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unit disc in the {x1,x2)-plane. Similar constructions provide harmon-

ic morphisms and conformal foliations by geodesics of open subsets of

˜

3

and (Baird & Wood. 1988, 1989a).

11-1

We note that in relativity theory, a conformal foliation by geodesics

is a geodesic shear-free congruence (Penrose & Rindler, 1984; Huggett

& Tod, 1985) and there are analogies between the theory of conformal

foliations by geodesics of a space form and that of geodesic shear-

free congruences of null geodesics in space-time.

252 Wood: Hannonic Morphisms

3. HARMONIC MORPHISMS AND THURSTON'S GEOMETRIES

r?

geometr,y we shall mean a simply-connected

By a 3-dimensional

Riemannian homogeneous space which admits a compact quotient. see

Thurston showed that there are exactly eight geomet-

(Scott, 1983).

3 3

Three of these are the space forms

ries. (1) 1H (2) R (3) b.:J ;

, ,

3

we have described canonical harmonic morphlsms of these in Â§ 1 (1H

having two), we shall now list the other geometries together with a

canonical metric and canonical harmonic morphism n to a surface

(which are all Riemannian submersions with geodesic fibres). The

fibres of the latter give a canonical conformal foliation by geodesics

in fact all these foliations are Riemannian.

geometry˜

of the

= projec-

with the product of standard metrles and

(4) .fxR 1C

tion onto the first factor.

2

= projec-

with the product of standard metrles and

1H xIR

(5) 1C

tion onto the first factor.

2

3

Nil; this can be identified with with the metric

(6) 1R ds

= dx2 +dyÂ·2+(dz-xdy)2 ˜ and then 1l:Nil ˜ 1R 2 is defined by (X,Jl,Z)

˜ (x,y) .

The universal cover SL2(1R) ; this can be

(7) SL2(1R)- of

=l(xJy,z) â‚¬ 1R 3 :

identified with the upper half-plane IR˜ OJ with

y)

=

the metric ds 2 2

n:SL2(1R)'" ˜

(dx +dy2 }/y.2 + (dx!y + dZ>2 , and then

z

1H is defined by (x,Y,z) t-+ (X,Jl) â€¢

$3 ˜ ˜ â€¢ have

Note that the last two examples, as for the Hopf map

non-integrable horizontal spaces and these three geometries with

their canonical metrics and foliations defined above are naturally

Sasakian manifolds. We have omitted the last (8) Sol

geometr˜T

because of the following:

(Baird-Wood 1989b)

Theorem 3 .. 1

Any non-constant harmonic morphism from an open subset of a 3-

dimensional geometry not of constant curvature to a surface Is the

restriction of one of the four maps listed above in (4)-{7) f01-

1l

Wood: Hannonic Morphisms 253

lowed by a weakly conformal map of its codomain to a surface. In

particular, there is no harmonic morphism to a surface from any open

subset of Sol.

Theorem 3.2 (loc.cit.)

Any conformal foliation by geodesics of an open subset of a 3-

dimensional geometry not of constant curvature is the restriction of

one of the four standard examples induced by the maps in (4)-{7)

1C

as explained above. In particular, no open subset of Sol supports a

conformal foliation by geodesics.

The last result is shown in the following way: Given any conformal

foliation by geodesics, let be an orthonormal frame with

U, ..:'{, Y U

˜˜+l Y.

along the foliation. Set Then from the Jacobi equation

Z=

we can show that

=0 (3)

Ricci(Z,Z)

But for this equation, one of the following holds: if the eigen-

(i)

values of the Ricci curvature are all distinct there is essentially

one solution two of the eigenvalues of the Ricci cur-

(ii) if

Z ;

vature are the same there are essentially two solutions; (iii) if

all three eigenvalues of the Ricci curvature are the same, in which

case the 3-manifold is of constant curvature, any is a solution .

Z

In the case of a Riemannian product or a Sasaklan 3-manifold,

YxlR

it can be shown that case (i) applies. in particular this applies to

the geometries (4)-(8). Thus. in these cases there is only one pos-

sible direction for a conformal foliation by geodesics. In the cases

(4)-(7) this really does gi ve a conformal foliation bj'" geodesics,

however in the case of Sol the foliation is not. in fact, conformal.

Note that the local uniqueness (Theorem 3.2) of a conformal follation

by geodesics of a geometry not of constant curvature is in contrast to

the abundance of such foliations on an open subset of 3-dimensional

254 Wood: Hannonic Morphisms

space form (see Â§2) where Theorem 2.3 asserts global uniqueness.

Furthermore, any isometry of a geometry (4)-{7) must preserve any

conformal foliation by geodesics, this is not the case for the space

a˜ovet

forms. Lastly note that the analysis together with' a unique

continuation theorem for harrnonic morphisms of conformal foliations by

(,˜

geodesics, shows that no Riemannian 3-manifold of non-constant

curvature can support more than two distinct non -constant harmonic

morphisms to surfaces or conformal foliations by geodesics.

4. HARMONIC MORPHISMS AND SEIFERT FIBRE SPACES

2

2

D xlll

we mean with the end

By a fibred solid torus Tp â€¢q D xI

2

D xlOl by a twist through for some

identified to the end 2nq/ p

1

2

integers with 1#0. If this gives the solid torus

P.q q=O D xS

with its product foliation by circles. By a fibred solid Klein bottle

2

we mean with its ends identified by a reflection. By a Seifert

D xI

we mean a 3-manifold with a decorn-

fibre space' (without reflections)

posJtion into circles (called fibres) such that each circle has a

neighbourhood and a fibre preserving homeomorphisn1 to a flbred solid

q=O the fibre is called regular, otherwise we

torus Tp â€¢q If

â€¢

shall call it singular, the integers are called the

(p, q)

(unnormalized) orbit or Seifert invariants of the fibre. If we also

allow fibred solid Klein bottles we obtain a Seifert fibre space with

It is result of Epstein that any foliation by circles of

reflections.

a compact 3-rnanifold is a Seifert fibre space possibly with reflec-

tions. If the manifold is orientable, this is necessarily without

reflections. We shall here concentrate on the latter sort of Seifert

fibration for reasons explained after Theorem 4.1. As an example, for

'-P.q

any non-zero co-prime integers p,q, consider the foliation by

=I(Z1, Z2)â‚¬

$3 I zd 2 + t Z2 f 2= 1.

circles of given by

J

(['2:

= constant (4

ZIP/ Z2 q

255

Wood: Hannonic Morphisms

=q = 1

For this is the foliation associated to the Hopf fibra-

p

=0 = ()

tion. For ,p, Q> 1 , the circles are singular

and

Z1 Z2

f1 bres wi th (unnorrnallzed) orbIt in variants and ( qtp) res-

(p, q)

pectivel˜",

Now it follows from Scott (1983) that any closed Seifert fibre space

3

3

1E If

is of the form where is one of the geometries (2)-{7) with

E

=$3

3

'-P.q

its canonical foliation, or 1E wi th the foliation for

r

some p, q ,and is a group of isometries which acts freely and

3

properly discontinuously on Note that 1H and Sol do not

JE3.

give rise to any Seifert fibre spaces. For example, if we factor the

˜xlR

geometry by the group generated by where is a

(cx,fJ) <X

s2

rotation of by an angle about the dIameter joining pojnts

21Cqi p

f3 is translation of IR by 1 unit, we obtain a Seifert

Â±a and

1 1

1

fibration on with singular fibres lalxS and l-a}xS both

s2xs

with orbit invariants (p, q).

Associated to any Seifert fIbre space without reflections Is the

˜.

natural projection onto its leaf space The latter is an orbifold

with cone points of angle corresponding to singular fibres with

21f.1p

orbit invariants For example, for the foliation on

(p,q). '-p,q

s2

$3 above, the leaf space is homeomorphic to but with two cone

2nlp

points of angles and 21Â£/ q respectively; in Scott's notationÂ·,

= s2(p,q). ˜xS1

Z For the folIation on descrIbed above, the leaf

˜p

space is Now in all cases apart from the metric

s2{ptp) q,

1

induced from the standard metric on the geometry makes the Seifert

fibre space a Riemannian foliation by geodesics. In the case

JE3If

$3 a metric

of we can give such that this becomes a

g

,"p, q

($3 ,g) be the

Riemannian foliation by geodesics. Indeed, we let

3 2'" 2 2 2

+ 1Z21 /q = I}

f(Z1,Z2)E([ : 1Z11˜/p

ellipsoid with the

QPoq=

metric induced from the ambient Euclidean space (see Eells & Ratto,

1988; Baird & Wood. 1989b). The metric on the Seifert fibre space

˜

induces a metric, and thus a conformal structure on the leaf space

except at the cone points. In fact we can give the leaf space a

Wood: Hannonic Morphisms.

256

conformal structure even at the cone points as follows: At a cone

2 2

point. the leaf space is of the form where is a disc with

D IIp D

rotationall˜'Â·

a symmetric metric g. Then by the Uniformlzation Theo-

rem (Ahlfors, 1979), there is a conformal diffeomorphism

˜ (D2 ,standard). z˜

2

z?

After composing this with the map

(D g)

t

V /71 p ˜ (D ,standard)

2

2

of the codomain it factors to a homeomorphism

which we use to give the leaf space a conformal structure at the cone

points. Our construction gives a hOIneomorphism from the leaf

h

space Z to a smooth surface with conformal structure which we

Zs

call the smoothed leaf space. We can use this smoothing procedure to

establish the following:

Theorem 4.1 (Baird & Wood, 1989b)

Jy3

Let be a closed Seifert fibre space without reflections. Then

˜

there exists a smooth metric on and a smooth harmonic morphism qJ

to a smooth surface with conformal structure such that the fibres of

˜.

coincide with the fibres of

â‚¬I'

Let be the composItion of the natural projection onto

rp

Proof.

the leaf space with the map above. This certainly has geodesic

h

fibres and is horizontally conformal away from critical fibres; it is

thus a harmonic morphism off the polar set given by those fibres. It

Is also continuous everywhere; it follows (cf. Baird & Wood, 1989a)

that it is a smooth harmonic rnorphlsm ever˜Â·where.

Note that each point of a singular fibre of . JyP Is a

RemarlfS.

critical point of In fact, at such a poInt. has the form of

qJ. qJ

zP.

z˜

a submersion followed by Note also that. given a Seifert

Y

fibre space with reflections, there cannot be any harmonic morphism

qJ:˜ --+ 11 to a surface. with fibres coinciding with the fibres of

Ji3 for, again, all poin ts of the singular fibres would have to be

t

critical points of But now the singular fibres corresponding to

q>.

fibred solid Klein bottles are whole surfaces and so not polar contra-

dicting the fact (Fuglede, 1978) that the critical points of a har-

257

Wood: Harmonic Morphisms

monic morphism form a polar set.

˜P.q on the ellipsoid

As an example, consider the foliation Q;.q.

?

The leaf space Is the orbifold with a rotationally symmetric

S"'(p, q)

˜F , which

metric. Composing with the slTloothing Inap 11:F(p,q)

will be of the form (*) (r,8) ˜ (.t{r),f3) in spherical polar co-

ordinates˜ gives (up to conformal self-mappings of S- of the form

Q;.q ˜ 52

the harmonic morphism of Eells & Ratto (1988). Note

(*Â»

that our method does not require discussion of any ordinary differen-

tial equation. this being implicit in the use of the Uniformization

Theorem.

Q;.q actually has two Riemannian foliations, the

The ellipsoid

second one given by replacing These foliations

q -q in (4).

by

(Zt,Z2) ˜ (Z1.Z2).

are interchanged under the reflection Now for

any non-zero coprime integers we have a free properly discon-

(r.s)

3

tinuous action of on Qp˜q given by (Zt,Z2) t---+

Z...

which preserves these foliations. Then the

(Zte 2", i ! r ,Z2e 2w is / r)

foliations factor to foliations on the lens space with ellipsoidal

= Q;,q/Z",. These last foliations are not isometri-

metric L(r,s)

cally related unless r=2 in which case the lens space is real projec-

In (Baird & Wood. 1989b) the closed oriented Rieman-

tive 3-space.

nian 3-manifolds supporting a pair of oriented Riemannian foliations

˜ $3 , any lens

by geodesics are classified. They are the 3-torus t

space and S?xS', all with one of an infinite family of non-standard

metrics. The constructions above then yield harmonic filorphisms ill

various homotopy classes from these manifolds to 2-tori and the 2-

sphere.

5 THE PRODUCT OF A SEIFERT FIBRE SPACE WITH A CIRCLE

i˜r

Let be an oriented RIemannian 3-manifold with an oriented confor-

mal foliation by geodesics. Then there is a natural alrnost Hermitian

258 Wood: Hannonic Morphisms

rxs'

structure on the product described as follows. At any

J

1

rxs eat e2. e3

point let be an oriented orthonormal basis

(x,..y) E

Txr e3

for along the foliation and let be the unit

e4

with

J( ˜)=- e,

1

Then set J( 6'1)= ez ,

at

8

positive tangent to Jl. t

J{˜)=e4 and Then a straightforward calculation of the

J(e4)=-e:J.

Nljenhuis tensor shows that is integrable. Now suppose that M'

J

is a closed oriented Seifert fibre space with oriented fibres. Then,

as in Â§4, it can be given the structure of a conformal (in fact Rie-

mannian) foliation by geodesics and we can construct a harmonic mor-

cp:Y ˜ Ji2

phism to a Riemann surface whose fibres are the fibres of

r. YxS' ˜ r --+,;

The composition of the projection onto the

first factor with is clearly holomorphic and has tori as fibres.

qJ

so we have shown in a very natural way the well-known:

Theorem 5.1

The product of a closed oriented Seifert fibre space with oriented

1

fibres and the circle 8 is naturally an elliptic surface.

S3 ˜ ˜.

As an example. start with the Hopf rnap Then we get a

1

$3xs

Hermitian structure on the well-known Hopf surface (see for

, 1

1981 L together with a surjective holomorphic map

example, Besse˜

trxs1 g3. If we take. instead, the Eells-Ratto harmonic morphism

-4

Q;.<I ˜ SZ , we get a Hermitian structure on Q˜tqXS1 which gives

$3xs1 a non-standard metric and Hermitian structure as an elliptic

surface. Similarly, using the harmonic morphisms discussed at the end

of Â§4, we obtain non-standard metrics and Hermitian structures on

1

1

1 1

S'xS xS xS' and together with holomorphic maps

SZxS'xS L{p,q)XS

â€¢

sz. .

SZ, 1

˜xS1xS1 ˜ S'xS xS xS ˜ S1 xS1

1

1 1

and L(p,Q)xS --+

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'ERGODIC TH˜9˜˜_˜!'!.˜

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