connections with exotic holonomy
TRANSCRIPT
CONNECTIONS WITH EXOTIC
HOLONOMY
Lorenz Johannes Schwachhofer
A Dissertation
in Mathematics
Presented to the Faculties of the University of Pennsylvania in
Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
August, 1992
Wolfgang Ziller, Thesis Advisor
Ted Chinburg, Graduate Group Chairman
In Dankbarkeit meinen Eltern
und Heike.
ii
ACKNOWLEDGEMENTS
First and foremost I thank my advisor Wolfgang Ziller for his tremendous sup-
port throughout my stay at the University of Pennsylvania, for his willingness to
schedule appointments whenever I felt the need, for his enthusiasm and love for
mathematics which were an inspiration in every phase of my studies, and finally for
his encouragement during the nervewracking months waiting for a job offer. Over
the years I learned to see a friend in him as well as an advisor.
I extend thanks also to my defense committee: Eugenio Calabi, Steven Shatz and
especially to Herman Gluck for his advice and support during the job application
process.
I wish to thank Bruce Kleiner and J.M. Landsberg for taking an interest in my
work and for the many inspiring conversations. I am grateful also to Karl-Heinrich
Hofmann from the Technische Hochschule Darmstadt whose support made it possi-
ble for me to come to the United States for Graduate studies.
Finally, I thank my friends and fellow students both at Tulane and the University
of Pennsylvania. It would be impossible to list them all, but there are some people
that were especially close to me: Christine Escher and Tom Schmidt with whom I
shared many good times, my roommates Gustavo Comezana who introduced me to
non-stop classical music, and Moez Alimohamed (the ultimate genius).
Last but not least I wish to thank my wife Heike whose ’transatlantic’ support
was always a source of solace I could depend on.
iii
ABSTRACT
CONNECTIONS WITH EXOTIC HOLONOMY
Lorenz Johannes Schwachhofer
Wolfgang Ziller (Supervisor)
In 1955, Berger [Ber] partially classified the possible irreducible holonomy repre-
sentations of torsion free connections on the tangent bundle of a manifold. However,
it was shown by Bryant [Br2] that Berger’s list is incomplete. Connections whose
holonomy is not contained on Berger’s list are called exotic.
We investigate examples of a certain 4-dimensional exotic holonomy representa-
tion of Sl(2,R). We show that connections with this holonomy are never complete,
give explicit descriptions of these connections on an open dense set and compute
their group of symmetry. Finally, we give strong restrictions for their existence on
compact manifolds.
iv
Was im Endlichen gilt,
gilt im Unendlichen schon lange.
Thomas Horsch
v
Contents
1 Introduction 1
2 Examples of Singular H3-connections 7
2.1 Example I : The homogeneous case (type Σ00) . . . . . . . . . . . . . 9
2.2 Example II : Types Σ±0 and Σ1
c . . . . . . . . . . . . . . . . . . . . . 12
2.3 Example III : Type Σ2c . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 The Structure Equations 16
4 Non-completeness 35
5 The Singular H3-connections 37
5.1 H3-connections of type Σ00 . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 H3-connections of type Σ±0 . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 H3-connections of type Σ1c . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 H3-connections of type Σ2c . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Regular H3-connections . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 H3-connections on compact manifolds 50
vi
1 Introduction
Let Mn be a smooth connected n-dimensional manifold. Let P(M) denote the set of
piecewise smooth paths γ : [0, 1] → M , and for x ∈ M , let Lx(M) ⊆ P(M) denote
the set of x-based loops, i.e. paths for which γ(0) = γ(1) = x.
Let ∇ be a torsion free affine connection on the tangent bundle of M . For each
γ ∈ P(M), the connection ∇ defines a linear isomorphism Pγ : Tγ(0)M → Tγ(1)M ,
called parallel translation along γ. For each x ∈ M , we define the holonomy group
of ∇ at x to be Hx := Pγ | γ ∈ Lx ⊆ Gl(TxM).
It is well known that Hx is a Lie subgroup of Gl(TxM), and that for any γ ∈
P(M), Pγ induces an isomorphism of Tγ(0)M with Tγ(1)M which identifies Hγ(0) with
Hγ(1) [KN].
Choose an x0 ∈ M and an isomorphism i : Tx0M → R
n. Then, because M is
connected, the conjugacy class of the subgroup H ⊆ Gl(n,R) which corresponds
under i to Hx0⊆ Gl(Tx0
M) is independent of the choice of x0 or i. By abuse of
language, we speak of H as the holonomy group and of the Lie algebra h of H as
the holonomy algebra of (M ,∇).
The following is a basic question in the theory:
Which (conjugacy classes of) subgroups H ⊆ Gl(n,R) can occur as the holonomy of
some torsion free connection ∇ on some n-manifold M?
Note that the condition of torsion freeness makes this problem non-trivial. In
fact, it is not hard to see that any representation of a connected Lie group can be
realized as the holonomy of some connection (with torsion) on an arbitrary manifold.
1
A necessary condition on the holonomy algebra of a torsion free connection was
derived by M. Berger [Ber] in his thesis as follows.
Let V be a vector space and define for a given Lie algebra g ⊆ gl(V ):
K(g) :=
φ : Λ2(V ) → g∣
∣ φ linear,∑
σ∈A3
φ(uσ(1), uσ(2))uσ(3) = 0 for all ui ∈ V
,
and
K1(g) :=
ψ : V → K(g) | ψ linear,∑
σ∈A3
ψ(uσ(1))(uσ(2), uσ(3)) = 0 for all ui ∈ V
.
Given (M,∇) as above and x0 ∈M , the curvature tensor of ∇ at x0, i.e. the map
Rx0: Λ2(Tx0
M) → gl(Tx0M) defined by Rx0
(u, v)w = ∇u∇vw −∇v∇uw −∇[u,v]w,
is known to have its values in the holonomy algebra hx0⊆ gl(Tx0
M), and to satisfy
the first and second Bianchi identities. This is equivalent to saying
Rx0∈ K(hx0
),
and
∇Rx0∈ K1(hx0
).
In this notation, Berger’s criterion is:
If g′ ⊂ g is a proper sub-algebra, and K(g
′
) = K(g), then g cannot be
the holonomy algebra of any torsion free connection on any n-manifold
M .
This criterion is a consequence of the Ambrose-Singer Holonomy Theorem, which
states that the holonomy algebra hx0is generated by the image of the curvature map
Rx0and its parallel translations [KN, II.8.1].
2
The study of locally symmetric connections, i.e. connections with ∇R = 0, can
be reduced to certain problems in the theory of Lie algebras. We therefore wish to
exclude this case from our discussion. A second necessary condition for g to be the
holonomy of a torsion free connection which is not locally symmetric is therefore
K1(g) 6= 0.
These two criteria are also referred to as Berger’s first and second criterion.
Using these, Berger [Ber] was able to partially classify the possible Lie algebras of
holonomy groups of torsion free connections which are not locally symmetric. His
classification falls into three parts:
• The first part classifies all possible Riemannian holonomies, i.e. connections
with holonomy group H ⊆ O(n,R). It turns out that the possible holonomy
groups of non-symmetric connections are those subgroups which act transi-
tively on the unit sphere in Rn. In fact, it is by now well known which elements
in Berger’s list actually do occur as holonomies of Riemannian metrics. We
mention in this context the work of Simons [S], Calabi [C], Alekseevskii [A]
and Bryant [Br1].
• The second part classifies all possible irreducible pseudo-Riemannian holono-
mies, i.e. connections with holonomy group H ⊆ O(p, q). In this case, the
question whether or not these candidates actually do occur as holonomies has
been resolved except for the group SO∗(2n) ⊆ Gl(4n,R) for n ≥ 3.
• The third part classifies the possible irreducibly acting holonomy groups of
affine torsion free connections, i.e. holonomy groups which do not leave in-
3
variant any non-degenerate symmetric bilinear form. These connections are
the least understood. In fact, Berger’s list in this case is incomplete, conceiv-
ably omitting a finite number of possibilities. The holonomies which are not
contained in Berger’s list are referred to as exotic holonomies.
R.Bryant [Br2] showed that exotic holonomies do, in fact, exist. He investigated
the irreducible representations of Sl(2,R) which can be described as follows:
For n ∈ N, let Vn := homogeneous polynomials in x and y of degree n which
is an n + 1-dimensional vector space. There is an Sl(2,R)-action on Vn induced by
the transposed action of Sl(2,R) on R2, i.e. if p ∈ Vn and A ∈ Sl(2,R) then
(A · p)(x, y) := p(u, v) with (u, v) = (x, y)A.
It is well known that this action is irreducible for every n and moreover that -
up to equivalence - this is the only irreducible n + 1-dimensional representation of
Sl(2,R). [BD]
Let Hn ⊆ Gl(Vn) be the image of this representation and let hn ⊆ gl(Vn) be the
Lie algebra of Hn. Bryant showed that hn does not satisfy Berger’s first and second
criterion if n ≥ 4.
For n = 3, however, he proved the existence of torsion free connections on 4-
manifolds whose holonomy group is H3, even though this group does not appear on
Berger’s list. We shall refer to these connections as H3-connections.
A diffeomorphism φ : M → M preserving the connection will be called a sym-
metry of ∇.
4
It turns out that locally there are very few examples of H3-connections. In fact,
the local classification given by Bryant can be summarized as follows:
• There is one example of a homogeneous H3-connection whose symmetry group
is five-dimensional.
• There is a finite set of H3-connections with a three-dimensional symmetry
group.
• There is a 1-parameter family of H3-connections with a one-dimensional sym-
metry group.
This classification is obtained by the methods of Exterior Differential Systems.
This approach, however, makes a concrete description of these connections very
difficult. In this thesis, we will describe the H3-connections more explicitly and will
also investigate their global behavior.
• The homogeneous H3-connection can be described globally.
• The singular H3-connections can be described on the dense open subsets on
which the action of their symmetry groups is locally free.
• For the regular H3-connections we can only give a description on some open
subset which will not be dense in general.
In particular, we will compute the Lie algebras of the symmetry groups of all
H3-connections.
5
Following this introduction, we will first give the descriptions of the singular
H3-connections (chapter 2).
In chapter 3, we derive the structure equations for H3-connections, following
closely [Br2].
As a first global result we will show in chapter 4 the
Theorem 4.1 H3-connections are never complete.
In chapter 5, we then solve the structure equations for the singular H3-connec-
tions under the generic assumption that the group action of the symmetry group
is locally free, as well as the structure equations for the regular H3-connections on
some open subset. This will show that the examples given in chapter 2 form in fact
a complete list of singular H3-connections under these restrictions.
Finally, in chapter 6 we shall be concerned with H3-connections on compact man-
ifolds, and we will prove
Theorem 6.1 Let M be a compact 4-manifold with an H3-connection. Then M is
locally homogeneous.
In fact, we will prove that any H3-connection on a compact manifold is given as a
biquotient of a certain Lie group (Corollary 6.8). We suspect that these biquotients
do not exists, i.e. that there are no H3-connections on compact manifolds.
6
2 Examples of Singular H3-connections
We first introduce some notational conventions. Recall that V3 is the vector space of
homogeneous polynomials of degree 3 in the variables x and y, and Sl(2,R) acts on
V3 by the action induced by the transposed action of Sl(2,R) on R2 = spanx, y.
We let ρ3 : Sl(2,R) → H3 ⊆ Gl(V3) and (ρ3)∗ : sl(2,R) → h3 ⊆ gl(V3) denote
the representation homomorphisms, and define a basis E1, E2, E3 of h3 by Ei :=
(ρ3)∗(Ei) where the basis E1, E2, E3 of sl(2,R) is given by
(
a cb −a
)
= aE1 + bE2 + cE3.
Furthermore, we let e0, . . . , e3 with ei = x3−iyi be a basis of V3, and ei | 0 ≤
i ≤ 3 be the standard basis of R4.
We fix once and for all a linear isomorphism
λ : V3 → R4
ei 7→ ei.
Then h3
:= λh3λ−1 ⊆ gl(4,R) has E1, E2, E3 with Ei = λEiλ
−1 as a basis, and
one sees that
r1E1 + r2E2 + r3E3 =
3r1 r33r2 r1 2r3
2r2 −r1 3r3r2 −3r1
.
We shall from now on use λ to identify h3 with h3.
Recall that if M is a manifold with an H3-connection ∇ then a diffeomorphism
φ : M → M preserving the connection ∇, i.e. satisfying φ∗ (∇XY ) = ∇φ∗(X)φ∗(Y )
for all vector fields X and Y on M , will be called a symmetry of ∇.
7
In order to describe a connection on a manifold M , we shall in each case give a
frame on M and the connection form θ w.r.t. this frame which takes values in h3.
This form is the pullback of the connection form on the frame bundle of M under
the section given by the frame.
Let X0, . . . , X3 be the given frame and let ω0, . . . , ω3 denote the dual coframe.
We define the V3-valued 1-form ω by
ω =∑
i
ωi ei,
which establishes an isomorphism between TpM and V3 for all p ∈M .
Then we can describe the covariant derivative associated to the connection by
ω(∇XiXj) := −θ(Xi) · ω(Xj) = −θ(Xi) · ej . (1)
The connection being torsion free is equivalent to the condition that
θ(Xi) · ej − θ(Xj) · ei = dω(Xi, Xj) for all i, j. (2)
The holonomy algebra of these connections is contained in h3 by the Ambrose-
Singer-Holonomy Theorem mentioned earlier. In fact, we will show (Corollary 3.2)
that the holonomy algebra of any such connection is actually equal to h3, provided
the connection is not flat.
As we shall see in chapter 3, to every H3-connection we can associate a pair of
homogeneous polynomials a ∈ V2 and b ∈ V3 as well as a constant c ∈ R. For future
reference we shall give these polynomials, called the structure polynomials, and this
constant in each particular case.
The description of these examples will be motivated in chapter 5.
8
2.1 Example I : The homogeneous case (type Σ0
0)
Let G = ASl(2,R) =
Axy
0 0 1
∣
∣
∣
∣
∣
∣
A ∈ Sl(2,R), x, y ∈ R
be the group of
unimodular affine motions of R2 and let g be the Lie algebra of G. Let Aij stand
for the 3 × 3-matrix with (i, j)-th entry 1 and all other entries 0. Then we define a
basis of the Lie algebra g of G by
Z0 = −9A23
Z1 = −3
2A13 + 6A21
Z2 = 3(A11 − A22)
Z3 = −9
2A12
Y = A13 + 2A21
and the subgroup H ≤ G as H = exp(tY ) | t ∈ R. Furthermore, we decompose g
as
g = h ⊕ m
with h = span(Y ) and m = span(Z0, . . . , Z3).
One checks that [h,m] ⊆ m, i.e. the homogeneous space G/H is reductive.
Moreover, if we define
ı : m → V3
Zi 7→ ei
then ad(Y ) = (ı)−1 E3 ı.
9
We wish to define an H3-connection on G/H such that the canonical action of
G on G/H is an action by symmetries. To do this, we need to define a map
λ : m → h3 ⊆ gl(V3)
such that
λ([Y, Zi]) = [E3, λ(Zi)] for i = 0, . . . , 3
and
λ(Zi) · ej − λ(Zj) · ei = ı([Zi, Zj]m)
with the isomorphism ı defined above [KN, X 2.1,2.3,4.2]. The covariant derivatives
are then defined by
ı(∇ZiZj) = λ(Zi) · ej ,
where TpM is identified with m.
One then checks that the following map satisfies these two conditions and thus
defines an H3-connection on M .
λ(Z0) = 0, λ(Z1) = E3, λ(Z2) = −E1, λ(Z3) = −3E2.
The structure polynomials and the structure constant in this case are
a = −2x2, b = 4x3, and c = 0.
It is easily seen that G acts transitively on the set of parabolas in R2. Also, note
that H is the group of unimodular affine motions which leave the standard parabola
y = x2 invariant. Therefore, we can naturally identify M = G/H with the space of
parabolas in R2.
10
Let ω0, . . . , ω3, θ be the coframe on G dual to the vector fields Z0, . . . , Z3, Y . Let
τi := ωi ∧ . . . ∧ ω3 for 0 ≤ i ≤ 3. It is easily seen that τi is invariant under the
isotropy representation of H , and therefore there are induced forms on G/H which
we also denote by τ0, . . . , τ3.
One checks easily that these forms satisfy the Frobenius condition
dτi = αi ∧ τi for some 1-form αi,
and hence the flag of distributions D0 ⊆ . . . ⊆ D3 on the tangent space of G/H
given by τi(Di) = 0 is integrable [EDS, II.1.1].
Let L : ASl(2,R) → Sl(2,R) be the homomorphism which maps an affine motion
to its linear part. Then we can define an angle function
θ : G/H −→ S1
gH 7−→L(g) ·
(
01
)
||L(g) ·(
01
)
||.
This function assigns to each parabola its direction, and it is easily checked
that θ is a fibration over S1 whose fibers are the maximal connected integral leaves
(m.c.i.l.’s) of D3, and that the fibers are totally geodesic. Also, since these fibers are
all parabolas with a given direction it is easy to see that they are diffeomorphic to R3.
Now let us investigate the level sets of θ. By homogeneity all level sets are
equivalent. Thus, let Y3 := (y = ea x2 + b x+ c) | a, b, c ∈ R. Then the function
r3 : Y3 → R which assigns the slope to a parabola, i.e. r3 (y = ea x2 + b x+ c) = a,
satisfies dr3 = ω2|Y3. Again, one can check that r3 is a fibration of Y3 over R whose
fibers are totally geodesic and diffeomorphic to R2.
11
Next, let Y2 be a level set of r3, e.g. Y2 := r−13 (0) = (y = x2+b x+c) | b, c ∈ R,
i.e. a set of parabolas with given direction and slope. We compute the connection
on Y2 and obtain that Y2 is flat. Note that a curve γ in Y2 can be described by
the curve of the vertices of γ(t). Then we compute that - up to parametrization -
the vertices of a geodesic in Y2 either move along a parabola of slope −2 or along a
vertical line.
Also, note that the restriction ω1|Y2is closed, hence there exists a function r2 :
Y2 → R such that dr2 = ω1|Y2. One finds that the level sets of r2 are the parabolas
in Y2 whose vertices lie on a fixed vertical line. By the previous, these level sets are
geodesics.
Finally, if Y1 is a level set of r2, i.e. a geodesic, we find some parametrization
r1 : Y1 → R such that dr1 = ω0|Y1.
Therefore, on every m.c.i.l. Yi of Di, i ≤ 3 we have a function ri : Yi → R
satisfying dri = ωi−1|Yi, and all these functions are totally geodesic fibrations of Yi.
2.2 Example II : Types Σ±0
and Σ1
c
Let c ∈ R be a given constant and define the Lie algebra g±c = span(Z1, Z2, Z3) with
the bracket relations
[Z1, Z2] = Z3
[Z1, Z3] = −c8Z2
[Z2, Z3] = ∓4cZ1.
It is easily seen that g±0∼= n3, the Lie algebra of the 3-dimensional Heisenberg
12
group N3, g−c∼= su(2) if c > 0 and g±
c∼= sl(2,R) in the remaining cases. We let
G±c be a Lie group corresponding to g±
c such that G±0 = N3, G
±c = Sl(2,R) or
G−c = SU (2) if c > 0.
Let M±c := R
+ × G±c and let t0 : M±
c → R+ be the projection onto the first
factor. On G±c , we have the left invariant vector fields Z1, Z2, Z3 corresponding to
the basis of g±c . There exists a vector field Z0 on M±
c satisfying Z0(t0) ≡ 1 and
[Z0, Zi] = 0 for i = 1, 2, 3.
We then define a frame on M±c by
X0 = −(12t20 ± c)
2t0Z1 +2Z3
X1 =3
4(4t20 ∓ c)Z0 +Z2
X2 = ± 1
t0Z1
X3 = ±3
2Z0
Now we give a connection w.r.t. this frame by the h3-valued 1-form
θ±c = θ1E1 + θ2E2 + θ3E3
whereθ1 = −2t0 ω1,
θ2 = c4t20 ∓ c
8t0ω0 −8t20 ∓ c
4t0ω2, and
θ3 = −12t20 ∓ c
4t0ω0 ∓ 1
2t0ω2.
The covariant derivative is defined by (1). Moreover, one checks that (2) is
satisfied, i.e. ∇ is torsion free. Also, ∇ is not flat, thus by Corollary 3.2 we conclude
that ∇ is an H3-connection.
Note that the natural left action of G±c on M±
c is an action by symmetries since
it fixes the level sets of t0 and therefore preserves θ±c .
13
The structure polynomials are
a = ±x2 − 12
(4t20 ∓ c) y2, and b = −t0y (±6x2 + (4t20 ∓ c) y2)
We shall see later that Mε1
c1and Mε2
c2with εi ∈ ± are equivalent iff ε1 = ε2
and sign(c1) = sign(c2). Thus the M±c give 6 different examples of H3-connections.
2.3 Example III : Type Σ2
c
Let k ∈ R\0 be a given constant and define the Lie algebra g±k = span(Z1, Z2, Z3)
with the bracket relations
[Z1, Z2] = Z3
[Z1, Z3] = ∓9Z2
[Z2, Z3] = ±24kZ1.
It is easily seen that g+k
∼= su(2) if k > 0 and g±k∼= sl(2,R) in the remaining
cases. We let G±k be a Lie group corresponding to g±
k such that G±k = Sl(2,R) or
G+k = SU (2).
Let M±k := R
+ × G±k and let t0 : M±
k → R+ be the projection onto the first
factor. On G±k , we have the left invariant vector fields Z1, Z2, Z3 corresponding to
the basis of g±k . There exists a vector field Z0 on M±
k satisfying Z0(t0) ≡ 1 and
[Z0, Zi] = 0 for i = 1, 2, 3.
We then define a frame on M±k by
X0 = − 3
2t0(2kt20 ∓ 1)Z1 −Z3
X1 = −1
2(6kt20 ± 1)Z0 +Z2
X2 =1
2t0Z1
X3 = −3
2Z0
Now we give a connection w.r.t. this frame by the h3-valued 1-form
θ±k = θ1E1 + θ2E2 + θ3E3
14
where
θ1 = 2kt0 ω1,
θ2 = − 3
2t0(±6kt20 + 1) ω0 +
1
2t0(4kt20 ∓ 1) ω2, and
θ3 =3
2t0(2kt20 ± 1) ω0 +
1
2t0ω2.
Again, the covariant derivative is defined by (1). Moreover, one checks that (2)
is satisfied, i.e. ∇ is torsion free. Also, ∇ is not flat for k 6= 0, thus by Corollary
3.2 we conclude that ∇ is an H3-connection.
Note that the natural left action of G±k on M±
k is an action by symmetries since
it fixes the level sets of t0 and therefore preserves θ±k .
The structure polynomials and the structure constant are
a = k (x2 ± y2) − 2 (kt0y)2, b = 2kt0y (3k (x2 ± y2) + 2 (kt0y)
2) and c = ±6k2.
We shall see later that Mε1
k1and Mε2
k2with εi ∈ ± are equivalent iff ε1 = ε2
and sign(k1) = sign(k2). Thus the M±k give 4 different examples of H3-connections,
and we will also show that they are indeed different from the connections given in
the previous section.
15
3 The Structure Equations
In this section we will mainly recall the results of Bryant [Br2] on H3-connections
and introduce a notation convenient for our purposes.
First, we shall define the bilinear parings
< , >p: Vn ⊗ Vm → Vn+m−2p
by
< u, v >p=1
p!
p∑
k=0
(−1)k
(
pk
)
∂pu
∂kx∂p−ky
∂pv
∂p−kx∂kyfor u ∈ Vn, v ∈ Vm.
It can be shown that these pairing are Sl(2,R)-equivariant and therefore are the
projections onto the summands of the Clebsch-Gordan formula.
Now we explicitly describe the spaces
K(g) :=
φ : Λ2(V ) → g∣
∣ φ linear,∑
σ∈A3
φ(uσ(1), uσ(2))uσ(3) = 0 for all ui ∈ V
,
and
K1(g) :=
ψ : V → K(g) | ψ linear,∑
σ∈A3
ψ(uσ(1))(uσ(2), uσ(3)) = 0 for all ui ∈ V
.
By straightforward computation we get as a basis for K(h3) the maps
φ0 : Λ2(V3) → h3 φ1 : Λ2(V3) → h3 φ2 : Λ2(V3) → h3
φ0(e0, e1) = 0 φ1(e0, e1) = 0 φ2(e0, e1) = 6E3
φ0(e0, e2) = 0 φ1(e0, e2) =−3E3 φ2(e0, e2) =−3E1
φ0(e0, e3) =−9E3 φ1(e0, e3) = 9E1 φ2(e0, e3) = 9E2
φ0(e1, e2) = 5E3 φ1(e1, e2) = −E1 φ2(e1, e2) =−5E2
φ0(e1, e3) =−3E1 φ1(e1, e3) = 3E2 φ2(e1, e3) = 0φ0(e2, e3) =−6E2 φ1(e2, e3) = 0 φ2(e2, e3) = 0
16
Also, we compute
K1(h3) := ψb0,...,b3 | b0, . . . , b3 ∈ R,
with
ψb0,...,b3(ei) := (3 − i)bi+1 φ0 − (3 − 2i)bi φ1 − ibi−1 φ2, 0 ≤ i ≤ 3.
As Sl(2,R) acts both on V3 and on h3 via the adjoint representation, there is an
induced action of Sl(2,R) on Hom(Λ2V3, h3) and K(h3) is easily seen to be invariant
under this action. Moreover, the basis φ0, φ1, φ2 was chosen such that the map
ıK : K(h3) → V2
φi 7→ x2−iyi
is an Sl(2,R)-equivariant isomorphism.
Furthermore, K1(h3) can be seen to be invariant under the Sl(2,R)-action on
Hom(V3,K(h3)), and the map
ıK1 : K1 −→ V3
ψb0,b1,b2,b3 7−→ b3 x3 − 3b2 x
2y + 3b1 xy2 − b0 y
3
is an Sl(2,R)-equivariant isomorphism.
The following Lemma is very simple but will be useful later on.
Lemma 3.1 Every φ ∈ K(h3)\0 is surjective.
Proof: Let φ = a0 φ0 + a1 φ1 + a2 φ2 and suppose a0 6= 0. Then φ(e2, e3) =
−6a0 E2, thus E2 ∈ im(φ). Also, φ(e1, e3) = −3a0 E1 + 3a1 E2, thus E1 ∈ im(φ).
Finally, φ(e0, e3) = −9a0 E3 + 9a1 E1 + 9a2 E2, thus E3 ∈ im(φ). Therefore, φ is
surjective if a0 6= 0. Similar arguments show that φ is surjective whenever φ 6= 0.
q.e.d.
17
Corollary 3.2 If the holonomy group of a torsion free connection ∇ on a manifold
M is contained in H3 then either the holonomy group of ∇ is equal to H3 or ∇ is
flat.
Proof: Suppose ∇ is not flat. Then there is a p ∈ M at which the curvature
map Ωp : Λ2TpM → h3 is not 0. By Lemma 3.1, Ωp is surjective and thus by the
Ambrose-Singer-Holonomy Theorem, the holonomy group of ∇ is equal toH3. q.e.d.
Suppose M is equipped with a torsion free H3-connection ∇. Let π : F → M
denote the total frame bundle of M which is a principal Gl(4,R)-bundle. On F, the
tautological 1-form ω takes values in R4, the connection 1-form θ takes values in
gl(4,R) and both are Gl(4,R)-equivariant [KN].
Then there exists a reduction F of F whose structure group is isomorphic to
Sl(2,R) [KN, II.7.1] and such that θ|F takes values in h3∼= h
3⊆ gl(4,R). Such a
reduction will be called an h3-reduction of F.
By abuse of notation we will denote the restrictions ω|F , θ|F and π|F also by ω, θ
and π resp.
Of course, h3-reductions are not unique. In fact, one sees easily that F , F ′ are
h3-reductions iff there is some g ∈ Norm(h3) ⊆ Gl(4,R) with F ′ = Lg(F ), where
Lg denotes the principal action of g on F. Now one computes that
Norm(h3) = H3 ×N
with
N =
tεt
tεt
∣
∣
∣
∣
∣
∣
∣
∣
t ∈ R\0, ε = ±1
.
18
It follows that any two h3-reductions F, F ′ satisfy F = Lg(F′) for some g ∈ N ,
and two such structures are called homothetic to each other.
Clearly, the property of being homothetic forms an equivalence relation on the
set of h3-reductions of F, and it is not hard to see that there is in fact a 1-1 corre-
spondence between H3-connections and homothety classes of h3-reductions with a
torsion free connection [Br2, 2.1].
We now derive the structure equations for H3-connections on a manifold M .
Let M be as above and fix an h3-reduction F of the total frame bundle F. On
F , we have the 1-form ω+ θ with values in R4 ⊕h
3∼= V3 ⊕h3 and may hence regard
ω + θ as an Sl(2,R)-equivariant V3 ⊕ h3-valued 1-form. It is well known that this
form gives a coframe on F , i.e. the real-valued 1-forms ω0, . . . , ω3, θ1, θ2, θ3 given by
ω(X) =∑
i
ωi(X) ei and θ(X) =∑
i
θi(X) Ei for all X ∈ TF
are linearly independent.
The frame of F dual to the coframe ω0, . . . , ω3, θ1, θ2, θ3 will be denoted by
X0, . . . , X3, Y1, Y2, Y3 and will be called the canonical frame on F .
A tangent vector X ∈ TF will be called vertical if ω(X) = 0 and horizontal if
θ(X) = 0. Thus, Xi is horizontal and Yi is vertical for all i.
The curvature 2-form Ω on F takes values in h3 and vanishes in vertical directions
which means that for every u ∈ F , there is a unique linear map φu : Λ2V3 → h3 such
that
Ω(X, Y ) = φu(ω(X), ω(Y )) for all X, Y ∈ TuF.
19
Furthermore, the first Bianchi identity implies that φu ∈ K(h3), and this defines
an Sl(2,R)-equivariant map
a : F → V2
u 7→ ıK(φu).
We then define the real-valued functions a0, a1, a2 on F by the equation
φu =∑
i
ai(u) φi or, equivalently, a(u) =∑
i
ai(u) x2−iyi
Now we can describe the structure equations on F .
The first structure equation - using that ∇ is torsion free - yields
dω = −θ ∧ ω (3)
or, equivalently,
dω0 = −3θ1 ∧ ω0 − θ3 ∧ ω1
dω1 = −3θ2 ∧ ω0 − θ1 ∧ ω1 − 2θ3 ∧ ω2
dω2 = − 2θ2 ∧ ω1 + θ1 ∧ ω2 − 3θ3 ∧ ω3 (4)
dω3 = − θ2 ∧ ω2 + 3θ1 ∧ ω3
The second structure equation is
dθ = −θ ∧ θ + Ω (5)
or, equivalently,
dθ1 = θ2 ∧ θ3 − 3a2 ω0 ∧ ω2 + 9a1 ω0 ∧ ω3 − a1 ω1 ∧ ω2 − 3a0 ω1 ∧ ω3
dθ2 = 2 θ1 ∧ θ2 + 9a2 ω0 ∧ ω3 − 5a2 ω1 ∧ ω2 + 3a1 ω1 ∧ ω3 − 6a0 ω2 ∧ ω3 (6)
dθ3 = −2 θ1 ∧ θ3 + 6a2 ω0 ∧ ω1 − 3a1 ω0 ∧ ω2 − 9a0 ω0 ∧ ω3 + 5a0 ω1 ∧ ω2
20
Note that these equations determine the brackets of the canonical vector fields
using the relation
dα(X, Y ) = X(α(Y )) − Y (α(X)) − α([X, Y ]) for any 1-form α.
For u ∈ F , we define a linear map ψu : V3 → Hom(Λ2V3, h3) by the equation
ψu(ei)(ω(Y ), ω(Z)) = (LXiΩ)(Y, Z) − φu(θ(Y ) · ei, ω(Z)) − φu(ω(Y ), θ(Z) · ei)
for all Y, Z ∈ TuF .
To see that this is well defined note that the right hand side is tensorial in Y and
Z and vanishes if either Y or Z are vertical. Moreover, the first Bianchi identity
implies that ψu(ei) ∈ K(h3), and finally, the second Bianchi identity implies that
ψu ∈ K1(h3). Therefore, we get a map
b : F → V3
u 7→ ıK1(ψu),
and define the real-valued functions b0, . . . , b3 on F by the equation
ψu = ψb0(u),...,b3(u)
or, equivalently,
b(u) = b3(u) x3 − 3b2(u) x
2y + 3b1(u) xy2 − b0(u) y
3.
Next, we compute the exterior derivatives of the functions a0, a1, a2. For this,
let Xi, Xj and Xk be horizontal vector fields of the canonical frame on F . Then, at
some point u ∈ F ,
Xi(Ω(Xj , Xk)) = (LXiΩ)(Xj , Xk) + Ω([Xi, Xj], Xk) + Ω(Xj , [Xi, Xk])
= ψu(ei) (ej, ek)
= ((3 − i)bi+1(u) φ0 − (3 − 2i)bi φ1 − ibi−1(u) φ2) (ej , ek).
21
Here we used the fact that the Lie bracket of two canonical horizontal vector
fields is always vertical since the connection is torsion free.
On the other hand,
Xi(Ω(Xj , Xk)) = Xi(
2∑
r=0
ar(u) φr (ej , ek)) = (
2∑
r=0
Xi(ar) φr) (ej , ek).
Setting these two equal, we get the horizontal derivatives of a. The vertical
derivatives follow from the Sl(2,R)-equivariance of a, and we obtain
da0 =
3∑
i=0
(3 − i)bi+1 ωi − 2a0 θ1 − a1 θ3
da1 = −3
∑
i=0
(3 − 2i)bi ωi − 2a0 θ2 − 2a2 θ3 (7)
da2 = −3
∑
i=0
ibi−1 ωi + 2a2 θ1 − a1 θ2
Taking exterior derivatives of these equations and solving for dbi, we can compute
that there exists some function c such that
db0 = 3( a22ω1 −a1a2ω2 + b03ω3 + b0θ1 + b1θ2 )
db1 = −3a22ω0 − b12ω2 − 3a0a1ω3 + b1θ1 + 2b2θ2 + b0θ3
db2 = 3a1a2ω0 + b12ω1 + 3a20ω3 − b2θ1 + b3θ2 + 2b1θ3
db3 = −3( b03ω0 − a0a1ω1 + a20ω2 + b3θ1 − b2θ3)
(8)
with
b03 = 3(a2
1
2− a0a2) + c and b12 =
a21
2− 5a0a2 + c
Taking exterior derivatives once again, we find that
dc = 0, (9)
i.e. c is a constant.
22
Definition 3.3 Let π : F →M be a principal Sl(2,R)-bundle over a 4-dimensional
manifold M . Suppose there exist 1-forms ω and θ on F with values in V3 and h3 resp.
and functions a and b with values in V2, V3 resp. such that the structure equations
(3) - (9) are satisfied for some constant c. Moreover, assume that ω(ker(π∗)) ≡ 0
and θ((Ei)∗) = Ei where Ei is the basis of sl(2,R) described earlier and (Ei)
∗
denotes the fundamental vector field corresponding to Ei [KN, Vol I p. 51]. Then
(π, F,M, ω, θ, a, b, c) is called a solution structure over M .
By the previous we know that any h3-reduction of an H3-connection on M gives
rise to a solution structure over M .
Now suppose that F ′ = Lg(F ) with g ∈ N is an h3-reduction homothetic to F .
Then if we let a = Lg a, b = Lg b and c = Lg c where Lg denotes the action of g
on V2, V3 resp., then (F ′, (Lg−1)∗(ω), (Lg−1)∗(θ), a, b, c) is also a solution structure.
In terms of the associated real valued functions we see that if
g =
tεt
tεt
∈ N
thena0 = t2a0, a1 = εt2a1, a2 = t2a3,
b0 = εt3b0, b1 = t3b1, b2 = εt3b2, b3 = t3b0c = t4c
(10)
Definition 3.4 Two solution structures (π, F,M, ω, θ, a, b, c) and
(π, F ,M, ω, θ, a, b, c) over the same manifold M are called homothetic if there exists
a bundle isomorphism L : F → F such that ω = L∗(ω), θ = L∗(θ), and moreover
the real valued functions associated to a, b, a, b satisfy (10) for some t 6= 0, ε = ±.
Clearly, homothety is an equivalence relation of solution structures and homo-
thetic h3-reductions give rise to homothetic solution structures. Therefore, by our
23
discussion earlier, to any H3-connection on a manifold M we can associate a homo-
thety class of solution structures over M .
We will now show that this correspondence between H3-connections and ho-
mothety classes of solution structures over a manifold M is in fact bijective. In
particular, we need to show the sufficiency of the structure equations (3) - (9).
Proposition 3.5 Let M be a 4-dimensional manifold and let π : F →M denote the
total frame bundle of M . Suppose there exists a solution structure
(π, F ,M, ω, θ, a, b, c) over M .
Then there exists a unique H3-connection ∇ on M and a unique embedding
ı : F → F such that
1) F := ı(F ) is an h3-reduction of F,
2) the diagram
F F
M
-ı
@@
@@Rπ
π
commutes and
3) ı∗(ω + θ) = ω + θ, where ω and θ denote the restrictions of the tautological
form and the connection form to F .
Proof: Let F, ω, θ, a and b be as above. Define the real-valued 1-forms and
the real-valued functions ω0, . . . ω3, θ1, . . . , θ3, a0, a1, a2, b0, . . . , b3 as before, and let
X0, . . . , X3, Y 1, Y 2, Y 3 be the frame on F dual to the coframe ω + θ.
24
By hypothesis, π∗((X0)u, . . . , (X3)u) is a basis of Tπ(u)M for all u ∈ F , hence a
point in F. Therefore, we define the map
ı : F → F
u 7→ π∗((X0)u, . . . , (X3)u)
By construction, we have ı∗(ω) = ω. Moreover, the structure equations and the
condition θ((Ei)∗) = Ei imply easily that ω and θ are Sl(2,R)-equivariant w.r.t. the
actions of Sl(2,R) on V3 and h3 induced by ρ3. This means that
- ı is an embedding whose image F is an h3-reduction of F w.r.t. the connection
defined by θ.
- if we define on F := ı(F ) the h3-valued 1-form θ′ := (ı−1)∗(θ) then there exists
a unique gl(4,R)-valued Gl(4,R)-equivariant connection 1-form θ on F such
that θ′ = θ|F ,
It is then straightforward that the map ı satisfies the desired properties and is
uniquely determined by them. q.e.d.
From the construction above it is easily seen that the images of the embeddings
ı1 and ı2 associated to two homothetic solution structures (πi, F i,M, ωi, θi, ai, bi, ci)
over M are in fact homothetic h3-structures. In particular, the connection ∇ on M
only depends on the homothety class of the solution structure.
We summarize the discussion so far, including Corollary 3.2, in the
25
Corollary 3.6 There is a 1-1 correspondence between H3-connections on a 4-man-
ifold M and homothety classes of solution structures over M for which a, b do not
vanish identically.
For the rest of this section, let F be an h3-reduction of an H3-connection ∇ on
a connected 4-manifold M . If we let V := V2 ⊕ V3, we get a map
K : F → V
u 7→ a(u) + b(u)
The structure equations (3) - (9) imply that w.r.t. the functions a0, a1, a2 and
b0, . . . , b3 on V and the canonical frame X0, . . . , X3, Y0, Y1, Y2 on F and we can write
the differential of K
K∗ : TF → TV
as
K∗ =
3b1 2b2 b3 0 −2a0 0 −a1
−3b0 −b1 b2 3b3 0 −2a0 −2a2
0 −b0 −2b1 −3b2 2a2 −a1 00 3a2
2 −3a1a2 3b03 3b0 3b1 0−3a2
2 0 −b12 −3a0a1 b1 2b2 b03a1a2 b21 0 3a2
0 −b2 b3 2b1−3b30 3a0a1 −3a2
0 0 −3b3 0 3b2
.
We compute that det(K∗) ≡ 0, i.e. rank(K∗) ≤ 6. Let L be the cofactor matrix
of K∗, consisting of the 6 × 6 minors of K∗. L has the property that
LK∗ = K∗L = 0.
26
and L has rank at most 1. Thus, there are polynomials r0, . . . , r6, s0, . . . , s6 in the
variables a0, a1, a2, b0, . . . , b3 such that
L =
r0...r6
(s0, . . . , s6).
Furthermore, if we define the polynomial
Rc =1
6912(< pc, pc >2 − 36 < a, q2
c >2)
with
pc = 18(4c− < a, a >2)a + < b, b >2 and qc =< a, b >2,
then we compute that
si =∂Rc
∂ai
for 0 ≤ i ≤ 2, and si =∂Rc
∂bifor 3 ≤ i ≤ 6,
and
ri = si+3 for 0 ≤ i ≤ 3, r4 = s1, r5 = −s0 and r6 = s2.
This means that
- d(Rc K) ≡ 0, hence K maps F into some level set of Rc,
- for u ∈ F , rank(K∗(u)) = 6 iff L(u) 6= 0 iff (r0(u), . . . , r6(u)) 6= 0 iff
(s0(u), . . . , s6(u)) 6= 0 iff K(u) is a regular point of Rc.
27
We now describe the set Σc ⊆ V of critical points of Rc. Using the Sl(2,R)-
invariance of Σc, we compute that
- for c 6= 0, Σc = Σ1c ⊔ Σ2
c , where
Σ1c = (a, b) ∈ V | pc = qc = 0
and
Σ2c = (v − 2u2, 2u (3v + 2u2)) ∈ V | u ∈ V1, v ∈ V2 with < v, v >2=
4
3c.
- for c = 0, Σ0 = Σ+0 ∪ Σ−
0 , where
Σ±0 = (±v2 − 2u2, 2u (±3v2 + 2u2)) ∈ V | u, v ∈ V1
We also define
Σ00 := Σ+
0 ∩ Σ−0 = (−2u2, 4u3) ∈ V | u ∈ V1.
Remark: The reader who is familiar with [Br2] might wonder why the descrip-
tions of the sets Σc and the function Rc there involves different constants than here.
This is due to the fact that the isomorphisms ıK and ıK1 are not determined canon-
ically but can be multiplied by constants. In fact, if we compose K with the map
: V → V , (a, b) := (− 3√
18a, b), then we will get the same constants as in [Br2].
However, to avoid roots in our formulas we will stick to the map K as defined.
28
We wish to determine the topology of the Σ’s. To do this we introduce the
Sl(2,R)-equivariant maps
φ2c : V1 × V2,c −→ Σ2
c
(u, v) 7−→ (v − 2u2, 2u (3v + 2u2)),
φ±0 : V1 × V1 −→ Σ±
0
(u, v) 7−→ (±v2 − 2u2, 2u (±3v2 + 2u2)),
φ00 : V1 −→ Σ0
0
u 7−→ (−2u2, 4u3),
where V2,c := v ∈ V2 | < v, v >2=43c.
One then computes that φ2c is a diffeomorphism for all c.
φ±0 is a branched double cover: in fact, φ+
0 |V1×(V1\0) is a double cover of Σ+0 \Σ0
0
whose non-trivial deck transformation is given by (u, v) 7→ (u,−v).
Finally, φ00|V1\0 is a diffeomorphism onto Σ0
0\0.
Note that Σ1c is smooth by the implicit function theorem. However, the number
of its connected components is unclear.
For Σ2c , we conclude from the above that it is smooth 4-dimensional and has
29
one or two connected components depending on the sign of c. In the case c > 0
we let V ±2,c = ±(u2 + v2)|u, v ∈ V1 and let Σ2,±
c := φ2c(V1 × V ±
2,c) be the connected
components of Σ2c .
Σ0\Σ00 is 4-dimensional and smooth and has two connected components, namely
Σ±0 \Σ0
0.
The set Σ00\0 is a smooth 2-dimensional manifold.
Moreover, for the ranks of K∗ we get:
- If c 6= 0 then rank(K∗(u)) = 4 for all u ∈ F with K(u) ∈ Σc.
- If c = 0 then
- rank(K∗(u)) = 4 for all u ∈ F with K(u) ∈ Σ0\Σ00,
- rank(K∗(u)) = 2 for all u ∈ F with K(u) ∈ Σ00\0,
- rank(K∗(u)) = 0 for all u ∈ F with K(u) = 0.
Theorem 3.7 The differential K∗ has constant rank on F, and
rank(K∗) ∈ 0, 2, 4, 6.
We start by proving the following
Lemma 3.8 Any two points in F can be joined by a piecewise differentiable path γ
such that γ′(t) = ±Xi or γ′(t) = ±Yi for some i, wherever γ′ is defined.
Proof of Lemma: On F , define the equivalence relation p ∼ q if p, q ∈ F
can be joined by such a path. Let ΦtXi,Φt
Yidenote the flow along the vector field
30
Xi, Yi resp. for time t. Then, given any p ∈ F , we can find some ε > 0 such
that ΦtXi
(p),ΦtYi
(p) exists for all t ∈ (−ε, ε). Define a map Φ : (−ε, ε)7 → F by
Φ(t0, . . . , t6) := (Φt6Y3 . . . Φt4
Y1Φt3
X3 . . . Φt0
X0)(p). Clearly, Φ is differentiable and
its derivative Φ∗(0, . . . , 0) is invertible. Thus, by the inverse function theorem, we
may assume that Φ is a diffeomorphism after shrinking ε if necessary. Now given
any (t0, . . . t6) ∈ (−ε, ε)7, there is a polygonal path γ : (a, b) → (−ε, ε)7 with a
subdivision a = u0 ≤ . . . ≤ u7 = b such that γ(a) = 0, γ(b) = (t0, . . . , t6) and
γ′(u) = ± ∂∂ti
for all u ∈ (ui−1, ui). Then γ := Φ γ is a path satisfying the desired
condition, showing that p ∼ q for all q ∈ im(Φ). But since Φ is a diffeomorphism,
im(Φ) is a neighborhood of p, showing that the equivalence classes of ∼ are open
in F . Since F is connected, there is only one equivalence class which proves the
Lemma. q.e.d.
Proof of Theorem: By the previous discussion we need to show that
1) either K(F ) ⊆ V \Σc or K(F ) ⊆ Σc.
2) if c = 0 and K(F ) ⊆ Σ0 then
- either K(F ) ⊆ Σ0\Σ00 or
- K(F ) ⊆ Σ00\0 or
- K(F ) = 0.
1) On F , define the vector field Z :=∑3
i=0 riXi +∑2
i=0 ri+4Yi with the functions
r0, . . . , r6 as on page 27. Then clearly, Zu 6= 0 iff K(u) ∈ V \Σc. Surprisingly, after
an enormous calculation it turns out that [Xi, Z] = [Yi, Z] = 0 for all i. Therefore,
31
if γ is a path in F with γ′ = ±Xi or γ′ = ±Yi for some i, then Z vanishes either
nowhere or everywhere along γ. This together with Lemma 3.8 yields that Z van-
ishes either nowhere or everywhere on F , in other words, either K(F ) ⊆ V \Σc or
K(F ) ⊆ Σc.
2) Now suppose that c = 0 and K(F ) ⊆ Σ0.
Claim: Let γ : [0, ε) → F be a continuous path which is differentiable for t > 0
and with γ′ = ±Xi or γ′ = ±Yi for some i and K(γ(t)) ∈ Σ0\Σ00 for t > 0. Then
K(γ(0)) ∈ Σ0\Σ00.
We will prove this claim only in the case where γ′ = X0 and K(γ(t)) ∈ Σ+0 \Σ0
0.
The other cases are treated similarly.
It is easily seen that there is a lifting path γ : [0, ε) → V1 ×V1 such that φ+0 γ =
K γ with φ+0 as defined earlier. Define the functions u1, u2, v1, v2 by the equation
γ(t) = (u1(t) x+ u2(t) y, v1(t) x+ v2(t) y).
Note that these functions are continuous everywhere and differentiable for t > 0.
One then computes that 2a30 + b23 = 2v2
1 (6u21 + v2
1)2, and 2a3
2 + b20 = 2v22 (6u2
2 + v22)
2.
Taking derivatives yields
d
dt(2v2
2 (6u22 + v2
2)2) = ±X0(2a
32 + b20) = 0,
which implies that either v2 vanishes identically or never.
32
Suppose that v2 ≡ 0. Then we get
d
dt(2v2
1 (6u21 + v2
1)2) = ±X0(2a
30 + b23) = ∓144v2
1u1u22 (6u2
1 + v21),
i.e.
d
dt(v1 (6u2
1 + v21)) = ∓36v1u1u
22 (11)
for all t > 0.
Next, observe that a2 = −2u22 and X0(a2) = 0. Therefore, u2 is constant along
γ.
If u2 = 0 then equation (11) implies that either v1 vanishes identically or never.
If u2 6= 0 then a1 = −4u1u2 and X0(a1) = 12u32 implies that du1
dt= ∓3u2
2. In this
case, equation(11) implies that
(2u21 + v2
1)dv1
dt= 0.
But as du1
dtis constant, u1 vanishes at most at one point, hence we get that
dv1
dt= 0
and hence that v1 either vanishes identically or never.
By hypothesis, γ(t) /∈ Σ00 for t > 0, which is equivalent to saying that either
v1(t) 6= 0 or v2(t) 6= 0 for t > 0. By the above discussion this implies that the same
holds at t = 0, i.e. γ(0) /∈ Σ00, and this proves the claim.
Using the claim it is then clear that any path γ in F as in Lemma 3.8 with
K(γ) ⊆ Σ0 has the property that either K(γ) ⊆ Σ0\Σ00 or K(γ) ⊆ Σ0
0. This to-
gether with Lemma 3.8 implies that either K(F ) ⊆ Σ0\Σ00 or K(F ) ⊆ Σ0
0.
33
Finally, suppose that K(F ) ⊆ Σ00. One sees easily from the structure equations
that a = b = 0 at some point implies that a = b = 0 everywhere, i.e. either
K(F ) ⊆ Σ00\0 or K(F ) = 0. This finishes the proof of the Theorem. q.e.d.
Of course, if K(F ) = 0 then the connection form θ is flat, i.e. the holonomy
group is not equal to H3.
We can easily check that the following are independent of the homothety class of
F , and hence by Corollary 3.6 are invariants of the connection:
- the rank of K∗,
- the sign of the constant c,
- for the case that rank(K∗) = 4 and c = 0, the value of ε = ± in K(F ) ⊆ Σε0,
and in this case the connection is said to be of type Σε0,
- for the case that rank(K∗) = 4 and c 6= 0, the value of i ∈ 1, 2 inK(F ) ⊆ Σic,
and the connection is then said to be of type Σic,
- for a connection of type Σ2c with c > 0, the value of ε = ± in K(F ) ⊆ Σ2,ε
c ,
and the connection is then said to be of type Σ2,εc .
34
4 Non-completeness
The purpose of this section is to prove the following
Theorem 4.1 H3-connections are never complete.
Proof: Let M be a manifold with an H3-connection ∇, and let
(π, F,M, ω, θ, a, b, c) be an associated solution structure over M . By [KN, III.6.3.],
the geodesics on M are precisely those curves γ in M with the property that if γ is
a horizontal lift of γ then γ′ =∑
i kiXi for some constants k0, . . . , k3.
Now suppose that ∇ is complete. Let γ be an integral curve of the vector field
X1 which projects down to the geodesic γ = π γ and therefore by hypothesis is
defined on all of R. From the structure equations we get that
X1(a2) = −b0 and X1(b0) = 3a22,
hence −3a2 satisfies on γ the differential equation
y′′ = y2. (12)
We shall see that the only solution of (12) which is defined on all of R is y = 0
and therefore a2 = 0 along γ.
Since γ was an arbitrary integral curve of X1 we conclude that a2 ≡ 0 on F .
But then the structure equations easily imply that all functions a0, a1, a2, b0, . . . , b3
vanish identically on F which means that the connection is flat, hence not an H3-
connection, and this contradiction will finish the proof.
35
It remains to show that (12) has no non-trivial solution. Note that any global
solution y is convex and hence either constant or unbounded. Suppose that y is a
global non-constant solution, i.e. y′(t0) 6= 0 for some t0. Replacing y(t) by y(−t) if
necessary, we may assume that y′(t0) > 0 and hence y′(t) ≥ y′(t0) > 0 for all t ≥ t0.
We get
d
dt((y′)2 − 2
3y3) = 0,
hence
(y′)2 = C +2
3y3
for some constant C.
As y is unbounded, i.e. limt−>∞ y = ∞, we may assume that y(t)3 > 3|C| for all
t ≥ t0 by increasing t0 if necessary. Then we get for all t ≥ t0
(y′)2 >1
3y3
which implies
(y−1
2 )′ < − 1
2√
3.
Thus we get for all t ≥ t0
y(t)−1
2 < C1 −1
2√
3t
for some constant C1. But this is a contradiction as the left hand side of this
inequality is always positive whereas the right hand side is negative for large t, and
this finishes the proof. q.e.d.
36
5 The Singular H3-connections
We now will solve the structure equations of the various types of solution structures.
As we mentioned in chapter 2, it will be important in each but the homogeneous case
to determine the subset U ⊆ M on which the identity component of the symmetry
group of ∇ acts locally free.
Let Z be a vector field on M such that ΦtZ , the flow along Z, is a local symmetry
for small time t. Let Z be the vector field on F whose flow ΦtZ
is the differential
of ΦtZ . Clearly, this implies that Z ∈ ker(K∗) since Φt
Zpreserves the curvature
and hence K. On the other hand, ΦtZ is locally free on the set where Z 6= 0 or,
equivalently, Z /∈ ker(π∗).
Therefore, to determine U we obtain as a sufficient condition that U contains
all points p ∈M satisfying
(ker(K∗) ∩ ker(π∗))|TxF = 0 for all x ∈ π−1(p). (13)
By Bryant’s uniqueness result of H3-connections it follows that this condition is
actually necessary and hence determines U [Br2]. In fact, he shows that the flow
along any vector field in ker(K∗) is a symmetry and therefore the dimension of the
symmetry group equals the corank of K∗.
5.1 H3-connections of type Σ0
0
Let M be a connected, simply connected 4-manifold with an H3-connection ∇ of
type Σ00. This means that we have an h3-reduction F of the total frame bundle F of
M and an Sl(2,R)-equivariant map K : F → Σ00\0 ⊆ V .
37
We define functions k1 and k2 by K(u) = φ00(k1(u) x+ k2(u) y) where φ0
0 is the
map defined on page 29. Then the structure equations imply that
dk1 = −3k22 ω0 + 2k1k2 ω1 − k2
1 ω2 − k1 ω4 − k2 ω6
dk2 = −k22 ω1 + 2k1k2 ω2 − 3k2
1 ω3 + k2 ω4 − k1 ω5.
rank(K∗) = 2, and therefore K is a submersion. Since Sl(2,R) acts transi-
tively on Σ00\0 it follows that K is surjective. F := K−1(φ0
0(x)) is a subman-
ifold of F which intersects all fibers π−1(p), p ∈ M , and the intersection with
each fiber is connected; for π(u1) = π(u2) iff u1 = Lh(u2) for some h ∈ H where
H :=
ρ3
(
1 0t 1
)∣
∣
∣
∣
t ∈ R
⊆ H3 which is connected. It follows that F is a prin-
cipal H-bundle. Hence π|F : F → M is a homotopy equivalence and in particular F
is connected and simply connected.
On F , we can now compute that the vector fields
Z0 = X0
Z1 = X1 +X6
Z2 = X2 −X4
Z3 = X3 − 3X5
Y = X6
span ker(K∗) and therefore give a frame on F .
Recall the H3-connection from section 2.1 on M = G/H where
G = ASl(2,R) =
Axy
0 0 1
∣
∣
∣
∣
∣
∣
A ∈ Sl(2,R), x, y ∈ R
is the group of unimodular affine motions of R2. Let µ be the g-valued 1-form on F
38
given by
µ(Z i) = Zi for 0 ≤ i ≤ 3,µ(Y ) = Y,
with the basis Z0, . . . , Z3, Y of g defined in section 2.1.
One checks now that µ satisfies the Maurer-Cartan equation
dµ+ µ ∧ µ = 0.
Therefore, since F is connected and simply connected, the second Cartan Lemma
applies and we can find an immersion : F → G such that µ = ∗(µG) where µG
denotes the Maurer-Cartan form of G.
Moreover, (H) = H and thus we get an induced immersion ı : F/H = M →
G/H .
Let θ denote the connection form on F . Then θ(Zi) = λ(Zi) for 0 ≤ i ≤ 3 and
θ(Y ) = E3. Thus, ı preserves the connection and we have the following
Theorem 5.1 Let M be a connected, simply connected 4-manifold with an H3-
connection ∇ of type Σ00. Then there exists a connection preserving immersion
ı : M → G/H
with G/H as in section 2.1, i.e. ı∗(∇XY ) = ∇ı∗(X)ı∗(Y ) for all vector fields X, Y
on M , where ∇ denotes the connection on G/H.
5.2 H3-connections of type Σ±0
Let M be a connected 4-manifold with an H3-connection ∇ of type Σ±0 . This means
that we have an h3-reduction F of the total frame bundle F of M and an Sl(2,R)-
equivariant map K : F → Σ±0 \Σ0
0 ⊆ V .
39
Let F := K−1(φ±0 (v1, x) | v1 ∈ V1) ⊆ F with φ±
0 as defined on page 29. Since
φ±0 is a cover and K has constant maximal rank, it follows that F is a smooth
5-dimensional submanifold of F . Moreover, the Sl(2,R)-equivariance of K implies
that F is a reduction of F with structure group
(
1 0t 1
)∣
∣
∣
∣
t ∈ R
⊆ Sl(2,R).
In particular, F is connected. We then define functions k1 and k2 on F by the
equation
K(u) = φ±0 (k1(u) x+ k2(u) y, x) for all u ∈ F .
Since rank(K|F ) = 2 we conclude that dk1 ∧ dk2 6= 0 on F . Also, k2 is constant
along the fibers of F and hence factors through to a function f : M → R.
Now we compute that condition (13) is satisfied iff k2(x) 6= 0 iff f(p) 6= 0 with
p = π(x). Since dk2 6= 0 and therefore df 6= 0, we conclude that k2 6= 0 (f2 6= 0
resp.) on an open dense subset of F (M resp.).
Note that the solution structures on f−1(R+) and on f−1(R−) are homothetic,
i.e. the connections on theses sets are equivalent. Thus, we only need to consider
the set U := f−1(R+) ⊆M , and we let U := F ∩ π−1(U) = k−12 (R+).
Let S := k−11 (0) ∩ U ⊆ F . Again, dk1 6= 0 implies that S is a smooth 4-
dimensional submanifold of U . Moreover, S intersects every fiber π−1(p) in exactly
one point for every p ∈ U . This means that S is the image of a smooth section
σ : U → U .
40
Using σ, we can describe the connection on U as follows. Let W0, . . . ,W3 be the
frame on U given by σ and let W := σ∗(Wi) for all i. Also, let ω0, . . . , ω3 be the
dual coframe and define the V3-valued 1-form ω =∑
i ωiei.
Of course, Wi = W i + riY1 + siY2 + tiY3 for some functions ri, si, ti where W i
denotes the horizontal lift of Wi. One easily computes these functions explicitly
using that Wi is tangent to F for all i. In particular, this determines the covariant
derivatives by
ω(∇WiWj) = −(riE1 + siE2 + tiE3) · ej.
Using the torsion freeness of ∇ we get from these the bracket relations
[Wi,Wj ] = ∇WiWj −∇Wj
Wi.
Moreover, we know the derivatives Wi(f) for all i and from there we can compute
Wi w.r.t. some local coordinate system one of whose coordinate functions is f .
It then turns out that locally the connection on U is equivalent to the connection
on Σ±0 given in example 2.2. Using the second Cartan Lemma we therefore get the
following
Theorem 5.2 Let M be a connected 4-manifold with an H3-connection ∇ of type
Σ±0 . Let U ⊆ M be the subset on which the symmetry group of M acts locally free
and let U =⊔
i Ui be its decomposition into connected components. Then there exist
connection preserving immersions
ı : Ui → M±0
with M±0 from section 2.2, where Ui denotes the universal cover of Ui.
41
We will call an H3-structure of type Σ±0 global if the image of the curvature map
K is all of Σ±0 . Suppose we have such a global H3-connection on M . Then f :
M → R is surjective and the symmetry group acts locally free on U = f−1(R\0).
Therefore, a global structure will contain two (equivalent) copies of M±0 , namely the
two components of U , glued together along the smooth submanifold f−1(0) of M .
We conjecture that f−1(0) is diffeomorphic to R3 and hence M is diffeomorphic to
R4. Note, however, that global structures need not exist.
5.3 H3-connections of type Σ1
c
Let M be a connected 4-manifold with an H3-connection ∇ of type Σ1c where c ∈
R\0. This means that we have an h3-reduction F of the total frame bundle F of
M and an Sl(2,R)-equivariant map K : F → Σ1c ⊆ V .
Again, we wish to determine the set where condition (13) is satisfied. One com-
putes that this is the case iff < b, b >2 6= 0.
Therefore, we let U := K−1 ((a, b) ∈ Σ1c | < b, b >2 6= 0) ⊆ F and let U :=
π(U) ⊆M . It is easy to verify that U (U resp.) is open and dense in M (F resp.).
Let f : M → R be given by f π = 4c − < a, a >2. Clearly, f |U 6= 0, and we
decompose U = U+ ∪ U−, U = U+ ∪ U−
by U± = f−1(R±) and U±
= π−1(U±).
Define a function t0 : U± → R+ by the equation f = ±(4t0)
2. Then one can
compute that - after changing to a homothetic solution if necessary - the Sl(2,R)-
42
orbit of each element (a, b) ∈ K(U±) contains precisely one pair of the form
(
±x2 − 1
2(4t20 ∓ c) y2,−t0y (±6x2 + (4t20 ∓ c) y2)
)
.
Let L± :=(
±x2 − 12
(4t20 ∓ c) y2,−t0y (±6x2 + (4t20 ∓ c) y2)) ∣
∣ t0 ∈ R+
⊆ Σ1c .
Then one sees easily that L± is a smooth 1-dimensional submanifold of V , hence
S± := K−1(L±) is a smooth 4-dimensional submanifold of F and is in fact the image
of a smooth section σ : U± → U±.
Using σ, we do a calculation similar to the one described in the previous section
and obtain the connection on U±. To describe the result we treat the two possibilities
for the sign of c separately.
If c > 0 then we see that the connection on U± is locally equivalent to the
connection on M±c described in example 2.2. However, the Lie algebras of the
symmetry group of these examples are different, and we conclude that either f ≥ 0
or f ≤ 0 on M . We say the connection is of type Σ1,±c , depending on the sign of f .
Thus we get the
Theorem 5.3 Let M be a connected 4-manifold with an H3-connection ∇ of type
Σ1,±c with c > 0. Let U ⊆ M be the subset on which the symmetry group of M
acts locally free and let U =⊔
i Ui be the decomposition of U into its connected
components. Then there exist connection preserving immersions
ı : Ui → M±c
with M±c from section 2.2, where Ui denotes the universal cover of Ui.
Now assume c < 0. In this case one computes that the connection is locally
equivalent to the connection on M±c described in example 2.2. Thus we get the
43
Theorem 5.4 Let M be a connected 4-manifold with an H3-connection ∇ of type
Σ1c with c < 0. Let U ⊆ M be the subset on which the symmetry group of M
acts locally free and let U =⊔
i Ui be the decomposition of U into its connected
components. Then there exist connection preserving immersions
ı : Ui →M+c ⊔M−
c
with M±c from section 2.2, where Ui denotes the universal cover of Ui.
The difference between these two results is that in the case c > 0 an H3-
connection of type Σ1,±c onM is either locally equivalent to M+
c or locally equivalent
to M−c since the symmetry groups in both cases are different.
In the case c < 0, however, it is very well possible that an H3-connection is
locally equivalent to M+c in some neighborhood, but equivalent to M−
c in some
other neighborhood on the same manifold M .
5.4 H3-connections of type Σ2
c
Let M be a connected 4-manifold with an H3-connection ∇ of type Σ2c where c ∈
R\0. This means that we have an h3-reduction F of the total frame bundle F of
M and an Sl(2,R)-equivariant map K : F → Σ2c ⊆ V .
Recall the diffeomorphism φ2c from page 29. Recall also that V2,c has one or two
connected components depending on the sign of c. We say that ∇ is of type Σ2,±c if
c > 0 and ((φ2c)
−1 K)(F ) ⊆ V1 × V ±2,c.
V2,c contains a polynomial k (x2 ± y2), where ” ± ” = sign(c) and k2 = 16|c|. In
the case c > 0 the choice of sign of k determines the component V ±2,c.
44
If we let F k := K−1(φ2c(v1, k (x2 ± y2)) | v1 ∈ V1) ⊆ F then F k is a smooth
5-dimensional submanifold of F and moreover a reduction of F with structure group
O(2) ⊆ Sl(2,R) if c > 0, and
O(1, 1) ⊆ Sl(2,R) if c < 0.
We then define functions r1 and r2 on F k by the equation
K(u) = φ2c(r1(u)x+ r2(u)y, k (x2 ± y2)) for all u ∈ F k.
Since rank(K|F k) = 2 we conclude that dr1 ∧ dr2 6= 0 on F k. Also, s := r2
1 ± r22
is constant along the fibers of F k and hence factors through to a function f : M → R.
We compute that the set Uk ⊆ M on which condition (13) is satisfied is the set
on which f 6= 0.
Let F0
k denote an SO(2), O(1, 1)e-reduction of F k resp. Let U±k := f−1(R±) ⊆
M . Then Uk := U+k ∪ U−
k = f−1(R\0) ⊆ M and we let U±
k := F0
k ∩ π−1(U±k ).
Since dr1 ∧ dr2 6= 0, we conclude that Uk is open dense in M and Uk is open
dense in F 0.
Now we let S := r−11 (0) ∩ Uk ⊆ F 0. Since dr1 6= 0, we conclude that S is a
smooth 4-dimensional submanifold of Uk. Moreover, S intersects every fiber π−1(p)
in exactly one point for every p ∈ Uk. This means that S is the image of a smooth
section σ : Uk → Uk.
Using σ, we do a calculation similar to the one described in the previous sections
and obtain the connection on Uk. To describe the result we treat the two possibilities
45
for the sign of c separately.
If c > 0 then f ≥ 0 on M , i.e. Uk = U+k . Define a smooth function t0 : Uk → R
+
by the equation f = (kt0)2. Then one sees that the connection described by the
section σ is locally equivalent to the connection on M+k described in example 2.3.
Thus we get the
Theorem 5.5 Let M be a connected 4-manifold with an H3-connection ∇ of type
Σ2,±c with c > 0 and let k such that sign(k) = ± and c = 6k2. Let U ⊆ M be the
subset on which the symmetry group of M acts locally free and let U =⊔
i Ui be
the decomposition of U into its connected components. Then there exist connection
preserving immersions
ı : Ui → M+k
with M+k from section 2.3, where Ui denotes the universal cover of Ui.
Now assume c < 0. In this case one sees easily that the solution structures of
U±k and U∓
−k are homothetic. Therefore, we only need to investigate U+k , i.e. we may
assume that f > 0. Again, define a smooth function t0 : Uk → R+ by the equation
f = (kt0)2. Then one computes that the connection described by the section σ is
locally equivalent to the connection on M−k described in example 2.3. Thus we get
the
Theorem 5.6 Let M be a connected 4-manifold with an H3-connection ∇ of type
Σ2c with c < 0 and let k such that c = −6k2. Let U ⊆ M be the subset on which the
symmetry group of M acts locally free and let U =⊔
i Ui be the decomposition of U
46
into its connected components. Then there exist connection preserving immersions
ı : Ui →M−k ⊔M−
−k
with M−±k from section 2.3, where Ui denotes the universal cover of Ui.
As in the last section, the difference between these two results is that in the case
c > 0 an H3-connection of type Σ2,±c on M is either locally equivalent to M+
k or
locally equivalent to M+−k since the image of K is contained in different components
of Σ2c in these cases.
In the case c < 0, however, it is very well possible that an H3-connection is
locally equivalent to M−k in some neighborhood, but equivalent to M−
−k in some
other neighborhood on the same manifold M .
5.5 Regular H3-connections
In this section we will discuss those H3-connections on M for which the map K :
F → V has maximal rank 6.
Unfortunately, in this case the structure equations involved are so complex that
they cannot be solved explicitly on an dense open subset of M . A (not very illumi-
nating) description of the connection on some open subset, however, can be obtained.
Let U := p ∈ M | < a, a >2 < 0, and a does not divide b on π−1(p). For
p ∈ U there is a unique frame u ∈ π−1(p) such that
b3(u) = 1, a(u) = f(u) xy with f(u)2 = −12< a(u), a(u) >2 > 0.
Using this section, we can describe the connection w.r.t. some coordinate system
on U as follows where the constants c and Rc are given.
47
Let x0, . . . , x3 be local coordinates and define the functions
t0 =1
x0, t1 =
x1
x20
, t2 =x2
1
x30
+x2
x0, and t3 =
x31
x40
+ 3x1x2
x20
− x20 + c+
1
x0r
where
r = ±2√
x31 − x3
2 + cx1x2 +Rc.
Furthermore, let
si :=∂
∂x0(x0ti) for all i.
Then we define a frame as follows:
X0 = 3x0t0∂
∂x0
X1 = x0t1∂
∂x0+t0 X1
X2 = −x0t2∂
∂x0+2t1 X1 +X2
X3 = −3x0t3∂
∂x0+3t2 X1 +3x0t1 X2 +X3
with
X1 = r∂
∂x2+
∂a
∂x2
∂
∂x3
X2 = −r ∂
∂x1− ∂a
∂x1
∂
∂x3
X3 = 2(3x22 − cx1)
∂
∂x1+2(3x2
1 + cx2)∂
∂x2
and where a = a(x1, x2) is some function satisfying X3(a) = 1.
This frame is defined on (x0, . . . , x3) | x0 6= 0, x31 − x3
2 + c x1x2 +Rc > 0 for the
given constants c and Rc.
The connection form is then given as
θ = θ1 E1 + θ2 E2 + θ3 E3
48
where
θ1 = −3
∑
i=0
si ωi
θ2 = −3
∑
i=0
i ti−1 ωi
θ3 =3
∑
i=0
(3 − i) ti+1 ωi
and where ω0, . . . , ω3 denotes the dual basis of X0, . . . , X3.
One can check that this connection is indeed torsion free and is a regular H3-
connection. Note in particular that the 1-dimensional symmetry group is given as
the flow along the vector field ∂∂x3
.
49
6 H3-connections on compact manifolds
Theorem 6.1 Let M be a compact 4-manifold with an H3-connection. Then the
H3-connection is of type Σ00, i.e. M is locally homogeneous.
Proof: Let π, F,M, a, b, c be an associated solution structure. Consider the
function
f : F −→ R
u 7−→ discr(a(u)) = −12< a(u), a(u) >2 .
Since f is constant along the fibers of F there is a unique function f : M → R
such that f π = f .
We shall now use that f must have both a maximum and a minimum on M .
Therefore, we shall investigate the critical points of f .
Using the structure equations (3) - (9) we find that
df = 2(
b0 . . . b3)
−3a1 2a0
−6a2 −a1 4a0
−4a2 a1 6a0
−2a2 3a1
ω0...ω3
.
The determinant of the matrix in this equation is 9f 2. So if u ∈ F is a critical
point of f and f(u) 6= 0 then b(u) = 0. We wish to compute the Hessian of f at a
critical point p ∈M w.r.t. some appropriate frame.
If f(p) > 0 then there is a frame u ∈ π−1(p) with a0(u) = a2(u) = 0 and
a1(u) 6= 0. We compute the Hessian w.r.t. this frame as
−9a1(2c+ 3a21)
a1(2c+ a21)
a1(2c+ a21)
−9a1(2c+ 3a21)
.
50
We see easily that this matrix has some positive eigenvalue regardless of the
values of a1 and c. Therefore, f cannot have a positive maximum, and we conclude
that f ≤ 0.
Let C := a ∈ V2 | discr(a) ≤ 0. One sees easily that we have C = C+ ∪ C−
with C± = ±(v21 + v2
2) | v1, v2 ∈ V1, and C+ ∩ C− = 0. Moreover, C± is invariant
under the Sl(2,R)-action on V2.
Suppose that f(p) = 0 for some p ∈ M and a 6= 0 on π−1(p). Then we can find
some frame u ∈ π−1(p) with a(u) = ±x2. Since u is a critical point of f , hence
df(u) = 0, we get that b(u) = bx3 for some b ∈ R. Computing the Hessian of f at u
we get
00 ±12c
∓8c
±12c 18(b2 ± 2)
The characteristic polynomial of this matrix is
λ(λ± 8c)(λ2 − 18(b2 ± 2)λ− 144c2)
The Hessian must be negative semidefinite, i.e. this polynomial cannot have any
positive root. It is easily seen that this is satisfied only if a(u) = −x2, c = 0 and
b2 ≤ 2. We conclude that f(u) = 0 implies a(u) ∈ C−.
Now suppose that p ∈ M with f(p) < 0 is a critical point of f . Then there is a
frame u ∈ π−1(p) with a1(u) = 0 and a0(u) = a2(u). We compute the Hessian w.r.t.
51
this frame as
36a30 0 −12a0(5a
20 − c) 0
0 4a0(13a20 − 2c) 0 −12a0(5a
20 − c)
−12a0(5a20 − c) 0 4a0(13a2
0 − 2c) 00 −12a0(5a
20 − c) 0 36a3
0
.
The characteristic polynomial of this matrix is
r(λ)2, where r(λ) = λ2 − 8a0(11a20 − c) λ− 144a2
0(2a20 − c)(6a2
0 − c).
If a(u) ∈ C+, i.e. a0(u) > 0, then r has at least one positive root since either
r(0) < 0 or r′(0) < 0. Therefore, f cannot have a negative maximum on π(f−1(C+)).
But this set is closed in M and hence compact. We conclude that if a(F ) ∩ C+ 6= ∅
then 0 ∈ a(F ).
Similarly, if a(u) ∈ C−, i.e. a0(u) < 0, then r has at least one negative root since
either r(0) < 0 or r′(0) > 0 if a0 < 0. Therefore, f cannot have a negative minimum
on π(f−1(C−)). But this set is a closed and hence compact subset of M , therefore f
must have a minimum in this set. We conclude that if a(u) ∈ C− then f(u) = 0.
Suppose now that a(U) ⊆ C−\0 for some open set U ⊆ F . By the above,
K(U) ⊆ (−v2, bv3) | v ∈ V1\0, b ∈ R. Since this set is 3-dimensional we get
rank(K) ≤ 3 on U . But then Theorem 3.7 implies that M is of type Σ00. In
particular, a(F ) ⊆ C−\0.
Thus, either M is of type Σ00 or a(F ) ⊆ C+.
But in the latter case, a(u) = 0 for some u ∈ F . Since a(F ) ⊆ C+, hence a0, a2 ≥
0 this means that u is a minimum for both a0 and a2, hence da0(u) = da2(u) = 0
52
and therefore b(u) = 0. Computing the Hessian of a0 at u yields the matrix
0 −3c2c
−3c 00
This matrix must be positive semidefinite which is satisfied iff c = 0. Thus,
a(u) = b(u) = c = 0, i.e. K(u) = 0, and again Theorem 3.7 implies that the con-
nection is flat, violating our assumption. Therefore, we get a contradiction from the
assumption a(F ) ⊆ C+ and this shows that M is of type Σ00. q.e.d.
Suppose now that M is a compact manifold with an H3-connection and let α :
M → M denote its universal cover. Then by Theorems 5.1 and 6.1 there is a
connection preserving immersion ı : M → G/H .
Recall the description of G/H in 2.1. Since the forms τi on G/H described there
are G-invariant, there are forms τ i, 0 ≤ i ≤ 3 on M such that α∗(τ i) = ı∗(τi) =: τi.
Also, the distributions Di given by τ i(Di) = 0 are integrable for all i.
Let Z0, . . . , Z3 be a frame on M such that Di = span(Z0, . . . , Z i−1) and
τ i(Z i, . . . , Z3) = 1. Such a frame can be easily constructed using an arbitrary
Riemannian metric on M . If we denote the dual 1-forms of Z0, . . . , Z3 by ω0, . . . , ω3
then we have τ i = ωi ∧ . . . ∧ ω3 for all i.
Passing to the universal cover M , let ωi := α∗(ωi), and let Zi be the lifts of Z i.
Note that Zi is complete for all i, i.e. the flow along Zi is defined for all times. This
follows from the completeness of the vector fields Z i which are defined on a compact
manifold.
53
Let Yi be a maximal connected integral leave (m.c.i.l.) of Di. Since ı∗(Di) = Di,
the image ı(Yi) is contained in some m.c.i.l. Yi of Di.
Lemma 6.2 The restriction ı : Yi → Yi is a diffeomorphism for i ≤ 3.
Proof: We proceed by induction on i. For i = 0, both Yi and Yi are points and
there is nothing to prove.
Suppose the claim is true for some i ≤ 2 and let Yi+1 and Yi+1 be m.c.i.l.’s such
that ı(Yi+1) ⊆ Yi+1. Recall the function ri+1 : Yi+1 → R defined in section 2.1, and
let si+1 : Yi+1 → R be the composition si+1 = ri+1 ı.
Suppose ı(p) = ı(q) for some points p, q ∈ Yi+1 and assume for convenience that
si+1(p) = si+1(q) = 0. Let γ : [0, 1] → Yi+1 be a smooth path joining p and q. Then
define a new path σ by
σ(t) = Φ−si+1(γ(t))
Zi(γ(t)),
where ΦtX denotes the flow along a vector field X for time t.
Note that σ(t) is well defined for all t ∈ [0, 1] by completeness of Zi. Also, since
Zi(si+1) ≡ 1 it follows that si+1(σ(t)) = 0, thus σ lies in a m.c.i.l. Yi of Di. In
particular, q ∈ Yi, and now the induction hypothesis implies that p = q, i.e. ı|Yi+1is
injective.
The induction hypothesis and the fact that the m.c.i.l.’s of Di+1 are precisely the
level sets of ri+1 imply that ı(Yi+1) = r−1i+1(I) for some open interval I ⊆ R. How-
ever, si+1 is surjective. In fact, given p ∈ Yi+1, we have si+1(ΦtZi
(p)) = si+1(p) + t,
and t is arbitrary. Therefore, I = R and this shows surjectivity of ı|Yi+1, and this
finishes the proof. q.e.d.
54
Unfortunately, since dω3 6= 0, we cannot continue the same proof to show global
injectivity of ı.
On M , we declare two points to be equivalent iff they lie in the same m.c.i.l. of
D3. Let π : M → M/∼ be the canonical projection. Then there is a unique function
θ : M/∼ → S1 making the diagram
M G/H
M/∼ S1
-ı
?π
?θ
-θ
(14)
commute where θ is the angle function defined in section 2.1.
Lemma 6.3 Consider the maps described above. Then
1) π : M → M/∼ is open,
2) θ : M/∼ → S1 is a local homeomorphism, and
3) M/∼ is Hausdorff.
Proof: 1) For this note that for a given open set U ⊆ M , we have
π−1(π(U)) =⋃
t0,...,t3
Φt0
Z0
. . . Φt3
Z3
(U),
which is clearly open.
2) Let p ∈ M/∼ and let x ∈ M such that π(x) = p. Choose a path γ : (−ε, ε) →
M with γ(0) = x which is transverse to the distribution D3. By the same argument
as above we see that V := π(γ) is an open neighborhood of p ∈ M/∼. Moreover,
the restriction π : γ(t) → V is an homeomorphism being an open and continuous
55
bijection. Since the restriction (θ ı)|γ is an homeomorphism, so is the restriction
θ|V , and this shows the claim.
3) Let p1, p2 ∈ M/∼ be two different points. If θ(p1) 6= θ(p2) then these two
points can be separated by open neighborhoods, so we assume that θ(p1) = θ(p2).
By 2), there are open neighborhoods Vi of pi such that the restrictions θ|Viis an
homeomorphism onto its image Ii ⊆ S1. Since singleton sets are closed in M/∼ we
may assume that pi /∈ Vj for i 6= j. Furthermore we may assume that I1 = I2 =: I
after shrinking one of the sets if necessary.
Let T := θ−1(I) ⊆ G/H and let Ui := π−1(Vi) ⊆ M . By Lemma 6.2 and the
choice of Vi it follows that the restrictions ıi : Ui → T are diffeomorphisms for
i = 1, 2. We therefore can define a diffeomorphism φ : U1 → U2 as φ := ı−12 ı1.
Note that the intersection U1 ∩ U2 is precisely the set x ∈ U1 | φ(x) = x. Thus,
U1 ∩ U2 is closed in U1 and we conclude that either U1 ∩ U2 = ∅ or U1 = U2. But
U1 = U2 implies that V1 = V2 which is impossible. Hence, U1∩U2 = ∅ which implies
V1 ∩ V2 = ∅. q.e.d.
Let G be the universal cover of the Lie group G = ASl(2,R) and let H ⊆ G
be the 1-parameter subgroup which covers H . Then G/H is the universal cover of
G/H .
56
Corollary 6.4 Let ı : M → G/H be the immersion described above. Then the lift
ı : M → G/H
is an embedding onto some subset T := θ−1(I) for some open interval I ⊆ R, where
θ : G/H → R denotes the lift of θ.
Proof: By Lemma 6.3 we know that M/∼ is a connected 1-manifold. Moreover
as a consequence of the proof of this Lemma we get that π is a fiber bundle whose
fibers are diffeomorphic to R3. Therefore, M/∼ is simply connected, i.e. M/∼ is
homeomorphic to R. θ being a local homeomorphism then implies that θ is a global
homeomorphism onto some open interval I ⊆ R.
Let T := θ−1(I) ⊆ G/H . Then the lifts of the maps in diagram (14) yield a
commutative diagram
M T G/H
M/∼ I R
-ı
?π
-⊆
?θ
?θ
-θ -⊆
Now use that θ is a homeomorphism and apply Lemma 6.2. q.e.d.
By virtue of this Corollary we can naturally identify M with T ⊆ G/H.
We now investigate the symmetries on M ∼= T .
Lemma 6.5 Every symmetry of T extends to a symmetry of G/H.
Proof: Recall the definition of the principal R-bundle π : F → G/H from
section 5.1, and let F T := π−1(T ) ⊆ F . Note that the symmetry group G acts
transitively on F .
57
Let φ : T → T be a symmetry and let φ : F T → F T be the map induced by the
differential of φ. Fix a point x ∈ F T and let p := π(x) ∈ T . There exists a g ∈ G
such that ψ(x) = x where ψ := g φ. This means that for ψ : T → G/H with
ψ = g φ we have ψ(p) = p and dψp = Id. But ψ is a symmetry, hence ψ is the
identity map on T , i.e. φ = g−1 ∈ G. q.e.d.
Suppose now that T 6= G/H, i.e. T = θ−1(I) with I 6= R. Let t0 be in the
boundary of I. By Lemma 6.5 the symmetries of T must leave the set θ−1(t0) in-
variant. If we precompose ı : M → T with an appropriate element of G we may
assume that t0 = π2.
Let S ⊆ G be the subgroup which fixes θ−1(π2). Let α : G → G be the covering
homomorphism. Note that any element g ∈ ker(α) ∼= Z acts on G/H such that
θ(g(x)) = θ(x) + 2πn with n ∈ Z. Thus, S ∩ ker(α) = e, i.e. α|S is injective.
Let S := α(S) ⊆ G. Then clearly S leaves θ−1(π2) invariant, i.e.
S =
ea x ze−a y
1
∣
∣
∣
∣
∣
∣
a, x, y, z ∈ R
.
From this we also conclude that S leaves all level sets θ( (2n+1)π2
), n ∈ Z invariant.
The fundamental group of M acts on M ∼= T by symmetries. Since α|S is an
isomorphism we conclude that π1(M) is isomorphic to a discrete subgroup Γ ⊆ S.
Lemma 6.6 Let N3 ⊆ S denote the Heisenberg group. Then Γ cannot be contained
in N3.
58
Proof: Suppose Γ ⊆ N3. The first step is to show that Ln := θ−1(− (2n+1)π2
) 6⊆
θ(T ). To see this note that Ln is Γ-invariant, hence Γ acts on Ln with compact
quotient. On Ln, we have the slope function r3 which was defined in section 2.1,
and it is easy to see that r3 is invariant under N3 and hence by hypothesis under
Γ. Thus, there is an induced function r3 : Ln/Γ → R. However, r3 has no critical
point on Ln and hence on Ln/Γ, but on the other hand Ln/Γ is compact. This is a
contradiction which finishes the first step.
Therefore, θ(T ) is a bounded interval of length < π and we may assume that
π2
is an upper bound of this interval. Let C ⊆ T ⊆ θ−1(−π2, π
2) be a compact
subset whose Γ-orbit is T . Note that the restriction of the covering map π|T is a
diffeomorphism, thus if we define T1 := π(T ) ⊆ θ−1(−π2, π
2) ⊆ G/H then the orbit
of C1 := π(C) ⊆ T1 must equal T1. Consider the function
s : G/H −→ R
gH 7−→< L(g) ·(
01
)
,
(
01
)
>
where as before L : ASl(2,R) → Sl(2,R) is the homomorphism assigning the linear
part to an affine motion and < , > is the standard inner product on R2.
Clearly, s(T1) = R+ since θ(T1) has π
2as its supremum. On the other hand, one
sees easily that s is invariant under N3 and hence under Γ. But s(C1) is compact,
hence bounded away from 0 which is a contradiction. q.e.d.
Lemma 6.7 Every finitely generated discrete subgroup Γ ⊆ S is either contained in
N3 or is cyclic and diagonalizable.
59
Proof: Suppose Γ is not contained in N3, i.e. there is some g0 ∈ Γ,
g0 =
ea r te−a s
1
with a 6= 0. Let Γ1 := [Γ,Γ] ⊆ N3. Then Γ1 is also finitely generated and is a
normal subgroup of Γ.
Let φ : N3 → R2 be the homomorphism given by
φ
1 x z1 y
1
:=
(
xy
)
,
and consider the finitely generated subgroup Γ2 := φ(Γ1) ⊆ R2.
If g ∈ Γ1 with φ(g) = (x, y) then one computes φ(gn0 gg
−n0 ) = (e2anx, e−any) ∈ Γ2.
However, there is no finitely generated subgroup of R2 which contains these elements
for all n ∈ Z unless x = y = 0.
We conclude that Γ1 ⊆ ker(φ). Using the same argument once again we can
conclude that Γ1 = 0, i.e. Γ is abelian.
Thus Γ is diagonalizable in S, hence we may assume that Γ consists of diagonal
matrices. Finally, the discreteness of Γ implies that Γ is cyclic. q.e.d.
We then arrive at the
Corollary 6.8 Let M be a compact manifold with an H3-connection. Then
M = Γ\G/H
for some discrete subgroup Γ ⊆ G.
Proof: Suppose Γ ⊂ S is cyclic and diagonal. Again, let C ⊆ T be a compact
subset whose Γ-orbit is T . Since S leaves T ′ := θ−1[−π2, π
2) invariant, the orbit of
60
the compact set C ′ := C ∩T ′ must equal T ′∩ ı(M). Note that the restriction of the
covering map π|T ′ is a diffeomorphism, thus if we define T1 := π(T ′) ⊆ θ−1[−π2, π
2) ⊆
G/H then the orbit of C1 := π(C ′) ⊆ T1 must equal T1 ∩ ı(M).
There exists a constant k ∈ R such that every parabola p ∈ C1 intersects the set
U := (x, y) | xy ≤ k ⊆ R2. But we see easily that U is invariant under Γ, hence
every parabola in the orbit of C1 intersects U . But this is impossible since there are
parabolas in T1 ∩ ı(M) which do not intersect U .
Since the fundamental group of a compact manifold is finitely generated, this
together with the previous two Lemmas implies that ı : M → G/H is a diffeo-
morphism. The fundamental group π1(M) acts on G/H by symmetries. Finally,
following the proof of Lemma 6.5 we see that G is the full symmetry group of G/H.
q.e.d.
Unfortunately, there are no known examples of such biquotients. Hence the
question of existence of compact manifolds with H3-connections remains open.
61
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