Abstract and Applied Analysis

Volume 2012 (2012), Article ID 315757, 8 pages

doi:10.1155/2012/315757

Research Article

Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

M. T. Mustafa

Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 30 January 2012; Revised 30 March 2012; Accepted 31 March 2012

Academic Editor: Saminathan Ponnusamy

Copyright © 2012 M. T. Mustafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 .

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 between Riemannian manifolds is called a harmonic morphism if, for every real-valued function which is harmonic on an open subset of with nonempty, is a harmonic function on .

These maps are related to horizontally (weakly) conformal maps which are a natural generalization of Riemannian submersions.

For a smooth map , let be its critical set. The points of the set are called regular points. For each , the vertical space at is defined by . The horizontal space at is given by the orthogonal complement of in .

Definition 1.2 (see [3, Section 2.4]). A smooth map is called horizontally (weakly) conformal if on and the restriction of to is a conformal submersion, that is, for each , the differential is conformal and surjective. This means that there exists a function such that

By setting on , we can extend to a continuous function on such that is smooth. The extended function is called the dilation of the map.

For , the assignments and define smooth distributions and on or subbundles of , the tangent bundle of . The distributions and are, respectively, called horizontal distribution (or horizontal subbundle) and vertical distribution (or vertical subbundle) defined by .

Recall that a map is said to be harmonic if it extremizes the associated energy integral for every compact domain . It is well known that a map 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 be a horizontally conformal submersion with dilation . If

(1), then is a harmonic map if and only if it has minimal fibres;(2), then two of the following imply the other:

(a) is a harmonic map,(b) has minimal fibres,(c) where denotes the projection of on the horizontal subbundle of , 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 with compact fibres for , we can use fibre integration to define the horizontal and vertical components of every integrable function on at regular points.

Definition 2.1. Let be a smooth map between Riemannian manifolds with compact fibres. Define the horizontal component of an integrable function , on , at a regular point as the average of taken over the fibre . Precisely, for any and integrable function , the horizontal component of at a regular point is defined as

where , , is the volume element of the fibre , is the volume of the fibre , and denotes the integral of .

The vertical component of is given by

Note that the horizontal component of a function depends only on the fibre and not the choice of .

Definition 2.2. Let be a submersion. A function is called horizontally homothetic if the vector field is vertical, that is, at each point is tangent to the fibre.

The components and have the following basic properties for submersions.

Lemma 2.3. Let be a submersion with compact fibres. Suppose that the fibres , are minimal submanifolds of . Consider and a function .

(1)If is horizontally homothetic at , then is also horizontally homothetic at .(2)If is horizontally homothetic at and either or (for all ) on the fibre through , then is horizontally homothetic, where is a local orthonormal frame for the horizontal distribution.(3)If is constant along the fibre through then .

Proof. The proof can be completed by following the calculations in Proposition 3.1.

Definition 2.4. Let be a submersion with compact fibres, and let be an integrable function. The horizontal component of defines a function as

where and . The function is called the projection of on , via the map .

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 , the vector fields on and on are said to be -related if for every . A horizontal vector field on is called basic if it is -related to some vector field on , and is called horizontal lift of . It is well known that for a given vector field on , there exists a unique horizontal lift of such that and are -related.

Proposition 3.1. Let be a nonconstant submersive harmonic morphism with dilation , having compact, connected, and minimal fibres. Then for any and ,

where is as defined in Definition 2.4 and , , are the Laplacians on , , , respectively, is the Levi-Civita connection on , denotes the vertical component of , and denote the horizontal lift of a local orthonormal frame for .

Proof. First notice from Theorem 1.3 that is horizontally homothetic, a fact which will be used repeatedly in the proof.

Choose a local orthonormal frame for . If denotes the horizontal lift of for , then is a local orthonormal frame for the horizontal distribution. Let be a local orthonormal frame for the vertical distribution. Then we can write the Laplacian on as

Now the Laplacian of the fibre is

If is the second fundamental form of the fibre as a submanifold in , we can write as

Let denote the mean curvature vector of given by

Setting , we obtain from (3.2)

where , denote the orthogonal projections of a vector field on the horizontal and vertical subbundles of TM, respectively.

Since is the horizontal lift of , we have

where denotes the Lie derivative along . The volume of the fibres does not vary in the horizontal direction because of the relation and the fact that the fibres are minimal.

Similarly, we obtain

The horizontal homothety of the dilation implies that is the horizontal lift of , cf. [9, Lemma 3.1]; therefore, we have

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 constant to have both and harmonic on and , respectively. In this case, by a homothety of we may suppose that and is a harmonic Riemannian submersion. We then have the following consequence.

Theorem 3.2. Let be a harmonic Riemannian submersion with compact, connected fibres. Then the projection (via ) of any harmonic function is a harmonic function. Moreover, . If denotes the class of harmonic functions on having the same horizontal component then each class has a unique representative in the space of harmonic functions on .

Proof. Since and the dilation , Proposition 3.1 leads to

where we have also used the fact that

for compact fibres.

Let be a local orthonormal frame for TN and be the horizontal lift of for . Then is a local orthonormal frame for the horizontal distribution. Let be a local orthonormal frame for the vertical distribution. Then using the standard expression for Levi-Civita connection, we have

Because are basic, are vertical we have vertical and therefore

Hence, from (3.10), is harmonic. The rest of the proof follows from the construction of .

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

Corollary 3.3. Let be a Riemannian manifold admitting a harmonic Riemannian submersion with compact fibres. Then

(1)every horizontally homothetic harmonic function on is horizontal, that is, , and so in particular is constant;(2)every nonhorizontally homothetic harmonic function on satisfies one of the following:

(a);(b) and for at least one ;(c), (for all ) and changes sign on the fibre, for at least one .

Proof. Equation (3.6) implies that a horizontally homothetic harmonic function on 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 -valued map is harmonic if and only if each of its component is harmonic, we see that Riemannian submersions with compact fibres project -valued harmonic maps from to -valued harmonic maps from .

(2) Given a Lie group and a compact subgroup of , the standard projection with -invariant metric provides many examples satisfying the hypothesis of Theorem 3.2. Further examples can be obtained from Bergery’s construction with and compact; see [10] for the details of the metrics for which 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.

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