The Structure of Harmonic Morphisms with Totally

Available at: http://www.ictp.trieste.it/~pub off IC/99/90

United Nations Educational Scienti c and Cultural Organization

and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE STRUCTURE OF HARMONIC MORPHISMS

WITH TOTALLY GEODESIC FIBRES

M.T. Mustafa

Faculty of Engineering Sciences,

GIK Institute of Enginneering Sciences and Technology,

Topi-23460, N.W.F.P., Pakistan

and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

The structure of lo cal and global harmonic morphisms b etween Riemannian manifolds, with

totally geo desic bres, is investigated. It is shown that non-p ositive curvature obstructs the

existence of global harmonic morphisms with totally geo desic bres and the only such maps from

compact Riemannian manifolds of non-p ositive curvature are, up to a homothety, totally geo desic

Riemannian submersions. Similar results are obtained for lo cal harmonic morphisms with totally

geo desic bres from op en subsets of non-negatively curved compact and non-compact manifolds.

During the course, of this work we prove the non-existence of submersive harmonic morphisms

with totally geo desic bres from some imp ortant domains, for instance from compact lo cally

symmetric spaces of non-compact typ e and op en subsets of symmetric spaces of compact typ e.

MIRAMARE { TRIESTE

August 1999

Regular Asso ciate of the Ab dus Salam ICTP. E-mail: [email protected]

1. Introduction

Harmonic morphisms are maps b etween Riemannian manifolds which preserve germs of har-

monic functions, i.e. these lo cally pull back real-valued harmonic functions to real-valued har-

monic functions. These are characterized as a sub class of harmonic maps, precisely, these are

harmonic maps which are horizontally weakly conformal. What is sp ecial ab out this character-

ization is that it endows harmonic morphisms with analytic as well as geometric prop erties. On

the other hand, it puts strong restrictions on their existence as solutions of an over-determined

system of partial di erential equations. The purp ose of this article is to study questions re-

lated to the existence and structure of harmonic morphisms, with totally geo desic bres, from

compact and non-compact Riemannian manifolds.

The Bo chner technique, in its natural setting, is a metho d to investigate obstructions to the

existence of geometric ob jects on p ositively curved compact manifolds. The technique mainly

relies on the development of a suitable Laplacian identity and its analysis to explore restrictions

on the existence of the ob jects under study. Following the usual Bo chner technique and the

work of Eells-Sampson [7], the author develop ed a Bo chner technique for harmonic morphisms

in [17 ] and presented restrictions on the existence of harmonic morphisms from p ositively curved

compact Riemannian manifolds. This was further extended in [18 ] to include some non-compact

domains.

The conventional Bo chner technique, though very powerful, is not a handy to ol to explore

restrictions on the existence of geometric ob jects on negatively curved compact domains. Until

now, the investigation of general restrictions on harmonic morphisms from compact negatively

curved manifolds is limited to the following cases.

m n

Case 1: [8] There exist no non-constant Riemannian submersions : M ! N m >n

with totally geo desic bres if M has negative sectional curvature.

Case 2: [2] Every harmonic morphism from the compact quotients of the hyp erb olic space

H to a Riemann surface is constant.

Case 3: [19 ] Any non-constant submersive harmonic morphism from a compact Riemannian

n+1 n

manifold M to N , such that Ric U; U 0 for U vertical, is totally geo desic.

Realizing that all the maps considered ab ove are particular cases of submersive harmonic

morphisms with totally geo desic bres, we take a uni ed approach and study the restrictions

on the existence of harmonic morphisms with totally geo desic bres and their structure, in case

these exist. In order to do so, we develop a variant of the usual Bo chner technique by develop-

ing a generalized Bo chner typ e formula which leads to obtaining restrictions on the existence of

harmonic morphisms, with totally geo desic, from negatively curved compact Riemannian man-

ifolds. These restrictions contain the ab ove results as particular cases. A comparitive study of

this generalized Bo chner typ e formula with the usual one [17 , Prop osition 2.1] provides lo cal

non-existence results for harmonic morphisms, with totally geo desic bres, from non-negatively

curved Riemannian manifolds. As a nal consequence, we obtain a classi cation of submer-

sive harmonic morphisms, having totally geo desic bres, from op en subsets of R to complete

manifolds of non-p ositive scalar curvature.

A conventional remark: The sign convention adopted for the curvature is the one that coincides

with the classical curvature tensor i.e. for vector elds X , Y , the curvature R of a connection

r is

RX; Y =r r + r r + r :

X Y Y X

[X;Y ]

2. Harmonic morphisms

The formal theory of harmonic morphisms between Riemannian manifolds b egan with the

work of Fuglede [9 ] and Ishihara [14 ].

m n

De nition 2.1. A smo oth map : M ! N between Riemannian manifolds is called a

harmonic morphism if, for every real-valued function f which is harmonic on an op en subset U

1 1

of N with U non-empty, f is a harmonic function on U .

Harmonic morphisms are related to horizontally weakly conformal maps which can b e de-

ned in the following manner.

m n

For a smo oth map : M ! N , let C = fx 2 M jrankd < ng be its critical set. The

p oints of the set M n C are called regular points. For each x 2 M n C , the vertical space at x is

de ned by V = Kerd . The horizontal space H at x is given by the orthogonal complement

x x x

of V in T M .

x x

m n

De nition 2.2. A smo oth map :M ; g ! N ; h is called horizontal ly weakly conformal

if d =0 on C and the restriction of to M n C is a conformal submersion, that is, for each

M ! T N is conformal and surjective. This means that x 2 M n C , the di erential d : T

there exists a function : M n C ! R such that

2 H

M: hdX ;dY = g X; Y 8X; Y 2 T

By setting =0 on C ,we can extend : M ! R to a continuous function on M such that

is smo oth. The extended function : M ! R is called the dilation of the map.

2 2 2

Let grad and grad denote the horizontal and vertical pro jections of grad .

H V

m n

De nition 2.3. A smo oth map : M ! N is called horizontal ly homothetic if it is a

horizontally conformal submersion whose dilation is constant along the horizontal curves i.e.

grad =0.

m n

Recall that a map : M ! N is said to b e harmonic if it extremizes the asso ciated energy

2 M

k k d for every compact domain M . It is well-known that a map integral E =

is harmonic if and only if its tension eld vanishes.

Harmonic morphisms can b e viewed as a sub class of harmonic maps in the light of the following

characterization, obtained in [9 , 14 ].

A smooth map is a harmonic morphism if and only if it is harmonic and horizontal ly weakly

conformal.

The following result of Baird-Eells [3 , Riemannian case] and Gudmundsson [12 , semi-Riemannian

case] re ects a signi cant geometric feature of harmonic morphisms.

m n

Theorem 2.4. Let : M ! N bea horizontal ly conformal submersion with dilation . If

1. n =2, then is a harmonic map if and only if it has minimal bres.

2. n 3, then two of the fol lowing imply the other,

a is a harmonic map

b has minimal bres

c is horizontal ly homothetic.

The notion of horizontally conformal maps is a natural generalization of Riemannian submer-

sions. The fundamental equations of Riemannian submersions were generalized for horizontally

conformal submersions in [11 ]. We state those results which will b e needed in the pro of of the

Bo chner formula and refer the reader to [11 ] for complete details.

If T and A denote the standard fundamental tensors of a horizontally conformal submersion

then the relation of the integrabilityof horizontal distribution with the tensor A is given by

the following expression.

1 1

2.1 V [X; Y ] g X; Y grad X; Y horizontal: A Y =

Moreover, the mixed sectional curvatures of the domain satisfy the following relation.

m n

Prop osition 2.5. [11 ] Let :M ; g ! N ; h be a horizontal ly conformal submersion with

total ly geodesic bres. If X , Y are horizontal vectors and U , V are vertical vectors then

2.2 g R X; U Y; V = g r A Y; V +g A U; A V

U X X Y

+ g A Y; U g grad ;V

where is the dilation.

For the fundamental results and prop erties of harmonic morphisms, the reader is referred to

[1 , 6, 9, 20 ] and for an up dated online bibliography to [13 ].

3. The Bochner type formula

In this section we develop the generalized Bo chner typ e formula, which will be the main

to ol in the next section. To establish the formula, we consider a horizontally homothetic map

with totally geo desic bres equivalently a submersive harmonic morphism with totally geo desic bres and compute the Laplacian of the dilation.

m n

Prop osition 3.1. Let :M ; g !N ; h be a non-constant horizontal ly homothetic map

with total ly geodesic bres. If denotes the dilation of then

n m

X X

2 2 M

= 3.1 g A e ;A e g R e ;e e ;e

e i e i i i

i=n+1

1 1 nn 4

g grad ; grad +

V V

2 2

n m

where e and e are local orthonormal frames for the horizontal and vertical distri-

i=n+1

butions respectively.

Proof. Taking the Laplacian on functions as

f = div grad f ;

wehave, for the dilation of

1 1 1

2 2 2

= +2g grad ; grad

V V

2 2 2

1 1 1

2 4

= g grad 2 ; grad

V V

2 2 2

m n

X X

1 1

2 2

g r grad g r grad ;e + ;e =

e e i

V V

2 2

i=n+1

1 1

2 g grad ; grad :

V V

2 2

Since

1 1

g r ;e =g grad ;A grad e ;

e e

V V

2 2

wehave

1 1 1 n 4 1

4 2 2

g grad g r grad 3.2 = ; grad + ;e :

e i

V V V

2 2 2 2

i=n+1

Now a straightforward computation using Equation 2.1 and the easily seen relation g r A e ;e =

e e i

g r A e ;e implies that

e e i

3.3

m m m

X X X

1 1

2 2

g A e ;e g grad g r A e ;e 2 ;e =2 ;e : g r grad

e i e e i i i e

i i

2 2

i=n+1 i=n+1 i=n+1

Using Equation 3.3 and Equation 2.2 we can write Equation 3.2 as

1 1 1 n 4

2 6 2

= g grad ; grad + g A e ;A e

e i e i

V V

2 2

i=n+1

2 M

3.4 g R e ;e e ;e :

i i

i=n+1

Since the ab ove identity holds for each e for =1;::: ;n, therefore, summing over completes

the pro of.

The ab ove formula will naturally be useful to obtain consequences for maps from compact

Riemannian manifolds. For the general domains, a comparison of this Bo chner typ e formula

with the Weitzenbock formula of [17 , Prop osition 2.1] yields the following identity.

m n

Prop osition 3.2. If the Riemannian manifolds M , N admit a submersive harmonic mor-

m n

phism : M ! N having total ly geodesic bres then the fol lowing identity is satis ed.

n n

X X

2 4 N 2 M

3.5 krd k + Scal g R e ;e e ;e

n m

X X

1 1 nn 4

6 2

g grad ; grad g A e ;A e + =

e i e i

V V

2 2

i=n+1

n m

where is the dilation of and e , e are local orthonormal frames for the hori-

i=n+1

zontal, vertical distributions respectively.

Proof. The Weitzenbock formula of [17 , Prop osition 2.1] says that the dilation of a harmonic

morphism satis es

2 2 4 N 2 M

= krdk + Scal Ric e ;e :

Writing

m n n n n

X X X X X

M M M

g R e ;e e ;e g R e ;e e ;e + Ric e ;e =

i i

=1 =1 =1

i=n+1

and comparing the ab oveWeitzenbock formula with Equation 3.1 gives the required identity.

4. Applications to harmonic morphisms with totally geodesic fibres

This section is devoted to the analysis of the structure of globally as well as lo cally de ned har-

monic morphisms with totally geo desic bres. Throughout the section we assume the following

range of indices:

1 ; n; n +1 i; j m

n m

Moreover, e , e will denote lo cal orthonormal frames for the horizontal, vertical

i=n+1

m n

distributions, resp ectively, induced by a horizontally conformal map : M ! N .

4.1. Global harmonic morphisms with totally geo desic bres. The main result, obtained

by applying Prop osition 3.1, is that the negative curvature of compact domains obstructs the

existence of harmonic morphisms with totally geo desic bres, thus making the class of such

harmonic morphisms very restricted on domains of non-p ositive curvature. Combining the

results of this section with Theorem 2.5 of [17 ] gives a much clearer picture of the structure

of harmonic morphisms from compact manifolds, having totally geo desic bres cf. remarks

following Corollary 4.4.

First we consider harmonic morphisms of dimension> 1.

m n

Theorem 4.1. Let : M ! N n 4;m n +2 be a non-constant submersive harmonic

morphism from a compact Riemannian manifold such that the sectional curvature K e ^ e

0 8 ; i.

1. If has total ly geodesic bres then it has constant dilation and integrable horizontal dis-

tribution i.e. up to a homothety it is a total ly geodesic Riemannian submersion.

2. If K e ^ e < 0 at some point for at least one pair of ; i then thereare no non-constant

submersive harmonic morphisms with total ly geodesic bres from M to N .

Proof. From Stokes' theorem and Prop osition 3.1

n m

X X

M M 2

0 = g A e ;A e g R e ;e e ;e

e i e i i i

i=n+1

1 1 nn 4

6 M

g grad ; grad : +

V V

2 2

Now the hyp othesis and a standard Bo chner typ e argument forces each term on the right-hand

side of Equation 3.1 to vanish, which completes the pro of.

For harmonic morphisms with one dimensional bres, the condition on the sectional curvature

can b e replaced by Ricci curvature to obtain b etter consequences.

n+1 n

Corollary 4.2. Let : M ! N n 4 be a non-constant harmonic morphism between

compact Riemannian manifolds. Let Ric j denote the Ricci curvature of M restricted to the

bres. i.e.

M M

Ric j = Ric e ;e

V n+1 n+1

for a vertical unit vector e .

n+1

1. If Ric j 0 then, up to a homothety, is a total ly geodesic Riemannian submersion.

2. If Ric j < 0 at some point then does not exist.

3. In particular, if M is Ricci- at then, up to a homothety, is a total ly geodesic Riemannian

submersion, and the bres, the horizontal submanifolds and N are al l Ricci- at.

n+1 n

Proof. First notice that : M ! N n 4 is submersive, as shown in [2]. Moreover, the

bres are automatically totally geo desic. NowParts 1,2 are similar to the pro of of Theorem 4.1,

since

M M

Ric e ;e = g R e ;e e ;e :

n+1 n+1 n+1 n+1

For Part 3 we see that , b eing a totally geo desic Riemannian submersion, makes the horizontal

and vertical foliations Riemannian with totally geo desic leaves. Hence, the Ricci curvatures of

the bres, the horizontal submanifolds and the target manifold vanish b ecause of Ricci- atness

of M .

The reader is referred to [19 , Corollary 3.6] where Part 1 of the ab ove result is proved by a

di erent approach.

Next, we discuss examples satisfying the hyp othesis of Theorem 4.1.

n+1

Corollary 4.3. Let M bea compact Riemannian manifold. Then there exists a metric g on

n+1 n

M such that there are no non-constant harmonic morphisms :M ; g ! N for n 4.

Proof. Due to Lohkamp [16 , Corollary 5.2], any manifold of dimension 3 carries a Riemannian

metric of negative Ricci curvature and hence, from ab ovewehave the non-existence result.

By considering compact lo cally symmetric spaces of non-compact typ e, we get a number of

examples where Theorem 4.1 can be applied to obtain restrictions or non-existence results for

harmonic morphisms with totally geo desic bres.

Corollary 4.4.

1. There are no non-constant submersive harmonic morphisms with total ly geodesic bres

from compact local ly symmetric spaces M m > n of non-compact type and rank 1 to

any Riemannian manifold N n 4.

2. Every non-constant submersive harmonic morphism with total ly geodesic bres, from a

compact irreducible local ly symmetric space M m>n of non-compact type and rank 2

to any Riemannian manifold N n 4 is, upto a homothety, a total ly geodesic Riemann-

ian submersion.

Proof.

1. Follows from the fact that all compact lo cally symmetric spaces M m 3 of non-compact

typ e and rank 1 have negative sectional curvature.

2. Since every compact irreducible lo cally symmetric space M of non-compact typ e and

rank 2 has the sectional curvature K 0 but K 6< 0, the pro of follows from ab ove.

We refer the reader to [12 ] where, in contrast to the ab ove result, Gudmundsson has given an

armative answer to the global existence question for harmonic morphisms from irreducible

symmetric spaces of non-compact typ e and rank 1.

If we fo cus only on manifolds whose sectional curvatures do not change sign i.e. either K 0

everywhere or K 0 everywhere. Then combining Theorem 4.1 with [17 , Theorem 2.5] we

have the following picture of the structure of submersive harmonic morphisms with totally

m n

geo desic bres from compact Riemannian manifolds M to compact Riemannian manifolds N

such that n 4 and m>n and the sectional curvatures of M , N do not change sign.

M M

1. If K 0 and K < 0 at some p oint then there are no non-constant submersive harmonic

morphisms with totally geo desic bres.

2. If K 0 then every submersive harmonic morphism with totally geo desic bres is, up

to a homothety, a totally geo desic Riemannian submersion, and the bres, the horizontal

submanifolds and the target manifold N are all at.

M M

3. If K 0 and K 6 0 then only p ossible non-constant submersive harmonic morphisms

N N

with totally geo desic bres are when Scal 0 and Scal 6 0. In which case, either the

horizontal distribution can b e integrable or the dilation can b e constant but b oth cannot o ccur.

4.2. Lo cal harmonic morphisms with totally geo desic bres. The obstruction theory for

lo cal harmonic morphisms with totally geo desic bres has an interesting contrast to the global

case in the sense that instead of negative curvature it is the p ositive curvature which imp oses

restrictions on the existence of lo cal harmonic morphisms with totally geo desic bres.

m n

For a horizontally conformal submersion : M ; g ! N , let Scal be the curvature

given by

M M

Scal = g R e ;e e ;e

; =1

for a lo cal orthonormal frame e of the horizontal distribution. When the horizontal

M H

distribution is integrable and totally geo desic Scal coincides with the scalar curvature Scal

of the horizontal submanifolds.

Theorem 4.5. Let M be a Riemannian manifold with Scal 0, U be a connected open

subset of M and N be a Riemannian manifold with Scal 0. Then every non-constant

n 4 having total ly geodesic bres, is up to submersive harmonic morphism : U ! N

a homothety, a Riemannian submersion with total ly geodesic bres and integrable horizontal

distributution. Furthermore,

1. the scalar curvature of the horizontal submanifolds is zero.

2. If Scal < 0 at some point, then such cannot exist.

Proof. The hyp othesis combined with Prop osition 3.2 makes each term in Equation 3.5 vanish,

which makes the dilation constant and the tensor A 0. The rest is immediate, since the

horizontal foliation b ecomes Riemannian with totally geo desic leaves.

Theorem 4.5 enables us to nd a lo cal analogue of the results of [17 ] where restrictions on the

existence of global harmonic morphisms from Riemannian symmetric spaces of compact typ e

were obtained.

Corollary 4.6. Let M be an irreducible Riemannian symmetric space of compact type, U a

connected open subset of M and N n 4 any Riemannian manifold.

N N

1. If Scal 0 but Scal 6 0 then there exists no non-constant submersive harmonic

morphism : U ! N with total ly geodesic bres.

2. In particular, there exists a metric h on N such that thereare no non-constant submersive

harmonic morphisms : U ! N ; h having total ly geodesic bres.

3. If Scal 0 then every non-constant submersive harmonic morphism : U ! N with

total ly geodesic bres, is up to a homothety, a total ly geodesic Riemannian submersion.

Proof.

1. The pro of follows from the fact that irreducible Riemannian symmetric space of compact

typ e have non-negative sectional curvature.

2. As n>3, there exists a Riemannian metric of negative Ricci curvature on N by [16 ].

3. Immediate from the ab ove.

A classi cation of lo cal and global harmonic morphisms from 3-dimensional simply-connected

space-forms was found in [4 , 5] which was generalized to higher dimensions in [15 ] for global

harmonic morphisms from R having totally geo desic bres. For lo cal case, a classi cation of

harmonic morphisms, with totally geo desic bres and integrable horizontal distribution, b etween

higher-dimensional simply-connected space-forms was obtained in [10 ]. Here, we extend the

classi cation of [15 ] to lo cal harmonic morphisms with totally geo desic bres.

m n

Corollary 4.7. Let U be a connected open subset of R and : U ! N n 4 be a non-

constant submersive harmonic morphism with total ly geodesic bres.

1. If N is complete, simply-connected with Scal 0 then is an orthogonal projection

fol lowed by a homothety.

> 0 then has a non-constant dilation. 2. If Scal

Proof.

1. From Theorem 4.5, up to a homothety, is a Riemannian submersion with totally geo desic

bres and integrable horizontal distribution. Therefore, the horizontal submanifolds are

totally geo desic and at, hence K 0, which completes the pro of.

2. If has constant dilation then the horizontal distribution is integrable from Equation 3.1.

This makes a totally geo desic map and then Equation 3.5 implies that Scal 0; a

contradiction.

Acknowledgments. This work was done within the framework of the Asso ciateship Scheme

of the Ab dus Salam International Centre for Theoretical Physics, Trieste, Italy. The author is

thankful to the Director of the Ab dus Salam ICTP for this supp ort.

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