Self-dual Zollfrei conformal structures with alpha-surface foliation

0.34 MB·English

by  Fuminori Nakata

Self-dual Zollfrei conformal structures with
7
0
0 -surface foliation
α
2

n
a
J Fuminori Nakata
∗

4

February 2, 2008
]
G
D
.
h
t Abstract
a
m
A global twistor correspondenceis established for neutral self-dual
[
conformal structures with α-surface foliation when the structure is

1 close to the standard structure on S2 S2. We need to introduce
v ×
some singularity for the α-surface foliation such that the leaves inter-
6
1 sect on a fixed two sphere. In this correspondence, we prove that a
1
natural double fibration is induced on some quotient spaces which is
1
0 equal to the standard double fibration for the standard Zoll projec-
7 tive structure. We also give a local general forms of neutral self-dual
0 metrics with α-surface foliation.
/
h
t
a
m 1 Introduction
:
v
i C. LeBrun and L. J. Mason investigated two kinds of twistor-type corre-
X
spondences in [6] and [7]. One of them is the correspondence for the Zoll
r
a projective structure on two-dimensional manifolds ([6]). A projective struc-
ture is an equivalence class of torsion-free connections under the projective
equivalence, where two torsion-free connections are called projectively equiv-
alent if they have exactly the same unparameterized geodesics. A projective
structure is called Zoll when all the maximal geodesics are closed. LeBrun
and Mason proved that there is a one-to-one correspondence between
∗This research is supported by the 21st century COE program at Graduate School of
Mathematical Sciences, the University of Tokyo.
1
equivalence classes of orientable Zoll projective structures (B,[ ]), and
• ∇
equivalence classes of totally real embeddings ι : RP2 CP2,
• →
when they are close to the standard structures. Here B is identified with the
moduli space of holomorphic disks on CP2 whose boundaries are contained
in N = ι(RP2).
The second twistor correspondence constructed by LeBrun and Mason
is the one for four-dimensional manifolds equipped with a neutral self-dual
Zollfrei conformal structure ([7]). An indefinite metric on a four-dimensional
manifold is called neutral when the signature is (++ ), and here we con-
−−
sider the indefinite conformal structures represented by such metrics. For a
neutral metric on a four-manifold, we can define the self-duality condition in
a similar way in the Riemannian case (cf.[3],[4],[7]). An indefinite metric is
called Zollfrei when all the maximal null geodesics are closed. In the neutral
four-dimensional case, the Zollfrei condition and the self-dual condition de-
pend only on the conformal class ([7]). LeBrun and Mason introduced the
notion of space-time orientation for a four-manifold with neutral metric, and
proved that there is a one-to-one correspondence between
equivalence classes of space-time oriented self-dual Zollfrei conformal
•
structures (M,[g]), and
equivalence classes of totally real embeddings ι : RP3 CP3,
• →
when they are close to the standard structures. Here M is identified with the
moduli space of holomorphic disks on CP3 whose boundaries are contained
in P = ι(RP3). They also proved that only S2 S2 admits a apace-time
×
oriented neutral self-dual Zollfrei conformal structure.
These two twistor correspondences are real, non-analytic and global ver-
sionsoftwoofthethreetwistorcorrespondencesexplainedin[5]byN.J.Hitchin.
The three are twistor correspondences for (1) complex surfaces with pro-
jective structure, (2) complex four-manifolds with anti-self-dual conformal
structureand(3)complexthree-manifoldswithEinstein-Weylstructure. The
corresponding twistor space is given by complex manifolds Z with an em-
bedded CP1 whose normal bundle is (1), (1) (1) or (2) respectively.
O O ⊕O O
The geometric structures (1), (2) and (3) are given as naturl structures on
the moduli spaces of such embedded CP1 in Z. Hitchin’s argument is local,
and the description is based on holomorphic category. The twistor space for
(1) is sometimes called mini-twistor space (cf.[2]).
2
The twistor correspondence for (2) is originally discovered by R. Penrose
([9]), and the Riemannian version of this twistor correspondence is given
by M. F. Atiyah, N. J. Hitchin and I. M. Singer ([1]). In the Riemannian
case, anti-self-dual conformal structure is automatically analytic since the
equation is elliptic. Moreover the family of rational curves on twistor space
forms a globally defined foliation, and, by this reason, it is streight fowerd to
translate the local description to the global case.
On the other hand, in the cases of LeBrun and Mason, the equations have
non-analytic solutions in general, and the family of CP1 in the twistor space
dose not form a foliation different from the Riemannian case. LeBrun and
Mason overcame these difficulties by using two techniques, the first one is to
usethefamilyofholomorphicdisksinsteadofthatofCP1,andthesecondone
is the setting in terms of Zollfrei condition. Notice that the Zollfrei condition
is an open condition in the space of neutral self-dual metrics (cf.[7]).
Recently, there are some development concerning to the reduction of the
neutral self-dual conformal structures on four-manifolds (M,[g]). M. Duna-
jski and S. West ([3]) proved that, if there is a null conformal Killing vec-
tor field on M, then there is a natural null surface foliation containing this
Killing field, and that a natural projective structure is induced on the leaf
space. D. Calderbank generalized this argument and weaken the assumption,
the weakened assumption is given as a property for a null surface foliation on
M. Both arguments are local, and formulated in smooth category. They also
studied the analytic case, then they showed that, under these conditions, a
twistor correspondence of case (2) induces a twistor correspondence of case
(1) as a reduction.
It would be natural to expect a theory of reduction for the two global
twistor correspondences of LeBrun and Mason. The local theory of Duna-
jski,West and Calderbank would suggest that the naturalclass of such theory
is neutral self-dual Zollfrei conformal structures with closed null surface foli-
ation. Even the standard example on S2 S2, however, does not contained
×
in this class. The purpose of this paper is to set a nice class of neutral self-
dual Zollfrei conformal structures equipped with an α-surface foliation with
some singularity explained later. Then we prove that there is a one-to-one
correspondence similar to the twistor correspondence of LeBrun and Mason,
and that the reduction works globally.
In our situation, the induced projective structure on the leaf space is
proved to be the standard Zoll projective structure. It would be interest-
ing problem to find some different formulations so that non-standard Zoll
3
projective structures are induced by the reduction.
The organization of the paper is as follows. In Section 2 and Section
3, we review the definitions and properties about projective structures and
neutral self-dual conformal structures respectively. In particular in Section
3, we prepare an explicit description without using spinor calculus, which
enables us to establish the general forms of a neutral self-dual metrics with
α-surface foliation in Section 4. In Section 5, we define a notion of basic
α-surface foliation which we need to carry out the reduction. Calderbank
defined the notion of self-dual α-surface foliation. In Appendix 1, we show
that basic is equivalent to self-dual under the assumption of self-duality for
the metric. The basic foliation, however, rather fits to our description. By
using this notion, we give a simple proof of above results of Dunajski and
West in Appendix 2.
We treat the global situation in Section 6 and Section 7. In Section 6,
We formulate a class of neutral self-dual conformal structures with a suitable
α-surface family, and we state the main theorem (Theorem 6.11). There
is a low dimensional or mini-twistor version of the main theorem, and we
prove this version in the rest of Section 6. The proof of the main theorem is
presented in Section 7.
In this article, we follow to the LeBrun and Mason’s conventions of orien-
tations and the terminology of α-surface and β-surface. We assume that all
the manifolds and metrics are C∞, and that the topology of maps between
manifolds is C∞-topology.
2 Projective structure
Let B be an oriented two dimensional manifold, and let = P(TB C) and
W ⊗
R = P(TB) be the projectivizations. Let p : B and pR : R B
W W → W →
be the projections. Then every w R corresponds to a complex line
L T B C, where b = p(w). ∈SiWnce\TWB C = L L¯ , w defines a
w b b w w
⊂ ⊗ ⊗ ⊕
complex structure on T B. Let ◦ be one of the two connected components
b W+
of R whoseelement definesanorientationpreserving complexstructure,
W\W
and we put ◦ to be the other component. Let be the closures of ◦,
W− W± W±
then we have :
W = W+◦ ∪W−◦ ∪WR = W+ ∪W−.
Let V B be a coordinate neighborhood with an oriented coordinate
⊂
4
(y0,y1). By putting ∂i = ∂∂yi, we can trivialize W = P(TCB) on V by
CP1 V ∼ : ([ζ : ζ ],b) [ ζ ∂ +ζ ∂ ] . (2.1)
V 0 1 0 0 1 1 b
× −→ W| 7−→
Noticethat (ζ,b) CP1 V : Imζ 0orζ = , whereζ = ζ /ζ
+ V 1 0
W | ≃ { ∈ × ≥ ∞}
is the fiber coordinate.
Let be a torsion-free connection on B, then the connection form re-
∇
specting the coordinate (y0,y1) is given by gl(2,R)-valued 1-form ω :
ω = (ωi), ∂ = ωi∂ .
j ∇ j j i
The horizontal lift of a tangent vector e on B at λ = λi∂ TB is
i
∈
∂
e˜= e ωiλj . (2.2)
− j ∂λi
Projecting to P(TB), the horizontal lift of e on at ζ = ζ /ζ is given by
1 0
W
∂
e˜= e ω1 +ζ(ω1 ω0) ζ2ω0 (e) . (2.3)
− 0 1 − 0 − 1 ∂ζ
(cid:0) (cid:1)
Now we define a rank 1 distribution LR on R as the tautological lifts,
W
i.e. L is the horizontal lift of the tangent line ∂ +ζ∂ , where x B
R,(x,ζ) 0 1
h i ∈
and ζ RP1 ∼= R,x is the local fiber coordinate. From (2.3), we obtain
∈ W
LR = n where
h i
∂
n = ∂ +ζ∂ ω1 +ζ(ω1 ω0) ζ2ω0 (∂ +ζ∂ ) . (2.4)
0 1 − 0 1 − 0 − 1 0 1 ∂ζ
(cid:0) (cid:1)
We can define a complex distribution on by = n , where n is
+
L W L h i
extended to the vector field on by the analytic continuation for ζ CP1.
+
W ∈
Bydefinition, wehaveL|WR = LR⊗C. Ifweputd = L+ ∂∂ζ¯ ,thend defines
an almost complex structure on + R since d satisfieDs TE + C = d d¯
W \W W ⊗ ⊕
on + R.
W \W
Torsion-free connections and ′ on B are called projectively equiva-
∇ ∇
lent, if and only if they defines exactly the same unparameterized geodesics.
We call a projective structure for a projectively equivalent class [ ]. The
∇
following proposition is prooved in [6]
Proposition 2.1. 1. L and d are defined only by [ ],
∇
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2. L and d are integrable.
Definition 2.2. A projective structure (B,[ ]) is called Zoll if and only if
∇
all of the maximal geodesics on B are closed.
Theorem 2.3 (LeBrun-Mason,[6]). There is a one-to-one correspondence
between
equivalence classes of oriented Zoll projective structures (B,[ ]), and
• ∇
equivalence classes of totally real embeddings ι : RP2 CP2,
• →
when they are close to the standard structures. The correspondence is charac-
terized by a double fibration B p ( +, R) q (CP2,N), where N = ι(RP2),
← W W →
p is the projection, and q is a surjection which is holomorphic on the interior
of .
+
W
Theroughsketch oftheproofisthefollowing. If(B,[ ])isgiven, thenwe
∇
can construct ( +, R) equipped with a rank 1 foliation on R. Collapsing
W W W
this foliation, we obtain the space (CP2,N). Conversely, if ι is given, then
thereisafamilyofholomorphicdisksinCP2 suchthattheboundariesofdisks
are contained in N and that this family defines a foliation on CP2 N. We
\
also remark that each holomorphic disk in this family is characterized by the
condition: therelativehomologyclassofthediskgeneratesH (CP2,N) = Z.
2 ∼
We define B to be the parameter space of this family. Then a Zoll projective
structure [ ] on B is induced so that each closed geodesic is written in the
∇
form p◦q−1(ζ) for some ζ N. Notice that such family of holomorphic disks
∈
is uniquely determined as a deformation of the standard family if ι is close
enough to the standard embedding.
3 Neutral metric
LetM beanorientedfour-dimensionalmanifold,andletg beaneutralmetric
on M where a neutral metric is an indefinite metric of split signature. An
oriented local frame (e ,e ,e ,e ) of the tangent bundle TM is called null
0 1 2 3
tetrad if and only if its metric tensor g = g(e ,e ) is given by
ij i j
1
1
g = (g ) =  − . (3.1)
ij 1
−
1 
 
 
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λ0 λ2
Notice that, if (e ) is a null tetrad, then we obtain g(λ,λ) = det
i λ1 λ3
(cid:18) (cid:19)
for a tangent vector λ = λie . When we make use of null tetrads, the
i
structure group of TM reduces to the Lie group
P
SO(2,2) := P SL(4,R) : tPgP = g . (3.2)
∈
SO(2,2) has two connected c(cid:8)omponents and we denot(cid:9)e SO (2,2) for the
0
identity component.
Definition 3.1 (cf.[7]). M is called space-time orientable when thestructure
group of TM reduces to SO (2,2).
0
Let SL(2,R) and SL(2,R) be copies of SL(2,R). For each (A,B)
+ −
∈
SL(2,R) SL(2,R) , the transformation
+ −
×
e e e e
0 2 A 0 2 tB
e e 7−→ e e
1 3 1 3
(cid:18) (cid:19) (cid:18) (cid:19)
defines an element of SO (2,2). In this way, we obtain a double covering
0
SL(2,R) SL(2,R) SO (2,2). The corresponding Lie algebra isomor-
+ − 0
× →
phism o(2,2) sl(2,R) sl(2,R) is given by
+ −
≃ ⊕
a b e 0
c d 0 e a−d b a+d e
  2 2 . (3.3)
f 0 d b 7−→ c d−a ⊕ f a+d
− (cid:18) 2 (cid:19) (cid:18) − 2 (cid:19)
 0 f c a
 − 
 
Taking M smaller, we can assume that M is space-time oriented and the
structure group of TM lifts of SL(2,R) SL(2,R) . Then we obtain a
+ −
×
decomposition TM = S+ S−, and the Levi-Civita connection on M
⊗ ∇
induces the connections ± on S±. S± are called the positive and negative
∇
spin bundles, and ± are called spin connections. If we take a local null
∇
tetrad (e ) on TM, then is represented by the connection form ω, where ω
i
∇
is a o(2,2)-valued 1-form, and the connection forms ω± of ± are sl(2,R) -
±
∇
valued 1-forms, which are defined as the component of the decomposition of
ω by (3.3).
There is an eigen space decomposition 2 = + − with respect to
∧ ∧ ⊕ ∧
Hodge’s -operator, where 2 = 2TM and ± are the eigen spaces for the
∗ ∧ ∧ ∧
eigen values 1. Let be a null 2-plane in T M and v ,v be the basis of
x 1 2
± V
. Then the bivector v v belongs to + or −.
1 2
V ∧ ∧ ∧
7
Definition 3.2. Let = v ,v T M be a null 2-plane. If v v +
1 2 x 1 2
V h i ⊂ ∧ ∈ ∧
then is called α-plane, and if v v − then is called β-plane. Let
1 2
V ∧ ∈ ∧ V
S M be a totally null embedded surface. If every tangent plane on S at
⊂
each point is α-plane, then S is called α-surface. β-surface is defined in the
similar way.
Let (M,g) be a space-time oriented neutral manifold, and (e ) be a null
i
tetrad on a open set U M. From now on, we denote e = φ ,e = φ for
2 0 3 1
⊂
a later convenience. The following lemma is checked by a direct calculation.
Lemma 3.3. + = ϕ ,ϕ ,ϕ , − = ψ ,ψ ,ψ , where
1 2 3 1 2 3
∧ h i ∧ h i
1
ϕ = e e , ϕ = φ φ , ϕ = (e φ e φ ),
1 0 1 2 0 1 3 0 1 1 0
∧ ∧ √2 ∧ − ∧
(3.4)
1
ψ = e φ , ψ = e φ , ψ = (e φ +e φ ).
1 0 0 2 1 1 3 0 1 1 0
∧ ∧ √2 ∧ ∧
g induces indefinite metrics on ± whose metric tensors are both given
∧
by the following matrix with respect to the frames (3.4) :
0 1 0
h = 1 0 0 . (3.5)
 
0 0 1
−
 
Let h be a Lie algebra defined by
a 0 c
h = X gl(3,R) : tXh+hX = 0 = 0 a b .
∈  − 
b c 0
 
(cid:8) (cid:9)
 
The Levi-Civita connection induces connections on ± whoseconnection
∇ ∧
forms are represented by h-valued 1-form with respect to the frames (3.4).
We can check that the exterior product representation associated to −
∧
is
a b e 0
a d 0 √2c
c d 0 e −
ρ− :   0 d a √2b . (3.6)
f 0 d b 7−→  − 
− √2b √2c 0
0 f c a
 −   
 
So the connection form of the induced connection on − is given by the
∧
h-valued 1-form θ = (θi) = ρ−(ω), where ω is the connection form of the
j
8
Levi-Civita connection. This connection naturally induces the connection on
−, where − = − C is the complexification. The horizontal lift of a
C C
∧ ∧ ∧ ⊗
tangent vector e on M at λiψ − is
i C
∈ ∧
∂
e˜= e θi(e)λj . (3.7)
− j ∂λi
Let = [ψ] P( −C) : g(ψ,ψ) = 0 and R = [ψ] P( −) : g(ψ,ψ) =
Z { ∈ ∧ } Z { ∈ ∧
0 . Let p : M and pR : R M be the projections. Then a trivializa-
} Z → Z →
tion of on the open set U M is given by :
Z ⊂
ι : CP1 U ∼ : ([ζ : ζ ],x) [ ζ2ψ +ζ2ψ +√2ζ ζ ψ ] . (3.8)
× −→ Z|U 0 1 7−→ 0 1 1 2 0 1 3 x
This is nothing but the correspondence between the fiber coordinate [ζ :
0
ζ ] CP1 and the complex β-plane ζ e +ζ e , ζ φ +ζ φ , since we have
1 0 0 1 1 0 0 1 1
∈ h i
(ζ e +ζ e ) (ζ φ +ζ φ ) = ζ2ψ +ζ2ψ +√2ζ ζ ψ .
0 0 1 1 ∧ 0 0 1 1 0 1 1 2 0 1 3
Restricting thefiber coordinate[ζ : ζ ]toRP1, wealso obtainatrivialization
0 1
of R,andeachpointin R correspondstoarealβ-planeinthesamemanner.
Z Z
Let βu TxM C be the complex β-plane corresponding to u R,
where x =⊂p(u). S⊗ince T M C = β β¯ , z defines a complex∈stZru\ctZure
x u u
⊗ ⊕
J on T M, and it is easy to check that J preserves the metric g. Let ◦
x Z+
be one of the two connected components of R whose element defines
Z \ Z
an orientation preserving complex structure, and we put ◦ to be the other
Z−
component. Let be the closures of ◦, then we have :
Z± Z±
Z = Z+◦ ∪Z−◦ ∪ZR = Z+ ∪Z−.
Let − π P( −) be the projectivization, then we obtain at (λ1,λ2,λ3) =
C C
∧ → ∧
(ζ2,ζ2,√2ζ ζ ),
0 1 0 1
∂ ∂
π θiλj = b+ζ(d a) ζ2c ι , (3.9)
∗ j ∂λi − − ∗ ∂ζ
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
where ζ = ζ /ζ is the non-homogeneous coordinate. From (3.7), the hori-
1 0
zontal lift of the tangent vector e on M to is
Z
∂
e˜= e b+ζ(d a) ζ2c (e) . (3.10)
− − − ∂ζ
(cid:0) (cid:1)
9
We can define a rank 2 distribution ER on R as the tautological lifts,
Z
i.e. E is the horizontal lift of the β-plane e +ζe ,φ +ζφ , where
R,(x,ζ) 0 1 0 1
h i
x M and ζ RP1 ∼= R,x. ER is called the twistor distribution [3] or the
∈ ∈ Z
Lax distribution [2]. From (3.10), we obtain ER = m1,m2 where
h i
m = e +ζe +Q (ζ)∂ , Q (ζ) = (b+ζ(d a) ζ2c)(e +ζe ),
1 0 1 1 ζ 1 0 1
− − −
m = φ +ζφ +Q (ζ)∂ , Q (ζ) = (b+ζ(d a) ζ2c)(φ +ζφ ).
2 0 1 2 ζ 2 0 1
− − −
(3.11)
We can define a complex distribution E on by E = m ,m , where m
+ 1 2 1
Z h i
and m are extended to the vector fields on analytically in ζ CP1. By
2 +
Z ∈
definition, we have E|ZR = ER ⊗C. If we put D = E + ∂∂ζ¯ , then D defines
an almost complex structure on + R since T + CD= DE D¯ on + R.
Z \Z Z ⊗ ⊕ Z \Z
The following theorem is basic and proved in [7], and see also [3].
Theorem 3.4. 1. E and D are defined only by the conformal class [g],
2. ER is Frobenius integrable if and only if [g] is self-dual. Moreover, the
almost complex structure on + R defined from D is integrable if and
Z \Z
only if [g] is self-dual.
Definition 3.5. Let(M,[g])beaneutralseld-dualconformalstructure, then
(M,[g]) is called Zollfrei if and only if all of the maximal null geodesics on
M are closed.
Theorem 3.6 (LeBrun-Mason,[7]). There is a one-to-one correspondence
between
equivalence classes of space-time oriented self-dual Zollfrei conformal
•
structures (M,[g]), and
equivalence classes of totally real embeddings ι : RP3 CP3,
• →
when they are close to the standard structures. The correspondence is charac-
terized by a double fibration M p ( +, R) q (CP3,P), where P = ι(RP3),
← Z Z →
p is the projection, and q is a surjection which is holomorphic on the interior
of .
+
Z
The proof is conceptually similar to Theorem 2.3. M is defined from ι as
the parameter space of the family of holomorphic disks in (CP3,P) foliating
CP3 P. Such family is uniquely determined if ι is close enough to the
\
standard embedding.
10