Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

Abstract

For Riemannian manifolds 𝑀 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.

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 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 [1] and Ishihara [2].

Definition 1.1. A smooth map 𝜙∶𝑀^𝑚→𝑁^𝑛 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 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. For each 𝑥∈𝑀⧵𝐶_𝜙, the vertical space at 𝑥 is defined by 𝑇^𝑉_𝑥𝑀=Ker𝑑𝜙_𝑥. The horizontal space 𝑇^𝐻_𝑥𝑀 at 𝑥 is given by the orthogonal complement of 𝑇^𝑉_𝑥𝑀 in 𝑇_𝑥𝑀.

Definition 1.2 (see [3, Section 2.4]). A smooth map 𝜙∶(𝑀^𝑚,𝐠)→(𝑁^𝑛,𝐡) is called horizontally (weakly) 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_𝑥𝑀→𝑇_𝜙(𝑥)𝑁 is conformal and surjective. This means that there exists a function 𝜆∶𝑀⧵𝐶_𝜙→ℝ⁺ such that 𝐡𝑑𝜙(𝑋),𝑑𝜙(𝑌)=𝜆²𝐠(𝑋,𝑌),∀𝑋,𝑌∈𝑇^𝐻_𝑥𝑀. (1.1)

By setting 𝜆=0 on 𝐶_𝜙, we can extend 𝜆∶𝑀→ℝ⁺₀ to a continuous function on 𝑀 such that 𝜆² is smooth. The extended function 𝜆∶𝑀→ℝ⁺₀ is called the dilation of the map.

For 𝑥∈𝑀⧵𝐶_𝜙, the assignments 𝑥↦𝑇^𝐻_𝑥𝑀 and 𝑥↦𝑇^𝑉_𝑥𝑀 define smooth distributions 𝑇^𝐻𝑀 and 𝑇^𝑉𝑀 on 𝑀⧵𝐶_𝜙 or subbundles of 𝑇𝑀|_{𝑀⧵𝐶𝜙}, the tangent bundle of 𝑀⧵𝐶_𝜙. The distributions 𝑇^𝐻𝑀 and 𝑇^𝑉𝑀 are, respectively, called horizontal distribution (or horizontal subbundle) and vertical distribution (or vertical subbundle) defined by 𝜙.

Recall that a map 𝜙∶𝑀^𝑚→𝑁^𝑛 is said to be harmonic if it extremizes the associated energy integral 𝐸(𝜙)=(1/2)∫_Ω‖𝜙_∗‖²𝑑𝜐^𝑀 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. Let 𝜙∶𝑀^𝑚→𝑁^𝑛 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)𝐠𝐫𝐚𝐝^𝐻𝜆²=0 where 𝐠𝐫𝐚𝐝^𝐻𝜆² 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 map 𝜙∶𝑀^𝑚→𝑁^𝑛 with compact fibres 𝜙⁻¹(𝜙(𝑥)) for 𝑥∈𝑀⧵𝐶_𝜙, 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. Let 𝜙∶𝑀^𝑚→𝑁^𝑛 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 (𝑥)=1ℋ𝑓vol𝜙⁻¹𝑦_𝜙⁻¹_(𝑦)𝑓𝑑𝑣^{𝜙−1(𝑦)}𝜙(𝑥), (2.1) where 𝑥∈𝑈, 𝜙(𝑥)=𝑦, 𝑑𝑣^{𝜙−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)}𝑓𝑑𝑣^{𝜙−1(𝑦)})(𝜙(𝑥)) denotes the integral of 𝑓|_{𝜙−1(𝜙(𝑥))}.
The vertical component of 𝑓 is given by 𝒱𝑓(𝑥)=𝑓−ℋ𝑓(𝑥). (2.2)
Note that the horizontal component of a function depends only on the fibre 𝜙⁻¹(𝑦) and not the choice of 𝑥∈𝜙⁻¹(𝑦).

Definition 2.2. Let 𝜙∶𝑀^𝑚→𝑁^𝑛 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 components ℋ𝑓 and 𝒱𝑓 have the following basic properties for submersions.

Lemma 2.3. Let 𝜙∶𝑀^𝑚→𝑁^𝑛 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)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 {𝑋_𝑖}^𝑛_𝑖=1 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 3.1.

Definition 2.4. Let 𝜙∶𝑀^𝑚→𝑁^𝑛 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 𝑓∶𝑉⊂𝑁→ℝ as 𝑓𝑦=ℋ𝑓(𝑥), (2.3) where 𝑥∈𝑈 and 𝑦=𝜙(𝑥). The function 𝑓 is called the projection of 𝑓 on 𝑁, via the map 𝜙.

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 submersion 𝜙∶𝑀^𝑚→𝑁^𝑛, the vector fields 𝑋 on 𝑀 and 𝑌 on 𝑁 are said to be 𝜙-related if 𝑑𝜙(𝑋_𝑥)=𝑌_𝜙(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. Let 𝜙∶(𝑀^𝑚,𝐠)→(𝑁^𝑛,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, Δ^𝑁1𝑓=vol𝜙⁻¹𝑦_𝜙⁻¹_(𝑦)1𝜆²Δ^𝑀𝑓−Δ^{𝜙−1(𝑦)}𝑓𝑑𝑣^{𝜙−1(𝑦)}+1vol𝜙⁻¹𝑦^𝑛_𝑖=1𝜙⁻¹_(𝑦)∇^𝑀_𝑋𝑖𝑋_𝑖^𝑉𝑓𝑑𝑣^{𝜙−1(𝑦)}, (3.1) where 𝑥∈𝑈,𝜙(𝑥)=𝑦,𝑓 is as defined in Definition 2.4 and Δ^𝑀, Δ^𝑁, Δ^{𝜙−1(𝑦)} are the Laplacians on 𝑀, 𝑁, 𝜙⁻¹(𝑦), respectively, ∇^𝑀 is the Levi-Civita connection on 𝑀, (∇^𝑀_𝑋𝑖𝑋_𝑖)^𝑉 denotes the vertical component of ∇^𝑀_𝑋𝑖𝑋_𝑖, and {𝑋_𝑖}^𝑛_𝑖=1 denote the horizontal lift of a local orthonormal frame {𝑋^′_𝑖}^𝑛_𝑖=1 for 𝑇𝑁.

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 𝑇𝑁. 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 Δ^𝑀 on 𝑀 as Δ^𝑀=^𝑛_𝑖=1𝜆𝑋_𝑖∘𝜆𝑋_𝑖−∇^𝑀_𝜆𝑋𝑖𝜆𝑋_𝑖+^𝑚_𝛼=𝑛+1𝑋_𝛼∘𝑋_𝛼−∇^𝑀_𝑋𝛼𝑋_𝛼=𝜆^2𝑛_𝑖=1𝑋_𝑖∘𝑋_𝑖−∇^𝑀_𝑋𝑖𝑋_𝑖+^𝑚_𝛼=𝑛+1𝑋_𝛼∘𝑋_𝛼−∇^𝑀_𝑋𝛼𝑋_𝛼. (3.2) Now the Laplacian of the fibre 𝜙⁻¹(𝑦) is Δ^{𝜙−1(𝑦)}=^𝑚_𝛼=𝑛+1𝑋_𝛼∘𝑋_𝛼−∇^𝜙−1_𝑋^(𝑦)_𝛼𝑋_𝛼. (3.3) If 𝐵 is the second fundamental form of the fibre 𝜙⁻¹(𝑦) as a submanifold in 𝑀, we can write ∇^𝑀_𝑋𝛼𝑋_𝛼 as ∇^𝑀_𝑋𝛼𝑋_𝛼=∇^𝜙−1_𝑋^(𝑦)_𝛼𝑋_𝛼+𝐵𝑋_𝛼,𝑋_𝛼. (3.4) Let 𝜇 denote the mean curvature vector of 𝜙⁻¹(𝑦) given by 1𝜇=𝑚−𝑛^𝑚_𝛼=𝑛+1𝐵𝑋_𝛼,𝑋_𝛼. (3.5) Setting 𝐻=(𝑚−𝑛)𝜇, we obtain from (3.2) Δ^𝑀=𝜆^2𝑛_𝑖=1𝑋_𝑖∘𝑋_𝑖−∇^𝑀_𝑋𝑖𝑋_𝑖^𝐻+Δ^{𝜙−1(𝑦)}−𝐻−𝜆^2𝑛_𝑖=1∇^𝑀_𝑋𝑖𝑋_𝑖^𝑉=𝜆^2𝑛_𝑖=1𝑋_𝑖∘𝑋_𝑖−∇^𝑀_𝑋𝑖𝑋_𝑖^𝐻+Δ^{𝜙−1(𝑦)}−𝜆^2𝑛_𝑖=1∇^𝑀_𝑋𝑖𝑋_𝑖^𝑉, (3.6) where 𝑋^𝐻, 𝑋^𝑉 denote the orthogonal projections of a vector field 𝑋 on the horizontal and vertical subbundles of TM, respectively.
Since 𝑋_𝑖 is the horizontal lift of 𝑋^′_𝑖(𝑖=1,…,𝑛), we have 𝑋^′_𝑖𝑓=1vol𝜙⁻¹𝑦_𝜙⁻¹_(𝑦)𝑋_𝑖𝑓𝑑𝑣^{𝜙−1(𝑦)}+_{𝜙−1(𝑦)}𝑓ℒ_𝑋𝑖𝑑𝑣^{𝜙−1(𝑦)}=1vol𝜙⁻¹𝑦_𝜙⁻¹_(𝑦)𝑋_𝑖𝑓𝑑𝑣^{𝜙−1(𝑦)}+^𝑚_{𝛼=𝑛+1𝜙−1(𝑦)}𝑓𝐠∇^𝑀_𝑋𝛼𝑋_𝑖,𝑋_𝛼𝑑𝑣^{𝜙−1(𝑦)}=1vol𝜙⁻¹𝑦_𝜙⁻¹_(𝑦)𝑋_𝑖𝑓𝑑𝑣^{𝜙−1(𝑦)}−_{𝜙−1(𝑦)}𝑓𝐠𝐻,𝑋_𝑖𝑑𝑣^{𝜙−1(𝑦)}, (3.7) where ℒ_𝑋𝑖 denotes the Lie derivative along 𝑋_𝑖. 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)𝑑𝑣^{𝜙−1(𝑦)} and the fact that the fibres are minimal.
Similarly, we obtain 𝑋^′_𝑖∘𝑋^′_𝑖𝑓=1vol𝜙⁻¹𝑦_𝜙⁻¹_(𝑦)𝑋_𝑖∘𝑋_𝑖𝑓𝑑𝑣^{𝜙−1(𝑦)}−_{𝜙−1(𝑦)}𝑋_𝑖𝑓⋅𝐠𝐻,𝑋_𝑖𝑑𝑣^{𝜙−1(𝑦)}−1vol𝜙⁻¹𝑦_𝜙⁻¹_(𝑦)𝑋_𝑖𝑓𝐠𝐻,𝑋_𝑖𝑑𝑣^{𝜙−1(𝑦)}−_{𝜙−1(𝑦)}𝑓𝐠𝐻,𝑋_𝑖²𝑑𝑣^{𝜙−1(𝑦)}. (3.8) The horizontal homothety of the dilation implies that (∇^𝑀_𝑋𝑖𝑋_𝑖)^𝐻 is the horizontal lift of ∇^𝑁_𝑋′𝑖𝑋^′_𝑖, cf. [9, Lemma 3.1]; therefore, we have ∇^𝑁_𝑋′𝑖𝑋^′_𝑖𝑓=1vol𝜙⁻¹𝑦×_{𝜙−1(𝑦)}∇^𝑀_𝑋𝑖𝑋_𝑖^𝐻𝑓𝑑𝑣^{𝜙−1(𝑦)}−_{𝜙−1(𝑦)}𝑓⋅𝐠𝐻,∇^𝑀_𝑋𝑖𝑋_𝑖^𝐻𝑑𝑣^{𝜙−1(𝑦)}. (3.9) 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 take 𝜆 constant to have both 𝑓 and 𝑓 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. Let 𝜙∶(𝑀^𝑚,𝐠)→(𝑁^𝑛,𝐡)(𝑛≥2) be a harmonic Riemannian submersion with compact, connected fibres. Then the projection 𝑓∶𝑉⊂𝑁→ℝ (via 𝜙) of any harmonic function 𝑓∶𝑈=𝜙⁻¹(𝑉)⊂𝑀→ℝ is a harmonic function. Moreover, ℋ𝑓=𝑓∘𝜙. If [𝑓_ℋ] denotes the class of harmonic functions on 𝑈=𝜙⁻¹(𝑉) having the same horizontal component then each class [𝑓_ℋ] has a unique representative in the space of harmonic functions on 𝑉.

Proof. Since Δ^𝑀𝑓=0 and the dilation 𝜆≡1, Proposition 3.1 leads to Δ^𝑁1𝑓=vol𝜙⁻¹𝑦^𝑛_𝑖=1𝜙⁻¹_(𝑦)∇^𝑀_𝑋𝑖𝑋_𝑖^𝑉𝑓𝑑𝑣^{𝜙−1(𝑦)}, (3.10) where we have also used the fact that _{𝜙−1(𝑦)}Δ^{𝜙−1(𝑦)}𝑓𝑑𝑣^{𝜙−1(𝑦)}=0 (3.11) for compact fibres.
Let {𝑋^′_𝑖}^𝑛_𝑖=1 be a local orthonormal frame for TN and 𝑋_𝑖 be the horizontal lift of 𝑋^′_𝑖 for 𝑖=1,…,𝑛. Then {𝑋_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 using the standard expression for Levi-Civita connection, we have ∇^𝑀_𝑋𝑖𝑋_𝑖^𝑉=^𝑚_𝛼=𝑛+1𝐠∇^𝑀_𝑋𝑖𝑋_𝑖,𝑋_𝛼𝑋_𝛼=12^𝑚_𝛼=𝑛+1𝑋_𝑖𝐠𝑋_𝑖,𝑋_𝛼+𝑋_𝑖𝐠𝑋_𝛼,𝑋_𝑖−𝑋_𝛼𝐠𝑋_𝑖,𝑋_𝑖−𝐠𝑋_𝑖,𝑋_𝑖,𝑋_𝛼+𝐠𝑋_𝑖,𝑋_𝛼,𝑋_𝑖+𝐠𝑋_𝛼,𝑋_𝑖,𝑋_𝑖𝑋_𝛼. (3.12) Because 𝑋_𝑖 are basic, 𝑋_𝛼 are vertical we have [𝑋_𝑖,𝑋_𝛼] vertical and therefore ∇^𝑀_𝑋𝑖𝑋_𝑖^𝑉=0. (3.13) Hence, from (3.10), 𝑓 is harmonic. The rest of the proof follows from the construction of 𝑓.

As an application, we give a description of harmonic functions on manifolds admitting harmonic Riemannian submersions with compact fibres.

Corollary 3.3. Let 𝑀^𝑚 be a Riemannian manifold admitting a harmonic Riemannian submersion 𝜙∶𝑀^𝑚→𝑁^𝑛 with compact fibres. Then(1)every horizontally homothetic harmonic function on 𝑈⊂𝑀 is horizontal, that is, 𝒱𝑓=0, and so in particular is constant;(2)every nonhorizontally homothetic harmonic function 𝑓 on 𝑈⊂𝑀 satisfies one of the following:(a)𝒱𝑓≠0;(b)𝒱𝑓=0 and 𝑋_𝑖(ℋ𝑓)≠0 for at least one 𝑖∈{1,…,𝑛};(c)𝒱𝑓=0, 𝑋_𝑖(ℋ𝑓)=0 (for all 𝑖) and 𝑋_𝑖(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 an ℝ^𝑁-valued map 𝑓=(𝑓¹,…,𝑓^𝑁) is harmonic if and only if each of its component is harmonic, we see that Riemannian submersions with compact fibres project ℝ^𝑁-valued harmonic maps from 𝜙⁻¹(𝑉) to ℝ^𝑁-valued harmonic maps from 𝑉.
(2) Given a Lie group 𝐺 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].

Acknowledgments

The author is thankful to the referee for valuable comments that have improved the quality of the paper. The author would also like to acknowledge the support and research facilities provided by King Fahd University of Petroleum and Minerals, Dhahran.