<<

. 9
( 10)



>>


== d˜˜x).
where £' The equations 5.18-5.20 are all pulled back to D. In the follo"ring
we shall write equations on C that are readily pulled back to D by applying maps
f.
induced from Thus the Euler-Lagrange equations may be written
Hartley & Tucker: Extremal immersions and the extended frame bundle
222

+ tJ>..i j3 + wi i A >..i p = 0
ep A >..i 5.26

(-1)(p-l)2.c'H i -Y-y7]ap*1 + ef3 A >..i a = 0 5.27

where for any form Ilai
= dJ.Lai - wf3 0 A Jl Pi
DJlai 5.28

is the exterior covariant on sections of A(T*C) and

w'i A f.lcxi
1. .
5.29
D f.loi = dJ-Lai -

is the exterior covariant derivative on sections of A«T*C)J..) induced from the nor-
mal bundle. From 5.27,5.34

5.30

D*e a
By the first structure equation, = 0, so

5.31

Then 5.26 becomes

5.32

with solution
5.33

We have the relations
5.34

5.35

where ˜.l.. is the Laplacian induced from the connection in the normal bundle on
any 0-form tl>:
5.36

and
A=VxaX a 5.37

>"i and>..i ex into 5.25 and using 5.34 and 5.35 we obtain finally
Inserting

=0
A1-(£'H;OI 01) £' Hi 01 Ot(RiPPi - Hj!J-y HiP-Y) 5.38
i£H;OI01
- -
Hartley & Tucker: Extremal immersions and the extended frame bundle 223

This is the Euler-Lagrange equation for the immersion C into M which extremalises

[ £(g(Trh,Trh»:n


A few specific examples may serve to illustrate the scope of this result. Consider
first
5.39
£,=1

The action above is the p-dimensional volume of C and since £" = 0 the Euler-
Lagrange equation is simply:
5.40

This is the familiar equation for an immersion with extremal volume.

As a more complicated example take

5.41

for some constant This is the generalisation to p-dimensional immersions into
K.

a curved spacetime of Polyakov's "rigid string "Lagrangian density [7]. We have

= 5.42
£,' K,



so the Euler-Lagrangian equation is:




Finally take the real root

==
{, = (±g(Tr h, Tr h) )C/2 ITrh IC 5.44

˜
for some real value c 1. Then 5.38 becomes




This equation agrees with that obtained by Chen and Willmore [3] for the special
= m - 1 with M flat" and with that obtained by Weiner [4] for the case
case p
(" = p = 2 in a space of constant sectional curvature.
Hartley & Tucker: Extremal immersions and the extended frame bundle
224


6. Boundary Conditions and Constants of the Motion


In this section we seek admissible variation vector fields VETB for the differ-
ential system S defined in section 4. This will enable us to specify appropriate
boundary conditions and derive conserved quantities for the variational problem
e
Adopt the coframe {ecW, w a ,8, w ij , Bi, Oi a, i 0,8} for B and let
discussed in 5.
{X , 'WOtIJ, 'Wii' Xi, 'W/lt , 'Wi°,8} be the dual frame with 'W01J = - 'Wfja, Wij =
o

= 'W i ,8o. Any vector V E TB can be written
-W ji , W/lt,8




with appropliate index symmetries understood in the summation. As described in
section 3 V generates an admissible variation iff

6.2

+ i"(rd, we can use the d(Ji and dO i a given in 4.6 and 4.14 to find
Writing £''9" = di'9"

6.3

and
+ W i j yi a
,8 a Vi ,8
......'9"{1 a I"'V dyi a -
ni -
F'
W


_(R i o,8j Hi o-yHj"'Y fj )efJyi - e,8yi OtIJ
-

˜ byia elJV i afj
(R i afjj Hia-yHj -y ,8 )eIJVj 6.4
- -
-


iJ is
where the mixed exterior derivative

i
,. i _ i
+W i j Y j a
{3
6.5
DV dV Y fj
a- W
0 - Q'




All the equations 6.2 to 6.5 hold under restriction by pullback to D or equivalently
to M. We shall suppress again pullback maps and write {eOt,e i } as a Darboux
frame for the immersion of D in M; with dual frame {Xa,Ni}. Putting 6.3 and 6.4
into 6.2 gives
6.6

6.7
Hartley & Tucker: Extremal immersions and the extended franlc bundle 225

These are the constraints which must be satisfied by V of the fortll 0.1 if it is t,O
generate admissible variations. We note that the va, yap and yij COIUpOllcut,s
are unconstrained. They merely induce leaf preserving diffeomorphisms on V( C).
Furthermore we see that 6.6 and 6.7 completely determine Vi a and Vi ap in terms
V
of the Vi. Thus apart from leaf preserving diffeomorphisms an admissible can
be given in terms of a single'vector yAt E (TC)1. given by

6.8


V, V..L, 6.6 and 6.7 can be written
In terms of the connections




It is interesting to note that the left-hand side of 6.10 is symmetric in 0.(3 whilst
the right-hand side appears at first to be asymmetric. However, the antisymmetric
part of the right-hand side vanishes by virtue if the Ricci equation:

R i Ja˜ - Rl. i JOtP _ (Hi Q-Y H J.#Ot[ {J Hi {Joy H. #Ot[
. . 6.11
_ )
- J Q




Having established the admissible variation vectors VETB, we can now seek spa-
tial boundary conditions. In pl-inciple such conditions need not bear any relation
to an extremalisation procedure. However the determination of the critical immer-
sion by such a process always involves integrals over possible boundaries and it is
therefore natural to elevate the variational principle in such a manner that both the
Euler-Lagrange equations and compatible boundary conditions be simultaneously
determined. It is worth noting that for the higher order Lagrangian systems con-
sidered in this paper, a determination of boundary conditions from a prescription
that considers only admissible variations vanishing on a boundary is no longer suffi-
cient to exhaust all possibilities. Specifying the derivatives of admissible boundary
variations (arbitrarily) enlarges the space of possible boundary conditions. In this
paper we have adopted the view that for immersions into a Lorentzian spacetime
the variational procedure determines spatial boundary conditions in any manner
Hartley & Tucker: Extremal immersions and the extended frame bundle
226

that results in the (lifted) variational integral being stationary modulo integrals
over (lifted) space-like chains. Thus from 3.26 the conditions should ensure that

f =0 6.12
j{r
Jj(Dr)

where
= iv(˜ + Ai A (}i + AiQ A (}iQ)
iy
+ TJAia A i'{r(}ia
˜ iv˜ + TJAi A iV(}i 6.13

[3 = .en
For our choice
= .cVaix n ˜ .e*liT
i v 13 6.14
Q




where the vector V T E TC is defined by

6.15

o.
and VT is the dual form related to it by the metric Substituting Ai, Aia from
5.33, 5.30 and using 1.6 j"(r becomes

6.16

Thus

Z *{.cV T +2(.c'H i P,sD.LVi-V i D.L(.c'H/ p »}
f
jy= 6.17
Ji(Dr) J!(D r )

If the spatial boundary f(D r ) of feD) has a unit normal field n E TC then the
induced volume element on f(D r ) is

6.18

Hiex aNi,
The integral 6.17 can be rewritten in terms of n, Vl.. and the vector Tr h =
so that the boundary condition 6.12 becomes

f + 2{.c'Tr h(V*"VA') - VA'(V*"(.c'Tr h»}}#1 = 0
{.cVT(n) 6.19
J!(Dr)


To satisfy these boundary conditions for arbitrary V T , VAl" and dVN" we can take
either:
on
#1 =0 6.20
Hartley & Tucker: Extremal immersions and the extended fnllllr hundll· 227


or

.c = 0 =0
=0 f(D r
£'Trh
and
V'-;«(,'Tr h) on
and )

6.21

For example, for an open membrane in Lorentzian spacetime with the Lagrangian
5.39 the boundary condition on Dr is

6.22
#1 =0

Condition 6.22 implies that the induced metric on the spatial boundary is degen-
erate. For strings, this is the familiar condition that the ends of the string move
at the speed of light. For (p - I)-dimensional membranes (p > 2), it implies that
the spatial boundary sweeps out a null surface, possessing a single null direction
and (p - 2) space-like directions. This is a stronger condition than requiring the
boundary to move at the speed of light (although the latter still applies).

For the Lagrangian with 5.41 the conditions 6.21 are incompatible for an immersion
with boundary and 6.22 may be imposed in the Lorentzian context. On the o·ther
hand fo˜ admissible variations that vanish on f(D r ), but not their derivatives we
see from 6.19 that an alternative boundary condition is

=0
(,' Tr h 6.23

Thus for the Lagrangian 5.41 with 5.42 we may impose

=0
Trh on 6.24


For the Lagrangian 5.44 we have the alternative boundary conditions on f(D r ):

= V';(Trh) = 0
Trh 6.25

To find the constants of the motion given by 3.28 we need to determine the admis-
r. Recall that V = t is an admissible symmetry vector ifit
sible symmetry vectors
it = it(˜ + AA /\ (}A)
satisfies 3.12 and 3.22. Thus for solutions of 3.17 the current
satisfies
6.26
Hartley & Tucker: Extremal immersions and the extended frame bundle
228

Since d COmtl1utes with pull backs this may be written on C as

6.27

Noting that

= D.1.(C' H i /3 ,8*DJ..ri) = nJ..(£' HiP ,8) 1\ *D.l.ri + £' HiP ,8˜l.ri*l
d*(C' HiP t3 D l.. r i)
6.28
and that
6.29

the symmetry vector condition 6.26 may be further rewritten

6.30

If this condition is satisfied, then by an argument similar to the one in section 3 the
quantity

{f*{.ctT(n)+2{.c'Trh(v˜rAf)-tAf(V˜(.C'Trh))}}#1
Q(l')= 6.31
iDa
is a constant of the motion for any space like (p - I)-chain DO' (with boundary in
Dr) with unit (timelike) normal vector field n E f(TD) and:

6.32



Conclusions


As can be seen from the preceeding calculations, the differential system of con-
straints on the extended frame bundle that we have used provides a convenient
framework for tackling a wide class of higher-order variational problems depending
on the extrinsic geometry of immersions. In this approach the Euler-Lagrange equa-
tions are expressed in terms of geometrical quantities relevant to the problem. As
illustrated in section 6, the "natural" boundary conditions which arise for higher-
order problems depend on a somewhat arbitrary choice of details in the variational
principle. We have adopted boundary conditions at the boundaries of space-like
Hartley & Tucker: Extremal immersions and the extended fntl11C hundlc 229

chains only, thus allowing for the specification of initial and final conditions. This
has also enabled us to construct constants of the motion in the presence of symmetry
vectors.

The general approach is both po,verful and flexible, and clearly lends itself to a
number of generalisations.


Acknowledgement


D H Hartley acknowledges support from the SERe.


References


1] T L Curtright, G I Ghandour and C D34 (1986) 3811.
!{ Zachos, Phys. Rev.
T L Curtright and P van Nieuwenhuizen, Nncl. Phys. B294 (1987) 125.
U Lindstrom, M Rocek and P van Nieuwenhuizen, Phys. Lett. 199B (1987) 219.
M Mukherjee and R W Tucker, Class. Quantum Grav. 5 (1988) 849.
M Onder and R W Tucker, Phys. Lett. 202B (1988) 501.
M Onder and R W 'lUcker, J. Phys. A: Math. Gen. 21 (1988) 3423.
U Lindstrom, Int. J. Mod. Phys. AlO (1988) 2410.
R W Tucker, "The Motion of Membranes in spacetime", Lancaster preprint, 1989.
D H Hartley, M Onder and R W Tucker, "On the Einstein-Maxwell equations for a
"stiff" membrane", to appear in Class. Quantum Grav.


1.] P A Griffiths, "Exterior Differential Systems and the Calculus of Variations",
Birkhauser, 1983.


q B-Y Chen, J. London Math. Soc. 6 (1973) 321.
T J Willmore, "Total curvature in Riemannian geometry", Ellis Horwood, 1982.


II J L Weiner, Indiana Univ. Math. J. 27 (1978) 19.
Hartley & Tucker: Extremal immersions and the extended frame bundle
230

[5] Y Choquet-Bruhat, C DeWitt-Morette and M Dillard..Bleick, "Analysis, Manifolds
and Physics", North-Holland, 1982.


[6] E Cartan, "Les systemes differentielles exterieurs et leurs applications
geometriques" , Herman, 1945.


[7] A Polyakov, Nucl. Phys. B268 (1986) 406.
Minimal surfaces in quaternionic symmetric spaces
F.E. BURSTALL
University of Bath




We describe some birational correspondences between the twistor spaces of quater-
nionic !(ahler compact symmetric spaces obtained by Lie theoretic methods. By
means of these correspondences, one may construct minimal surfaces in such sym-
metric spaces. These results may be viewed as an explanation and a generalisation
of some results of Bryant [1] concerning minimal surfaces in 8 4 •

This represents work in progress in collaboration with J.H. Rawnsley and S.M. Sala-
mon.

BACKGROUND
This work has its genesis in our attempt to understand the following result of Bryant
[1]:
Theorem. Any compact Riemann surface may be minimally immersed in 8 4 •
To prove this, Bryant considers the Penrose fibration 1(' : CP3 --+ 8 4 = HPI. The
perpendicular complement to the fibres (with respect to the Fubini-Study metric)
furnishes CP3 with a holomorphic distribution rl C T 1 ,oCP3 and it is well-known
that a holomorphic curve in CP3 tangent to rt (a horizontal holomorphic curve)
projects onto a minimal surface in 8 4 • Bryant gave explicit formulae for the horizon-
tality condition on an affine chart which enabled him to integrate it and provide a
"Weierstraf3 formula" for horizontal curves. Indeed, if j, 9 are meromorphic functions
on a Riemann surface M then the curve ˜(f, g) : M --+ CP3 given on an affine chart
by
iP(j,g) = (I - ig(dg/dj),g, i(dl/dg))
˜(f,9)
is an integral curve of 1i. For suitable j, 9, is an immersion (indeed, an
(˜mbedding) and the theorem follows.


In [6], Lawson gave an interesting interpretation of Bryant's method by introduc-
= U(3)/U(1)
ing the flag manifold D 3 x U(l) X U(l) which may be viewed as the
t.wistor space of CP2 (the twistor fibring being the non-±holomorphic homogeneous
fibration of D 3 over Cp2). Again we have a holomorphic horizontal distribution 1(,
Burstall: Minimal surfaces in quaternionic symmetric spaces
232

perpendicular to the fibres of the twistor fibration but, this time, horizontal curves
are easy to construct. Indeed, viewing D 3 as P(T1 ,OCP2), JC is just the natural con-
tact distribution and a holomorphic curve in CP2 has a canonical horizontal lift into
D3 given by its tangent lines.

The remarkable fact, implicit in Bryant'8 work and brought to the fore by Lawson, is
Theorem. There is a birational correspondence q> : D3 --. Cps mapping JC into 'H.
Recall that a birational correspondence of projective algebraic varieties is a holomor-
phic map which is defined off a set of co-dimension 2 and biholomorphic off a set of
co-dimension 1.

Thus it suffices to produce horizontal curves in D 3 which avoid the singular set of <b
and this may be done by taking the lifts of suitably generic curves in CP2.

Lawson gave an analytic expression for ˜ but a geometrical interpretation of the map
seemed quite hard to come by. An algebro-geometric interpretation has been given
by Gauduchon [4] but it is our purpose here to show how this map arises naturally
from Lie theoretic considerations.

QUATERNIONIC SYMMETRIC SPACES
The 4-sphere and CP2 may be viewed as the 4-dimensional examples of the quater-
nionic Ka.hler compact symmetric spaces. These are 4n-dimensional symmetric spaces
N with holonomy contained in Sp(I)Sp(n). Geometrically, this means that there is a
parallel subbundle E of End(TN) with each fibre isomorphic to the imaginary quater- I




nions. There is one such symmetric space for each simple Lie group; the classical ones
in dimension 4n being



Following [7], we consider the twistor space Z of N which is the sphere bundle of E ,
Of, equivalently,
Z = {j E E: j2 = -I}.
This twistor space is a Kahler manifold, indeed a projective variety, and once more'
the perpendicular complement to the fibres 11, is a holomorphic subbundle which is
called the horizontal distribution. Our main theorem is then
Theorem. Let Nt, N 2 are compact irreducible quaternionic [(ahler symmetric spaces
of the same dimension with twistor spaces Zl' Z20 Then there is a birational corre-
spondence Zl --+ Z2 which preserves the horizontal distributions.
For this we must study the homogeneous geometry of the twistor spaces: if N is
the symmetric space GJ [( then G acts transitively on Z and, moreover, this action
233
Burstall: Minimal surfaces in quatemionic symmetric spaces

extends to a. holomorphic action of the complexified Lie group GO. Further,' the
horizontal distribution is invariant under this GC action. In fact, Z is a special kind
of GC-space: it is a flag manifold, that is, of the form GC/ P where P is a parabolic
subgroup.

For any flag manifold GC / P, let p be the Lie algebra of P. We have a decomposition
of the Lie algebra of GC
gC = pEen
where n is the nilradical of p so that ii ˜ T:#Gc / P is a nilpotent Lie algebra. Let N
be the corresponding nilpotent Lie group and consider the N orbits in GO / P. The
orbit {} of the identity coset is a Zariski dense open subset of GC / P (it is the "big
cell" in the Bruhat decomposition of the flag manifold). In fact, the map ii' --+ n
given by
e˜expe·p

is a biholomorphism with polynomial components (since n is nilpotent) and so ex-
tends to give a birational correspondence of GC/ P with P(n EI1 C). Thus GC / P is
a rational variety: a classical result of Goto [5]. However, more is true: let Gf/Pl
and Gf / P2 be flag manifolds and suppose that the nilradicals nl and n2 are isomor-
phic as complex Lie algebras. Then we have an isomorphism </J : ii} --+ n2 which
we may exponentiate to get an isomorphism of Lie groups 4> : N 1 --+ N 2 and thus
a biholomorphism 0 1 --+ {12 which extends to a birational correspondence between
the flag manifolds. Moreover, on {}1' this biholomorphism is N t-equivariant and so
will preserve any invariant distribution so long as </J does when viewed as a map
T;˜ Gf/ P l --+ Telp˜ G˜ / P2 •

We now specialise to the case at hand: if Z is the twistor space of a quaternionic
!(ahler compact irreducible synunetric space then Z is a rather special kind of flag
manifold. In fact, Wolf [8] has shown that here ii' is two-step nilpotent with 1-
dimensional centre and so is. precisely the complex Heisenberg algebra. Thus any
two of our twistor spaces of the same dimension have isomorphic n and so the main
theorem follows.

APPLICATIONS TO MINIMAL SURFACES
rrhe relevance of these constructions to minimal surface comes from the well-known
fact that, just as in the 4-dimensional case, horizontal holomorphic curves in Z project
oato minimal surfaces in N. Moreover, in some of the classical cases, horizontal
holomorphic curves are quite easy to come by. For example, the twistor space of
()2(C n +2 ) is the flag manifold Z = U(n+2)/U(1) x U(n) x U(l) which we may realise
a,s the set of flags


= n + I}.
C C n +2 : dim.e = 1, diol1r
c
{f 1r
Burstall: Minimal surfaces in quatemionic symmetric spaces
234


Horizontal, holomorphic curves in this setting are just a special kind of ai-pair in the
sense of Erdem-Wood [3] and may be constl"Ucted as follows: if / : M -+ cpn+l is
a holomorphic curve, we may construct the associated holomorphic curves fr : M -+
Gr + 1 (C n +2 ) given locally by
a/ arf
f A -8 A ••. A -8·
fr = z zr

I 11,"" fn are defined and then the map t/J : M -+ Z
Generically, is full so that
given by


is horizontal and holomorphic. Note that for n = 1, t/J is just the lift of I : M -+ CP2
discussed above. Composing these curves with the birational correspondences of the
previous section, we then have horizontal holomorphic curves in all the other twistor
spaces of the same dimension so long as we can ensure that the curves avoid the
singular sets of the correspondences. Thus, for example, one has the possibility of
constructing minimal surfaces in the 8-dimensional exceptional quaternionic symmet-
ric space G2 /SO(4) from holomorphic curves in CP3.

However, to carry out such a programme, a rather more detailed analysis of these
singular sets is required so as to ensure that they are avoided for suitably generic f.
Work is still in progress on this issue.

EXTENSIONS
Many parts of the above development apply to arbitrary generalised flag manifolds.
Burstall-Rawnsley [2] have shown that any flag manifold fibres in a canonical way over
a Riemannian symmetric space of compact type and, moreover, any such symmetric
space with inner involution is the target of such a fibration. In this setting, the
perpendicular complement to the fibres is not in general holomorphic but there is a
sub-distribution thereof, the superhorizontal distribution, which is holomorphic and
GO-invariant. Again, holomorphic integral curves of this distribution project onto
minimal surfaces in the symmetric space.

The above discussion applies so that isomorphisms of nilradicals exponentiate to
give birational correspondences of flag manifolds which preserve the super-horizontal
distributions. However, apart from the quaternionic symmetric case, we have not yet
found any examples of differing flag manifolds with isomorphic nilradicals.

REFERENCES
1. R. L. Bryant, Conformal and minimal immersions of compact sUl'faces into the
4-sphere, J. Diff. Geom. 17 (1982), 455-473.
BurstaIl: Minimal surfaces in quaternionic symmetric spaces 235


2. F. E. Burstall and J. H. Rawnsley, Twistor theory for Riemannian symmet-
ric spaces with applications to harmonic maps of Riemann surfaces, Bath-Warwick
preprint, 1989.
3. S. Erdem and J. C. Wood, On the construction of harmonic maps into a Grass-
mannian, J. Lond. Math. Soc. 28 (1983),161-174.
4. P. Gauduchon, La correspondance de Bryant, Asterisque 154-155 (1987), 181-
208.
5. M. Goto, On algebraic homogeneous spaces, Amer. J. Math. 76 (1954), 811-818.
6. H. B. Lawson, Surfaces minimales et la construction de Galabi-Penrose, Asterisque
121-122 (1985), 197-211.
7. S. M. Salamon, Quaternionic !(ahler manifolds, Invent. Math. 67 (1982), 143-
171.
8. J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmet-
ric spaces, J. Math. Mech. 14 (1965), 1033-1047.
Three-dimensional Einstein-Weyl Geometry
K.P.TOD
The Mathematical Institute, Oxford
Abstract. I review what is known about 9-dimensional Einstein- Weyl spaces.



1. Introduction.
The Einstein-Weyl equations are a naturally-arising, conformally-invariant, gener-
alisation of the Einstein equations. In the special case of 3-dimensions, the Einstein
equations on a space force it to have constant curvature. The metric is then charac-
terised locally by a single number, the Ricci scalar. By contrast, the Einstein-Weyl
equations allow some functional freedom locally. Further, the equations fall into
the small class of non-linear partial differential equations which can be solved by
the twistor correspondence. -
In this article I shall review the definition of an Einstein-Weyl structure (§2) and
its twistor correspondence (§3). I also give some examples of Einstein-Weyl spaces
(§4), and describe some general and some particular properties (§5,§6).
I am very grateful to the organisers of the LMS Durham Symposium for inviting
me to take part in a most stimulating and worthwhile conference.
2. Einstein-Weyl spaces.
A Weyl space is a smooth (real or complex) manifold W equipped with
(1) a conformal metric
(2) a symmetric connection or torsion-free covariant derivative (the Weyl con-
nection )
which are compatible in the sense that the connection preserves the conformal metric.
This compatability ensures in particular that orthogonal vectors stay orthogo-
nal when parallel propogated in the Weyl connection and that the null geodesics,
which can be defined just given a conformal metric, are also geodesics of the Weyl
connection.
In local co-ordinates (or using the "abstract index convention" [8]) we write a
(·hosen representative 9 for the conformal metric as gab and we write the Weyl
covariant derivative as D a . The compatability condition becomes
238 Too: Three-dimensional Einstein-Weyl geometry

= W a 9bc
(2.1) Da 9bc

for some I-form W = wadx a • Under change of representative metric we have



(2.2)

where n is a smooth, strictly positive, function on W.
Note that the I-form w encodes the difference between the Weyl connection and
the Levi-Civita connection of the chosen representative metric. Thus we can think
of a Weyl space as a pair (g, w) modulo (2.2).
The Weyl connection has a curvature tensor and hence a contracted curvature
tensor or Ricci tensor. The skew part of the Ricci tensor is a 2-form which is
automatically a multiple of dw. To impose the Einstein condition on W we require
that the symmetric part of the Ricci tensor be proportional to the conformal metric,
[4]. In local co-ordinates (or abstract indices) the Einstein condition is


= Agah
(2.3) some A
W(ab)


where Wab is the Ricci tensor of the Weyl connection. When W is 3-dimensional,
as we shall assume from now on, we also have the identity


(2.4) = -3Fa b
W[ab]


where Fab is the 2-form dw. (As is customary in the relativity literature we
denote symmetrisation and anti-symmetrisation by round and square brackets re-
spectively. )
A Weyl space satisfying the Einstein condition we shall call an Einstein- Weyl
space.
Since the Weyl connection can be written in terms of the Levi-Civita connection
of the chosen representative metric g and the I-form w, we can rewrite (2.3) as an
equation on gab and W a • We find that the condition is

1 1
2'i7(aWb) - ˜WaWb ex
(2.5) Rab - gab


where R ab is the Ricci tensor of the Levi-Civita connection of 9 and V a is the
corresponding covariant derivative.
239
Too: Three-dimensional Einstein..Weyl geometry

Note that if the chosen representative metric g is actually an Einstein metric on
W then (2.5) is satisfied with the I-from w vanishing. Since (2.5) is invariant under
(2.2), any W conformal to an Einstein space is an Einstein-Weyl space. These
examples have Fab in (2.4) vanishing and, locally at least, are characterised by this
property. Thus the Einstein- Weyl equation (2.9) is a natural conformally-invariant
equation which generalises the Einstein equation.
That the Einstein-Weyl equation has a direct geometrical content is shown by
the following
H W is 3-complex dimensional and satisfies the Einstein-
PROPOSITION 2.1 [2].
Weyl equation then W admits a 2-complex dimensional family of totally geodesic
null hypersurfaces.
(That is, the hypersurlaces are totally geodesic with respect to the Weyl connection
and have normal which is of zero length or null with respect ,to the conformal metric.
Cartan calls the hypersurfaces "isotropes".) If W is 3 reaf-dimensional we need to
suppose that it is real analytic and then complexify it to use this Proposition. We
shall return to this point in §5. This result was re-interpreted by Hitchin [4] in a
way that we may conveniently call a corollary.
2.2 [4]. Such a W admits a twistor construction.
COROLLARY

To say what this means we shall consider twistor constructions in the next section.
To conclude this section we note that there is another characterisation of the
Einstein-Weyl condition equivalent to Proposition (2.1) and obtainable by a con-
sideration of the geodesic deviation equation in W :
[9]. The Einstein-Weyl condition is equivalent to the existence
PROPOSITION(2.3)
of a complex structure on the space of Jacobi fields along each geodesic.

We shall see the connection between these two characterisations below.
3. Twistor correspondences.
The idea of twistor constructions or twistor correspondences, as I shall use the
term here, is that one has two manifolds with a non-local correspondence; one space
has some differential geometric objects satisfying local equations; the other does not
but is a complex manifold, and under the correspondence the local equations on one
space are solved automatically by virtue of the the complex analyticity of the other
space.
The first example, which I shall call the Penrose correspondence [6], begins with
a. 4-complex dimensional manifold M with a conformal structure. The conformal
(˜urvature C of the conformal structure on M is required to be anti-self-dual as a
2-form:
240 Too: Three-dimensional Einstein-Weyl geometry

This condition is the integrability condition for M to admit a 3-complex dimen-
sional family of 2-dimensional surfaces on which the conformal metric degenerates
completely ("totally null" surfaces). Call these a-planes, then the space of a-planes
is a 3-dimensional complex manifold g, the twistor space of M. A point p of M
is represented in g by the a-planes through it, which define a holomorphic curve
.c
£1' in g. The curve p is a copy of Cpl and automatically has normal bundle
N ˜ 0(1) ‚¬I' 0(1) . This gives the non-local correspondence between g and M.
To define the conformal metric on M it is sufficient to be able to recognise which
vectors are null. A tangent vector v at a point p in M corrresponds to a section Lv
of the normal bundle N of L,p. Define this vector to be null if Lv has a zero. From
the character of N this is a quadratic condition. Remarkably, the conformal metric
obtained in this way automatically has anti-self-dual conformal curvature tensor.
This is the twistor correspondence; it becomes a twistor construction if one can
find a way of building complex manifolds like g, since these will in turn generate
"sp˜e-times" M which automatically satisfy local equations.
The second example, which I shall call the Hitchin correspondence [4], relates
a 3- complex- dimensional Einstein-Weyl space W to the 2-complex -dimensional
manifold T whose points are the totally geodesic null hypersurfaces in W, the
existence of which is guaranteed by Proposition 2.1. (T is sometimes called the
mini-twistor space of W, by analogy with the previous case.) A point p of W is
represented by the set of surfaces through it, which forms a holomorphic curve £p
in T. Again,.cp is a copy of Cpl, but now the normal bundle is N ˜ 0(2). This
defines the non-local correspondence between W and T.
To define the conformal metric on W, we note as before that a vector v at a
point p of W defines a section Lv of the normal bundle N of L,p. We define v to be
null if Lv has coincident zeros. Again, this is a quadratic condition by virtue of the
character of N.
To define the Weyl connection on W is more complicated. The idea is as follows:
take a curve £p in T corresponding to a point p in W; now take two points 0'1,0'2 on
.c
.cp and consider all the curves q which meet L,p in 0'1 and 0'2. The corresponding
point q in W lies on a geodesic 1=/(0'1,0'2). Geometrically, we can think of 1 as
being the intersection of the two null hypersurfaces in W corresponding to 0'1 and
(12 in T. (This intersection is automatically a geodesic, since the hypersurfaces are
totally geodesic.)
If the Einstein-Weyl space W is the complexification of a real space with positive-
definite conformal structure then one has an alternative description of T. A real
geodesic, will lie in the intersection of a complex conjugate pair of totally geodesic
null hypersurfaces, so symbolically 0'2 =Ut. We can define an orientation on 1 and
associate 0'1 with one orientation and (J2 with the opposite orientation. Then T
appears as the space of oriented geodesics of W. (This picture of T arises readily
from Proposition 2.3.)
The two twistor correspondences can be fitted into a larger picture (Figure 3.1),
241
Too: Three-dimensi˜nal Einstein-Weyl geometry

which we summarise as follows.


3.1.
PROPOSITION

(1) Suppose M has a I-parameter family of conformal isometries (by which
we mean self-diffeomorphisms preserving the conformal structure). The in-
finitesimal generator X is called a conformal Killing field. The space of
trajectories of X automatically has a conformal metric, and also has a nat-
ural Weyl connection which satisfies the Einstein condition ,[5J. The vector
field X induces a holomorphic vector field on the twistor space g, and the
quotient is T.
(2) H W is an Einstein-weyl space then one can construct conformally anti-
self-dual spaces M , as fibre bundles over W, from solutions (V, a) of the
monopole equation




Here V is a function and 0: is a I-form on M (more generally we can work
with a connection on a circle bundle over W [5J).
(3) Given a mini-twistOT space T, the proj'ective tangent bundle P(TT) is the
twistor space for an anti-sell-dual manifold M, and the boundary 8M at
infinity is the Einstein-Weyl3-manifold associated to 7,[4].




Penrose correspondence
g .. ' .. M
1t 1t (2)
(I) (3) (I)
Hitchin correspondence
•W
T..



Figure 3.1 Fitting together the Penrose and Hitchin correspondences. The
numerals refer to Proposition 9.1.
Too: Three-dimensional Einstein-Weyl geometry
242

4. Examples.
4.1 The quadric.



The simplest example of the Hitchin correspondence has for T the complex
quadric Q in 9 P3 . The curves £p are the conics in Q and a geodesic of the
Weyl structure is defined by the set of conics meeting a fixed line in CP3. The
Einstein-Weyl space W is the complexification of S3 or the hyperbolic space H3.
Suitable choices of reality structure give one or the other. Choosing S3 and applying
the construction of (3.1)(3) fills in S3 with the 4-dimensional hyperbolic space H4.
This example essentially just gives an Einstein space. The following Proposition
suggests that Einstein-weyl spaces may be rare:

4.2, [4].
PROPOSITION
H T is an open set in a c˜mpact surface then 7 is Q or a cone in Cp3, and W
is an Einstein space. (The cone gives Bat space R3.J

However, one can analyse an initial value problem, locally, to find:

4.3, [2], 4. An Einstein-weyl space W is determined by an initial
PROPOSITION
value problem with data 4 functions of two variables, or by a characteristic initial
value problem with data 2 functions of 2 variables.


4.4 Einstein- Weyl 3paces from 8 4 •

To find some more examples explicitly we may follow the suggestion of Proposition
3.1(1). The Einstein metric on 8 4 is conformally flat and has many conformal
Killing fields. In this way we obtain two classes of Einstein-Weyl spaces [9] :
.
a˜l

A 3-parameter family of Einstein-Weyl structures on S3, generalising the 1-
parameter family of Einstein metrics.

Class 2
A 2-parameter family of Einstein-Weyl structures on R 3 •

The corresponding conformal Killing fields are shown diagrammatically in Figur˜
4.1. In both classes we can take the equatorial S3 as a representative for W, althougl
in Class 2 we must remove a point.
243
Too: Three-dimensional Einstein-Weyl geometry




Class 1 Class 2




Figure 4.1



This is a manifold which admits no Einstein metric. It can be given an Einstein-
Weyl structure by rescaling the flat metric in R 3 following (2.2), [9]:


9 = dX 2 + d8 2 + sins 2 8diP 2
W = -2˜X,

n = r- 1 , and imposing periodicity in x-
with X = logr and

4.6 Other examples.


I3y ad hoc methods one finds that there are Einstein-Weyl structures on three more
of Thurston's eight homogeneous geometries [10], in which the metric is closely
related to the natural one. These are R X H2, twisted R X H2 ( or the universal
244 Ted: Three-dimensional Einstein-Weyl geometty
I
I
cover of 8£(2, R) ) and twisted R x R2 ( or "Nil"). However in all three of these
examples the metric is an indefinite version of the natural one [9]. ˜
It is also possible to produce these examples by constructing the mini-twistor space i
T directly, [7],[9]. 1

5. Ellipticity and analyticity.
An attractive feature of the Einstein equations is that positive-definite solutions are
automatically real-analytic in suitable co-ordinates, [3]. An analogous statement is
true here:
5.1, [9]. In co-ordinates which are harmonic with respect to the Weyl
PROPOSITION
connection, and with the conformal freedom (2.2) restricted by the condition d*w =
o the Einstem-Weyl equations in the positive-definite case are eniptic. Positive-
definite Einstein-Weyl spaces are therefore real analytic in a suitable gauge.
The construction of the mini-twistor space T requires the complexification of W and
hence is only possible when W is real analytic. The above Proposition therefore
has the corollary:
Every Einstein-Weyl space W has a mini-twistor space T.
COROLLARY 5.2.

It also follows from Proposition 5.1 that the linearisation of the Einstein-Weyl equa-
tions about, say, the Einstein metric on 8 3 gives an elliptic system for the pertur-
bation. In this way one can show the following:
5.3 , [9].
PROPOSITION
The Einstein-Weyl structures found in Class 1 of Section 4.4 include all the solu-
tions obtained by linearising about the Einstein metric.
Note however that there exists another family, disconnected from the Einstein
metric and bif}1Tcating from Class 1.
/
6. Geodesics.


In an Einstein space the behavior of geodesics is closely regulated by the Ricci
scalar, which is automatically constant, [1]. From a study of the geodesic devia-
tion equation one finds that a positive Ricci scalar forces all the geodesics to have
conjuagate points, and ultimately forces the space to be compact, while a negative
Ricci scalar forces geodesics to diverge from each other exponentially rapidly. In an
Einstein-Weyl space different things can happen, as we shall see from two examples.
The first is 8 1 X 52 from Section 4.5. We draw this as the region between two con-
centric spheres in R 3 (Figure 6.1). Geodesics are straight lines, and must reappear
on the inner sphere on the same radius as they meet the outer sphere. Therefore
any non-radial geodesic will close in asymptotically on the radial one parrallel to it,
[9].
The second example is the "Berger sphere" [5], which lies in Class 1 of Section
4.4. This has
245
Tocl: Three-dimensional Einstein-Weyl geometry




limiting geodesic




Figure 6.1




= 11˜ + q˜ + A2 qi
9
= A(1 - A2 )4 113 ,
W



where qi are the standard basis of left-invariant I-forms on S3.
By a direct integration of the geodesjc equations one finds that the 0'3 directions
form geodesics, and that geodesics in all other directions close in asymptotically on
these ones [9].
In these two examples the space of geodesics, which is the mini-twistor space T,
will be non-Ha'US do rff.
What one finds from a study of the geodesic deviation equation in a 3-dimensional
Einstein-Weyl space is that the space of Jacobi fields along each geodesic admits a
complex structure (Proposition 2.3) .. This is the fundamental property which leads
to the existence of the twistor correspondence.
Too: Three-dimensional Einstein-Weyl geometry
246

REFERENCES

[1] Besse,A.L. Einstein Manifolds Springer Verlag 1987
[2] Cartan,E. Sur une classe d'espace de Weyl Ann. Se. Ecole Normale Su-
perieure, 3e. Serie 60 1-16, 1943
[3] Deturek, D. and I<azdan, J. Some regularity theorems in Riemannian geometry
Ann. Se. Ecole Normale Superieure, 4e. Serie 14 249-260, 1981
[4] Hitchin,N.J. Complex manifolds and Einstein equations in Twistor geometry
and Nonlinear System3, Proc., Primorsko, Bulgaria, 1980. Ed. H.D. Doebner and
T.D. Palev Lect1.tre Notes in Maths. 970, Springer 1982
[5] Jones,P.E. and Tod,I<.P. Mini-twistor spaces and Einstein-Weyl spaces Class.
Quant. Gravity 2 565-577, 1985
[6] Penrose,R. Nonlinea.r gravitons and curvedctwistor theory Gen. ReI. Grav.
7 31-52, 1976
[7] Pedersen,H. Einstein- Weyl spaces and (l,n) curves in the quadric surface Ann.
Global Anal. Geom. 4 89-120, 1986
[8] Penrose,R a.nd Rindler,W. Spinors and Space-time, Vol I Ca.mbridge U.P. 1984
[9] Pedersell,H. and Tod,K.P. Three-dimensional Einstein-Weyl geometry Ad-
vances in Maths. To appear
[10] Thurston,W.P. Three dimensional manifolds, Kleinian groups and hyperbolic
geometry AMS Proc. Symp. Pure Maths. 39 1 87-111, 1983
CON˜'ORMAL
HARMONIC MORPHISMS, FOLIATIONS AND SEIFERT FIBRE SPACES
JOHN C. WOOD
University of Leeds


1 INTRODUCTION


A harmonic morphism is a mapping between Rielnannian manifolds which
preserves Laplace's equation in the sense that it pulls back harmonic
functions on the codomain to harmonic functions on the domain (see
below). The aim of this paper is to introduce harmonic morphisms to
the non-specialist and show how they occur naturally in low dimension-
al geometry. In particular. we shall show that each of Thurston's
geometries apart from has a natural map which can be character-
Sol
(§3)˜
ized as the unique non-constant harmonic morphism to a surface
and in §4, we shall show that, for any closed Seifert fibre space
Iv? , there is a harmonic morphism with fibres equal to the fibres of
MJ. Finally, in §5 we use these ideas to give a simple explanation
of why the product of a circle and a closed oriented Seifert fibre
space with oriented fibres is naturally an elliptic surface.


The author wishes to thank several participants at the conference for
useful conversations about this work. especially K.P. Tod, J.H. Rubin-
stein and J.D.S. Jones. Most of the work in this paper represents
work of the author with Paul Baird. It Is hoped that this informal
accoun t will prove useful for wishing to tas te the flavour of
thOSf)

the subject but unable to read the full account in (Baird & Wood,
1988. 1989a. 1989b).


2 DEFINITIONS AND BASIC PROPERTIES


2.1 Harmonic morphisms
kf"l em
pf7 be
and Riemannian manifolds of dimensions and
m
Let
tp:kfll ˜ JIl (f
n respectively, and let be a mapping. Then is
<p
called a harlnollic morphisrll if for any harmonic function .f: u˜ (R
t
Wood: Hannonic Morphisms
248

defined on an open subset of with non-empty,
cp-1 (U)
U N
-1
( U) -+ R is also a harmonic function. Because of the exis-
fQq>:f{)
tence of harmonic coordinates (Greene & Wu, 1962) any harmonic mor-
c!
phism is necessarily Harmonic morphisms can be characterized in
terms of harmonic maps as follows: Say that is horizontall.y
cp
M"'
(weakl.y) conformal if, at each point where dcp "# 0 , writing
xE
x
x = ker dip x x = VxJ. n T x M ,
and maps conformally
V H dq>x' H H
x
x
= A(x)1I vii
Itdcpx( V) !I
onto that is, for all for
Ttp(x)N H '
VE
t
x
=0
>o.
some Setting at points where dtp x ¢ 0 (called
A(X)
)..{x)
?
)":M ˜ IR ,with
critical points) we obtain a continuous function A""
smooth, called the dilatioll of Note that a horizontally weakly
qJ.

is just a Riemannian submersion.
conformal map with Ae 1 We have


Theorem 2.1 (Fuglede, 1978; Ishihara, 1979)
qJ:M"' ˜ JIl
A smooth map is a harmonic morphism if and only if it is
a harmonic map and horizontally weakly conformal.
(8) (b)



=<C
2
M = JR3 , N= 1R
We remark that in the case • this says that
tp:fR ˜ <L
3
is a harmonic morphism if and only if it satisfies


(1)




the first equation expressing harmonicity of and the second,
rp
horizon tal conforJnality.


m'2 n. Further˜
In particular. if is non-constant, if
q>
m= n = 2 ,then is a harmonic morphism if and only if it is a
tp
>2 , the only harmonic Inorphisms
weakly conformal map, and if m =n
are hornothetic maps. Thus we concentrate on the case m >n. Har-
monic morphisms to surfaces (Le. two-dimensional Riemannian mani-
folds) have many nice properties For example we have
249
Wrod: Harmonic Morphisms

Theorem 2.2 (Baird & Eells, 1981)
rp:Jifl ˜ t1
Let be a submersion to a surface. is a harmonic
Then qJ

morphism if and only if it is horizontally conformal and its fibres
are minimal submanifolds.


m = 3, minilnal fibres means that the fibres are
In the case that
geodesics. Another property of harmonic Illorphisms to a surface is
that composition of such a map with a conformal map of surfaces gives
In particular. the concepts of harmonic
another harmonic morphism.
morphism to a Cf'O 2-manifold with a conformal structure (i.e. an
equivalence class of smooth metrics) and harmonic morphism to a Rie-
mann surface (i.e. a one-dimensional comple.x manifold) are well-
defined without specifying a particular Hermitian metric on the sur-
˜$AJ
For this reason, by surface we shall henceforth mean a
face.
2-manifold with a conformal structure.


We can now give some examples of harmonic morphisms. concentrating on
ones from 3-manlfolds to surfaces.

3
denote hyperbolic 3-space thought of as the open
Let 1-1
(1 a)

=l{X1'J˜2 • .(3) : 3
3 2
<I}
I xl
unit disc of 1R (where
D
2 2
IxI 2 2
with the metric 4LdXi2/{1-lxI2)2. Then geode-
)
=X1 +.X2 +X3
3
sics are circular arcs hitting the boundary 8D orthogonally.
z
=0
Identify the hyperbolic plane 1H with the equatorial disc X3
3 3
of Draw all the geodesics of which cut this disc orthogon-
D • 11-1

rp:1I ˜ 1H
2
3
ally. Define a map by projection down these geodesics.
3
(I b) As a limiting case of (3). choose a point Xo E 8D .

=the
3
let point of intersection of the
Then, for each 1H
xE qJ(x)
,

fI':lH ˜ 8D \L\"oJ
3
3 3
aD
x with
geodesic through This defines a map


˜ ([; = ˜\{Xo}
3
where we identify with ([; by stereographic
8n \lxo}
projection.
1R ˜ 1R
3 2
(2) Orthogonal projection
SJ ˜ ˜.
The Hopf map With
(3)
250 Wood: Hannonic Morphisms

˜Z1/Z2E([;U{roJ
2 2
IZf1 +lz2 1 = 1}. This can be defined as (Z1,Z2)

a:s2
followed by the inverse of stereographic projection (fulrol .
---?-




These are essentially the only harmonic morphisms from a simply-
conneeted three-dimensional space form to a surface. PreclselY4


'fheorem 2.3. (Baird & Wood, 1988, 1989a)
Up to isometries of the domain, any non-constant harlnonic morphism of
3 3
g'J to a surface is one of the above tour examples followed
1H 1R
. 01'

by a weakly conformal mapping of the codomain to a surface.


˜ <L
3
In particular. the only solutions q>:1R to the equations (1) and
= where the
(2) above are linear ones qJ(X1.X2,X3) 81x1+a2x2+a3x3

a12+a22+a3˜=O, conlposed with a weakly conformal
satisfy
8i‚¬ft

<r. More interestingly from the geometrical point of
self-mapping of
SJ . . . . ;. s2 is characterized as the unique non-
the Hopf map
view t




s2
s3
constant harmonic morphism from to (up to conformal self map-
pings of $:'.



2.2 Conformal foliations
A smooth foliation on a Riemannian manifold is called conformal (resp.
Riemannian) If its leaves are locally the fibres of a horizontally
rp:˜ ˜ ti
conformal (resp. Riemannian) submers.ion. Let be a sub-
mersive harmonic morphism of a Riemannian 3-manifold to a surface.
p,j3
Then. by Theorem 2.2, the fibres give a conformal foliation of by
geodesics. Conversely, given a conformal foliation by geodesics of a
˜ (for example, geodesically convex), we can
suitable subset U of
form its leaf space˜ a surface. in fact a R.iemann surface if ˜ and
the foliation are oriented. Then the canonical projection of onto
U
its leaf' space is a harmonic morphism. In we can trans-
particular˜
3
late the above results into the theorem that, up to isometries, 1R
3
$3 have just one conformal foliation b,Y geodesics and 1H
and has two.


The above results on the three space forms are global; indeed. there
251
W00<1: Hannonic Morphisms

is a large supply of harmonic morphisms (or conformal foliations by
˜'\or ˜
geodesics) defined on open subsets of a space form. let be a
conformal foliation by geodesics of a geodesically convex subset of a
3
=1R 3 , SJ 3
JI
space form or 1H and let be its leaf space. Then
1E
i:˜ ˜ G
we have an injective map into the space of all geodesics
3
lE
which describes the foliation. can be given the structure
Now G3
IE
of a complex surface (Hitchin, 1982; Baird & Wood, 1989a). Then,
because the foliation is conformaC i is holomorphic. Conversely, a
holomorphlc injective map defines a conformal foliation by geodesics
3
on a suitable open subset of 1E Choosing this subset to be geodes-
.

ically convex, the natural projection of it onto .its leaf space is a
3 3
1E = 1R
For example, if may be identified
harmonic rnorphism. ,G)E3

˜.2.
with the unit tangent bundle of Using stereographic projection
Ttf I tf \lxol ˜
from a point Xo and its differential as a chart
(1

fLx<C ,a tangf˜nt vector at a point 0'-1 (x) E tf\fxol .is represented
by a pair We can then, for example, define a holomor-
(x, v) E <f,x<C.

z˜ (z, e ie z)
phic rnap <r -+ 7˜cf e
where is a constant. This
by
defines a 1- paralneter family of conformal foliations by geodesics of
3
e= 0
dense open subsets of For example, if this is the folia-
1R .

e =1(
3
tion of 1R \{O} by radii and if it is a foliation which
tWists through the unit disc (see Baird & Wood, 1988; Baird, 1987).
The corresponding harmonic morphisms are radial projection
1R \{OJ ˜ tf \:D 2 ˜ D2 where
3 3
D2
and a harmonic morphism is the

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