)→(Nn,h)(n≥2) be a harmonic Riemannian submersion with compact, connected fibres. Then the projection f̃:V⊂N→ℝ (via ϕ) of any harmonic function f:U=ϕ-1(V)⊂M→ℝ is a harmonic function. Moreover, ℋf=f̃∘ϕ. If [fℋ] denotes the class of harmonic functions on U=ϕ-1(V) having the same horizontal component then each class [fℋ] has a unique representative in the space of harmonic functions on V.Proof.
SinceΔMf=0 and the dilation λ≡1, Proposition 3.1 leads to
(3.10)ΔNf̃=1vol(ϕ-1(y))∑i=1n∫ϕ-1(y)(∇XiMXi)Vfdvϕ-1(y),
where we have also used the fact that
(3.11)∫ϕ-1(y)Δϕ-1(y)fdvϕ-1(y)=0
for compact fibres.
Let{Xi′}i=1n be a local orthonormal frame for TN and Xi be the horizontal lift of Xi′ for i=1,…,n. Then {Xi}i=1n is a local orthonormal frame for the horizontal distribution. Let {Xα}α=n+1m be a local orthonormal frame for the vertical distribution. Then using the standard expression for Levi-Civita connection, we have
(3.12)(∇XiMXi)V=∑α=n+1mg(∇XiMXi,Xα)Xα=12∑α=n+1m{Xi(g(Xi,Xα))+Xi(g(Xα,Xi))-Xα(g(Xi,Xi))-g(Xi,[Xi,Xα])+g(Xi,[Xα,Xi])+g(Xα,[Xi,Xi])}Xα.
Because Xi are basic, Xα are vertical we have [Xi,Xα] vertical and therefore
(3.13)(∇XiMXi)V=0.
Hence, from  (3.10), f̃ is harmonic. The rest of the proof follows from the construction of f̃.As an application, we give a description of harmonic functions on manifolds admitting harmonic Riemannian submersions with compact fibres.Corollary 3.3.
LetMm be a Riemannian manifold admitting a harmonic Riemannian submersion ϕ:Mm→Nn with compact fibres. Then(1)
every horizontally homothetic harmonic function onU⊂M is horizontal, that is, 𝒱f=0, and so in particular is constant;(2)
every nonhorizontally homothetic harmonic functionf on U⊂M satisfies one of the following:
(a)
𝒱f≠0;(b)
𝒱f=0 and Xi(ℋf)≠0 for at least one i∈{1,…,n};(c)
𝒱f=0, Xi(ℋf)=0 (for all i) and Xi(f) changes sign on the fibre, for at least one i∈{1,…,n}.Proof.
Equation (3.6) implies that a horizontally homothetic harmonic function on M is harmonic on the fibre and hence is constant on the fibre. Now using Lemma 2.3 we get the proof.Remark 3.4.
(1) Since anℝN-valued map f=(f1,…,fN) is harmonic if and only if each of its component is harmonic, we see that Riemannian submersions with compact fibres project ℝN-valued harmonic maps from ϕ-1(V) to ℝN-valued harmonic maps from V.
(2) Given a Lie groupG and a compact subgroup H of G, the standard projection ϕ:G→G/H with G-invariant metric provides many examples satisfying the hypothesis of Theorem 3.2. Further examples can be obtained from Bergery’s construction ϕ:G/K→G/H with K⊂H⊂G and K,H compact; see [10] for the details of the metrics for which ϕ is a harmonic morphism. Another reference for such examples is [11, Chapter 6].



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