<< стр. 10(всего 10)СОДЕРЖАНИЕ
1R
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 В§ 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 +dyВ·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) в‚¬ 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) вЂў

\$3 ˜ ˜ вЂў 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 В§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 вЂўq 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 вЂўq If
вЂў

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)в‚¬
\$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
В±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 21ВЈ/ q respectively; in Scott's notationВ·,
= 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˜'В·
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
в‚¬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˜В·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.

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
(*В»
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 В§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 В§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
вЂў

sz. .
SZ, 1
˜xS1xS1 ˜ S'xS xS xS ˜ S1 xS1
1
1 1
and L(p,Q)xS --+

REFERENCES

Ahlfors. L.V. (1979). Complex Analysis. 3rd edn. London: McGraw-Hill.
Baird. P. (1987). Harmontc morphisms onto Riemann surfaces and gener-
alized analytic functions. Ann. Inst. Fourier, Grenoble, 37,
Wood: Hannonic Morphisms. 259

135-173.
Baird, P. & Eells, J. (1981). A conservation law for harmonic maps. In
Geometry Symposium, Utrecht 1980, ed. E. Looijenga. D. Siersma,
F. Takens, pp. 1-25. Lecture Notes in Math. 894. Berlin: Springer
Verlag.
J.e. Bernst˜in
Baird. P. & Wood, (1988). theorems for harmonic mor-
3
s3.
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J.e. (1989a). Harmonic morphisms and conformal
Baird, P. & Wood,
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print, University of Leeds.
J.e. (1989b). Harmonic morphisms a.nd conformal
Baird, P. & Wood,
foliations bJ'В· geodesics of arbitrary three-dimensional manifolds.
Preprint, University of Leeds.
Besse, A. (1981). Geometrie riemannlenne en dimension 4. 8eminaire
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Fuglede, B. (1978). Harmonic morphism between Riemannian manifolds.
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44nn.
Greene, R.E. & Wu, H. (1962). Embeddings of open Riemannian manifolds
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Hitchin, N. (1982). Monopoles and geodesics. Corom. Math. Ph;yВ·s. 83,
579-602.
Huggett. S.A. & Tod. K.P. (1985). An Introduction to TUiistor Theor,v,
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Ishihara, T. (1979). A mapping of Riemannian manifolds which preserves
harmonic functions. J. Math. Kyoto University. 19. 215-229.
Penrose, R. & Rindler, W. (1984) Spinors and space-time vВ·ols. 1 & II.
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Scott, P. (1983) The geometries of 3-manifolds. Bul.l. London Math.
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REFERENCE LIBRARY FOR
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