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arXiv:1204.5657v2 [math.DG] 15 May 2012
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN
MANIFOLDS
HELGA BAUM, KORDIAN L¨
ARZ, AND THOMAS LEISTNER
Abstract. We study the full holonomy group of Lorentzian manifolds with a parallel null line
bundle. We prove several results that are based on the classification of the restricted holonomy
groups of such manifolds and provide a construction method for manifolds with disconnected
holonomy which starts from a Riemannian manifold and a properly discontinuous group of isome-
tries. Most of our examples are quotients of pp-waves with disconnected holonomy and without
parallel vector field. Furthermore, we classify the full holonomy groups of solvable Lorentzian
symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct
examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, dis-
connected full holonomy and a parallel spinor.
Keywords: Lorentzian manifolds, holonomy groups, isometry groups, parallel spinor fields, glob-
ally hyperbolic manifolds, pp-waves
Contents
1. Introduction 1
2. Algebraic preliminaries 5
3. Null line bundle and screen bundle 8
4. Holonomy groups and coverings 9
5. Construction of Lorentzian manifolds with disconnected holonomy 11
6. Examples with disconnected holonomy 16
7. Full holonomy groups of Lorentzian manifolds with parallel null spinor 23
References 28
1. Introduction
The aim of this paper is to study the holonomy group of Lorentzian manifolds with a parallel
bundle of null lines. The holonomy group of a semi-Riemannian manifold (M, g) at a point p∈M
is given as the group of parallel transports along loops1based at p,
(1) Holp(M, g ) := {Pγ:TpM→TpM|γ: [0,1] →Ma curve with γ(0) = γ(1) = p}.
Here Pγdenotes the parallel transport along γwith respect to the Levi-Civita connection ∇gof g.
The holonomy group is a subgroup of the orthogonal group O(TpM, gp), where TpMis the tangent
space of Mat pand gpthe scalar product induced by the semi-Riemannian metric gon TpM.
Holonomy groups are not necessarily closed nor connected. The connected component Hol0
p(M, g)
is given by restricting the definition (1) to curves that can be contracted to the point p. Indeed,
1991 Mathematics Subject Classification. Primary 53C29; Secondary 53C50, 53C27.
This work was supported by the Group of Eight Australia and the German Academic Exchange Service through
the Go8 Germany Joint Research Co-operation Scheme grant “Spinor field equations in global Lorentzian geometry”.
The third author acknowledges support from the Australian Research Council via the grant FT110100429, the first
one from the DFG-CRC 647 “Space-Time-Matter”.
1All curves we consider are piecewise smooth.
1
2 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
by contracting the loop γwe obtain a path in the holonomy group from Pγto the identity. Hence,
holonomy groups of simply connected manifolds are connected. The restricted holonomy group
Hol0
p(M, g) is a normal subgroup in the full holonomy group Holp(M, g). Moreover, the fundamen-
tal group of Msurjects homomorphically onto their quotient,
(2) π1(M, p)։Holp(M , g)/Hol0
p(M, g)
[γ]7→ [Pγ]
(for a proof see for example [20, Chap. II, Sec. 4]). The holonomy group is a very powerful tool,
for example, for determining parallel sections in geometric vector bundles. Knowing the holonomy
group of a given semi-Riemannian manifold allows to find the solution to the partial differential
equation for a parallel section by solving an algebraic problem, namely to determine the fixed
vectors of the corresponding representation of the holonomy group. The parallel section is then
obtained by parallel transporting the algebraic object at a point to the whole manifold and thus
defining a global section. For example, for finding a parallel vector field one has to find a fixed
vector under the holonomy group acting on the tangent space. For finding a parallel spinor field,
one has to find a spinor that is fixed under the image of the holonomy group in the corresponding
spinor group, or for simply connected manifolds, one that is fixed under the spin representation of
the holonomy algebra. Another important example is the parallel complex structure of a K¨ahler
manifold. Here the the holonomy group is reduced to the unitary group. These facts also show the
importance of manifolds with special holonomy, e.g. Calabi-Yau manifolds, in string theory, where
in some situations the underlying spacetime is required to have a covariantly constant, i.e. parallel,
spinor field. For these reasons, a classification of possible holonomy groups of semi-Riemannian
manifolds is a desirable result, but out of reach in full generality.
Classification results for holonomy groups are usually obtained only for the restricted holonomy
group (see for example [7, 26, 23]). Such results are based on the Ambrose-Singer holonomy
theorem [1] which states that the Lie algebra of the holonomy group is generated by curvature at
every point in M. More precisely, the Lie algebra of the holonomy group at pis generated as a
vector space by the following linear maps of TpM
(3) P−1
γ◦Rγ(1)(X, Y )◦Pγ,
where γ: [0,1] →Mis a curve starting at p,Rγ(1) the curvature tensor at γ(1), and X, Y ∈Tγ(1)M.
Since the Levi-Civita connection is torsion free, the curvature and hence all the maps in (3),
satisfy the Bianchi identities. Hence, via the Ambrose-Singer holonomy theorem, the Bianchi
identities impose strong algebraic conditions on the Lie algebra of the holonomy group which lead to
classification results, but only for the restricted holonomy groups, and mostly under the assumption
that it acts irreducibly, or at least indecomposably (see next paragraph for the definition). Similar
classification results for full holonomy groups are out of reach. For example, due to the complete
reducibility of the holonomy representation for Riemannian manifolds, the restricted holonomy
group of a Riemannian manifold is always closed and hence compact. But the example given in
[32] shows that Holp(M, g) can be non-compact even for compact Riemannian manifolds.
For Lorentzian manifolds the classification of restricted holonomy groups is obtained as follows:
Using the splitting theorems by de Rham [14] and Wu [34] one can decompose every simply
connected, complete Lorentzian manifold into a product of Riemannian manifolds and a Lorentzian
manifold, all simply connected and complete, and with indecomposably acting holonomy group. By
indecomposable we mean that the metric degenerates on every subspace of TpMthat is invariant
under the holonomy group. Of course, for Riemannian manifolds this implies that the holonomy
group acts irreducibly and one can apply Berger’s holonomy classification [7] to the Riemannian
factors. The remaining Lorentzian factor is either flat, irreducible or indecomposable. On the one
hand, the irreducible case is dealt with by the Berger’s list [7], on which SO0(1, n −1) is the only
possible irreducible restricted holonomy group of Lorentzian manifolds. This also follows from the
more fundamental result by Di Scala and Olmos in [15] that SO0(1, n −1) has no proper irreducible
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 3
subgroups. On the other hand, the classification in the indecomposable, non-irreducible case was
achieved recently by Berard-Bergery and Ikemakhen [6], the third author [23], and Galaev [17].
We will explain the classification in the following paragraph and in Section 2.
We say that a Lorentzian manifold (M , g) of dimension (n+ 2) has special holonomy, or simply
is special if its restricted holonomy group acts indecomposably but is not equal to SO0(1, n +1), the
connected component of the special orthogonal group in Lorentzian signature2. As SO0(1, n + 1)
has no proper irreducible subgroups, this means that the representation of the restricted holonomy
group of a special Lorentzian manifold cannot be irreducible but that the metric is degenerate on
all invariant subspaces. Hence, there is a Hol0
p(M, g)-invariant degenerate subspace W⊂TpM
which defines an invariant null line L:= W∩W⊥and the restricted holonomy group is contained
in the stabiliser in SO0(TpM, gp) of this line L. Identifying TpMwith R1,n+1 by fixing a basis
(ℓ, e1,...,en, ℓ∗) in TpMsuch that ℓ∈Land the metric at pis of the form
(4)
0 0 1
01n0
1 0 0
,
this stabiliser in O(1, n + 1) can be written as the parabolic subgroup
P:= StabO(1,n+1)(L) = (R∗×O(n)) ⋉ Rn
=
a xt−1
2a−1xtx
0A−a−1Ax
0 0 a−1
a∈R∗, A ∈O(n), x ∈Rn
,
whose connected component is given by the stabiliser of Lin SO0(1, n + 1), i.e.,
P0= StabSO0(1,n+1)(L) = (R+×SO(n)) ⋉ Rn.
This defines three projections of Ponto R∗, O(n) and Rn, and of P0onto R+, SO(n) and Rn
which we denote by
prR:P→R∗,prO(n):P→O(n),prRn:P→Rn.
For Lorentzian manifolds with restricted holonomy group H0acting indecomposably but not
irreducibly in [23] it was shown that prO(n)(H0)⊂SO(n) has to be the holonomy group of a
Riemannian manifold. Using results in [6, see our Section 2] this gave a full classification of
restricted holonomy groups of Lorentzian manifolds acting indecomposably and not irreducibly.
Galaev [17] then extended previous existence results and verified that indeed all groups on the list
can be realised as holonomy groups of Lorentzian manifolds. Together with the splitting theorems
by de Rham [14] and Wu [34] and the fact mentioned above that SO0(1, n + 1) has no proper
irreducible subgroups, this yields the classification of restricted holonomy groups of Lorentzian
manifolds.
Our first result about the full holonomy group is that it has the same Rn-part as the restricted
holonomy (see Proposition 1 for a more precise statement):
Theorem 1. Let (M, g )be a Lorentzian manifold of dimension (n+ 2) >2such that its restricted
holonomy group Hol0
p(M, g)acts indecomposably but not irreducibly. Then
1) the full holonomy group Holp(M, g)acts indecomposably but not irreducibly, and
2This is in accordance with the terminology in the Riemannian setting where special holonomy usually refers
to manifolds with restricted holonomy different from SO(n) but still acting irreducibly. Of course, in Riemannian
signature irreducibility is the same as indecomposability, but in other signatures indecomposability is the property
that is geometrically more important: If the restricted holonomy group acts decomposably, i.e. with a non-degenerate
invariant subspace, then, without further assumptions on M, the manifold is locally a semi-Riemannian product
[34, Proposition 3].
4 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
2) there is a subset Γ⊂R∗×O(n)such that
Hol(M, g) = Γ ·Hol0(M , g).
After recalling the basics on special Lorentzian geometry and proving this result, a large part
of the paper is devoted to the construction of Lorentzian manifolds with disconnected holonomy.
Our construction uses a method to obtain special Lorentzian manifolds of dimension (n+ 2) from
Riemannian manifolds of dimension n. Using this method for every group Gthat is a Riemannian
holonomy group, connected or disconnected, we obtain special Lorentzian manifolds with holonomy
G⋉ Rn,(R+×G)⋉ Rn,(R∗×G)⋉ Rn,(Z2×G)⋉ Rn
(see Proposition 4 for details). Further examples are obtained as quotients of Lorentzian manifolds
by a properly discontinuous group of isometries Γ. To this end, in Proposition 3 of Section 4 we
prove a generalisation of the fundamental formula (2) for general coverings π:f
M→M:= f
M/Γ,
which provides a surjective group homomorphism
Γ։Holp(M)/Holep(f
M)
σ7→ [Pγ],
where γis a loop at pthat, when lifted to a curve eγstarting at ep, ends at σ−1(ep), and yields a
formula of the parallel transport in Min terms of that in f
M. Applying this to our context in
Theorem 3 and Corollary 1 leads to a variety of examples of special Lorentzian manifolds with
disconnected holonomy in Section 6 including an example for which the quotient Hol/Hol0is
infinitely generated. These various examples illustrate the possible differences between the full and
the restricted holonomy group and feature a coupling between the O(n)-part and the R∗-part of
the holonomy group that is not present for the restricted holonomy group. Most of our examples
are quotients of pp-waves.
One of the class of examples we consider are solvable Lorentzian symmetric spaces, so-called
Cahen-Wallach spaces [11, 10]. In Proposition 6 we show that the full holonomy of a Cahen Wallach
space is either connected, in which case it is equal to Rn, or given as Z2⋉ Rn, where the Z2factor
is generated by a reflection in O(n).
In the last part of the paper we consider the full holonomy of Lorentzian manifolds that admit
a parallel spinor field. Such a spinor field induces a parallel vector field and hence the holonomy
stabilises a vector, which, for indecomposable manifolds, has to be null, i.e. lightlike. First we show
in Proposition 8 and Corollary 2 that, for a time- and space-orientable Lorentzian manifold with
holonomy G⋉ Rn, the existence of a spin structure with parallel spinors depends solely on G. This
result enables us to apply to the Lorentzian situation the classification of irreducible subgroups of
O(n) stabilising a spinor and having SU( n
2), Sp(n
4), G2or Spin(7) as connected component. This
classification was given by McInnes [24] and Wang [31] and it yields our
Theorem 2. Let (M, g)be a Lorentzian spin manifold of dimension (n+2) >2with full holonomy
group H= Holp(M, g)with a parallel spinor. Assume that
(i) the connected component H0of the holonomy group Hacts indecomposably, and
(ii) G0:= prO(n)(H0)acts irreducibly on Rn.
Then H=G⋉ Rn, where G⊂SO(n)is one of the groups listed in Theorem 4 and the dimension
of parallel spinors on (M, g )is equal to the dimension Nof spinors fixed under Gas given in
Theorem 4.
Finally, we study the existence problem for metrics with these holonomy groups and parallel
spinors. We use a method developed in [5] to construct globally hyperbolic Lorentzian manifolds
with complete spacelike Cauchy hypersurfaces, parallel spinors and holonomy G⋉ Rnfrom Rie-
mannian manifolds. We apply this method to examples given by Moroianu and Semmelmann in
[27] and obtain globally hyperbolic metrics with parallel spinors for the groups in Theorem 2.
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 5
2. Algebraic preliminaries
Let (M, g ) be a Lorentzian manifold of signature (1, n + 1). The holonomy group as defined
in (1) is an immersed Lie subgroup of O(TpM , gp) (for a proof see [20, Thm. II.4.2]). We denote
its Lie algebra by holp(M , g). For connected manifolds, holonomy groups at different points are
conjugated in O(1, n + 1) to each other. We assume from now on that all manifolds are connected.
Hence we may omit the point pand consider holonomy groups only up to conjugation.
We will first derive some purely algebraic results which will imply Theorem 1 of the introduction.
Let R1,n+1 be the (n+ 2)-dimensional Minkowski space, in which we fix a basis (ℓ, e1,...,en, ℓ∗)
such that the Minkowski inner product is of the form (4). Let Lbe the null line spanned by ℓ.
Furthermore let H⊂O(1, n + 1) be a subgroup and H0a normal subgroup of H. Obviously, His
contained in the normaliser in O(1, n + 1) of H0,
H⊂NorO(1,n+1)(H0).
In this situation we prove:
Lemma 1. Let H0⊂O(1, n + 1) be a subgroup that acts indecomposably and stabilises the null
line L. Then the normaliser of H0stabilises Las well, i.e.
NorO(1,n+1)(H0)⊂StabO(1,n+1) (L).
In particular, if H⊂O(1, n + 1) is an immersed Lie group and H0is the connected component of
H, then Hstabilises Land acts indecomposably if H0does.
Proof. Let g∈NorO(1,n+1) (H0) and L=R·ℓ. Then for each h∈H0we have that also ˆ
h:=
ghg−1∈H0. Since H0stabilises Lthere are ˆ
λsuch that ˆ
h(ℓ) = ˆ
λℓ. Multiplying this with g−1gives
hg−1(ℓ) = ˆ
λg−1(ℓ). Hence, g−1(ℓ) spans a null line that is fixed under all of H0. But since H0
was assumed to be indecomposable, Lis the only line that is fixed by H0. Hence, g−1(ℓ)∈L.
Remark 1. We should remark that we can have immersed subgroups that fix Land acting
indecomposably, but the connected component H0acts decomposably. An example of this is given
in Section 6.
From now on let H⊂O(1, n + 1) be an immersed subgroup with connected component
H0⊂P0= StabSO0(1,n+1)(L) = (R+×SO(n)) ⋉ Rn
in the stabiliser of a null line L. Lemma 1 ensures that
H⊂P= StabO(1,n+1)(L) = (R∗×O(n)) ⋉ Rn
and we can define
G:= prO(n)(H).
If G0denotes the connected component of Gwe have
G0= prO(n)(H0) = prSO(n)(H0).
We denote by h⊂so(1, n + 1) the Lie algebra of H0and recall the classification of subalgebras of
so(1, n +1) that act indecomposably but not irreducibly given in [6]. If his such a subalgebra, then
it is contained in the Lie algebra of the stabiliser pof the null line L, i.e. h⊂p:= (R⊕so(n)) ⋉ Rn.
We will write elements in pas triple (a, X, v)∈R×so(n)×Rn. Denote by gthe projection of h
onto so(n). Since gis reductive, it decomposes into its centre zand its derived Lie algebra g′, i.e.
g=z⊕g′. Then it was proven in [6] that, if hacts indecomposable, it is of one of the following
types, the first two being uncoupled and the last two coupled:
Type 1: h= (R⊕g)⋉ Rn,
Type 2: h=g⋉ Rn,
6 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
Type 3: There exists an epimorphism ϕ:z→R, such that h= (f⊕g′)⋉ Rn,where
f:= graph ϕ={(ϕ(Z), Z )|Z∈z} ⊂ R⊕z. Or, written in matrix form:
h=
ϕ(Z)vt0
0Z+X−v
0 0 −ϕ(Z)
Z∈z, X ∈g′, v ∈Rn
.
Type 4: There exists a decomposition Rn=Rk⊕Rn−k, 0 < k < n, and an epimorphism
ψ:z→Rk, such that h= (f⊕g′)⋉Rn−kwhere f:= {(Z, ψ(Z)) |Z∈z}= graph ψ⊂z⊕Rk.
Or, written in matrix form:
h=
0ψ(Z)tvt0
0 0 0 −ψ(Z)
0 0 Z+X−v
0 0 0 0
Z∈z, X ∈g′, v ∈Rn−k
.
Note that, since hacts indecomposably, its projection onto Rnis always all of Rn,
prRn(h) = Rn,
for all four types. However, in the second coupled type, hdoes not contain Rn, only Rn−k. The
connected Lie groups H0corresponding to hof types 1 and 2 are given as
(R+×G0)⋉ Rnor G0⋉ Rn.
Denote by G′0the connected subgroup in Pthat corresponds to the derived Lie algebra g′of g.
For the coupled type 3 we have that
H0= (F0×G′0)⋉ Rn,
where F0is the connected Lie group corresponding to the Lie algebra f={(ϕ(Z), Z )|Z∈z} ⊂ R⊕z.
For the last coupled type the connected component of His given by
H0= (F0×G′0)⋉ Rn−k,
where F0is the connected Lie group corresponding to the Lie algebra f={(Z, ψ(Z)) |Z∈z} ⊂
z⊕Rk. In all cases we have that prRn(H0) = Rnand G0:= prSO(n)(H0) is given by the the
connected Lie subgroup in SO(n) corresponding to g.
Proposition 1. Let H0⊂SO0(1, n + 1) be the connected component of an immersed Lie group
H⊂O(1, n+1), and assume that H0acts indecomposably and not irreducibly. If G:= prO(n)(H)⊂
O(n)is the projection of Honto O(n)and G0= prO(n)(H0)⊂SO(n)its connected component,
then, for the four different types, we have:
Type 1: H= (R∗×G)⋉ Rn, or H= (R+×G)⋉ Rn,
Type 2: H=ˆ
G⋉ Rn, where ˆ
G⊂R∗×Gwith connected component G0,
Type 3: There is a subset Γ⊂Z2×G⊂R∗×O(n)such that H= Γ ·H0.
Type 4: There is a subset Γ⊂R∗×Gsuch that H= Γ ·H0.
Proof. In order to prove the statement, we show in all four cases,
(5) for every P∈Hthere is an element Q∈H0such that P·Q∈R∗×O(n).
For the first three types for which we have Rn⊂H0⊂Hthe statement is obvious: Here we have
P=
a vt∗
0A∗
0 0 a−1
∈H
and we find
Q:= exp
0−a−1vt0
0 0 a−1v
0 0 0
=
1−a−1vt∗
01∗
001
∈Rn⊂H0
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 7
such that
P·Q=
a0 0
0A0
0 0 a−1
∈R∗×O(n).
This implies the form of Hin the types 1 and 2. For type 3 we can prove more. For an arbitrary
P=
±eavt∗
0A∗
0 0 ±e−a
∈H
we choose Z∈zsuch that a+ϕ(Z) = 0 and consider
Q1=: exp
ϕ(Z) 0 0
0Z0
0 0 −ϕ(Z)
=
eϕ(Z)0∗
0 exp(Z)∗
0 0 e−ϕ(Z)
∈F0⊂H0
and
Q2:=
1∓vteZ∗
01∗
0 0 1
∈Rn⊂H0.
Then
P·Q1·Q2=
±ea+ϕ(Z)vteZ∗
0AeZ∗
0 0 ±e−(a+ϕ(Z))
1∓vteZ∗
01∗
001
=
±1 0 ∗
0AeZ0
0 0 ±1
∈Z2×O(n).
In the case, when Rn6⊂ H0, the Lie algebra hof H0is of the second coupled type and we write
P=
a utvt∗
0A B ∗
0C D ∗
0 0 0 a−1
∈H,
with u∈Rkand v∈Rn−k. Since the linear map ψ: prso(n)(h)→Rkis surjective, we find an
X∈so(n−k) such that ψ(X) = −a−1u. Then for
Q1:= exp
0 0 −a−1vt0
0 0 0 0
0 0 0 a−1v
0 0 0 0
=
1 0 −a−1v∗
010∗
0 0 1∗
0 0 0 1
∈H0
Q2:= exp
1ψ(X)t0 0
0 0 0 −ψ(X)
0 0 X0
0 0 0 0
=
1ψ(X)t0∗
0 1 0 ∗
0 0 exp(X)∗
0 0 0 1
∈H0
we obtain
P·Q1·Q2=
a ut0∗
0A B ∗
0C D ∗
0 0 0 a−1
·Q2=
a aψ(X)t+ut0∗
0A B exp(X)∗
0C D exp(X)∗
0 0 0 a−1
∈R∗×O(n),
as ψ(X) = −a−1u. This verifies (5) also for type 4 and proves the proposition.
Note that Lemma 1 and Proposition 1 imply Theorem 1 from the introduction when applied to
the full and the restricted holonomy group of a Lorentzian manifold.
8 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
3. Null line bundle and screen bundle
In this section let Holp(M, g ) and Hol0
p(M, g) be the full and the restricted holonomy group of
a Lorentzian manifold (M, g ) of dimension n+ 2 >2, and let ∇denote the Levi-Civita connection
of g. We assume that the restricted holonomy group acts indecomposably and not irreducibly.
Then, from Theorem 1 we know that the same holds true for the full holonomy group. Hence,
by the fundamental principle of holonomy by which holonomy invariant subspaces correspond to
distributions on the manifold that are invariant under parallel transport [8, 10.19], the manifold
admits a global distribution Lof null lines that is invariant under parallel transport. Of course,
also the distribution L⊥whose fibres are orthogonal to the fibres of Lis invariant under parallel
transport. Hence, the tangent bundle is filtrated by parallel distributions
L ⊂ L⊥⊂T M.
The Levi-Civita connection induces a linear connection ∇Lon the bundle Lby ∇L:= prL◦ ∇|L,
where prLis the projection onto L. Moreover, the metric gand the Levi-Civita connection ∇
induce a bundle metric gSas well as a covariant derivative ∇Son the so-called screen bundle
S:= L⊥/L → M
by gS([X],[Y]) := g(X, Y ) and ∇S
X[Y] := [∇XY], where [ .] : L⊥→ S =L⊥/Ldenotes the
canonical projection. The following Proposition shows the relation between the holonomy groups
of (L,∇L) and (S,∇S) and the projections of Holp(M , g) onto R∗and O(n), respectively.
Proposition 2. Let (M, g)be a Lorentzian manifold with indecomposably, non-irreducibly acting
restricted holonomy group, let Lbe the corresponding distribution of null lines and S=L⊥/Lthe
sreen bundle on M. Then:
1) Holp(L,∇L) = prR∗(Holp(M, g )).
2) The line bundle Lis orientable, i.e., Ladmits a global nowhere vanishing section if, and
only if, prR(Holp(M, g)) ⊂R+. This is equivalent to time-orientability of (M, g).
3) The connection ∇Lis flat if, and only if, prR∗(Hol0
p(M, g)) = {1}, and the line bundle L
has a global parallel section iff prR∗(Holp(M, g)) = {1}.
4) Holp(S,∇S) = prO(n)(Holp(M, g )) and Hol0
p(S,∇S) = prO(n)(Hol0
p(M, g)).
Proof. The first statement is obvious since the parallelity of Limplies PL
γ=Pg
γ|Lfor any curve γ,
where PL
γis the parallel transport in (L,∇L) and Pg
γthat of (M, g ).
If Ladmits a global section V∈Γ(L), then PL
γ|t(V(γ(0))) = α(t)V(γ(t)) for all curves γ,
where αis a positive function with α(0) = 1. Hence prR∗(Holp(M , g)) ⊂R+. Conversally, let
prR∗(Holp(M, g )) ⊂R+. Then using the holonomy principle, the half-line R+ℓ⊂ Lpprovides a
well defined field of directions L+⊂ L. Thus, we have a covering M=SkUkand local sections
Vk∈Γ(Uk,L) such that Vk(x)∈ L+
xfor any x∈Uk. Using a partition of unity we derive a global
nowhere vanishing section of L. The orientablility of Lis equivalent to time-orientability of (M, g).
To see this, we choose a splitting s:S→ L⊥of the sequence
0→ L → L⊥→ S → 0.
Then E:= s(S)⊂ L⊥and E⊥⊂T M is a subbundle of signature (1,1) with L ⊂ E⊥. Hence,
the light-cone of E⊥
pat any p∈Mis a union of two lines, one of which is given by Lp. Thus,
we derive a second lightlike vector field Z∈Γ(M, E⊥) with g(V, Z ) = 1. Then 1
√2(V−Z) is a
nowhere vanishing timelike unit vector field and (M, g) is time-orientable. On the other hand, any
timelike unit vector field Ton (M, g) defines a global field of null direction L+⊂ L by requiring
g(T, L+)>0, hence, Lis orientable.
The third statement follows from the first one and standard facts of holonomy theory: ∇Lis
flat iff Hol0
p(L,∇L) = {1}, the line bundle (L,∇L) admits a global parallel section if, and only if,
Holp(L,∇L)) = {1}.
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 9
Finally, we proof the fourth statement. Let γ: [0,1] →Mbe a loop around the point p=
γ(0) = γ(1) ∈M. We fix a complement E⊂ L⊥
pof Lpthat is orthogonal to Lp. Then the
canonical projection [ .] : L⊥
p→ Spbecomes a linear isomorphism when restricted to E. For a fixed
non-vanishing vector ℓ∈ Lpwe denote by V(t) the parallel displacement of ℓalong γwith respect
to the Levi-Civita connection ∇gof g. We have to prove that the parallel transport Pg
γand the
parallel transport PS
γwith respect to ∇Scommute with the canonical projection [ .],
(6) Pg
γ(e)=PS
γ([e]),
for every e∈E. Indeed, if we write PS
γ|[0,t]([e]) = [U(t)] with U(t)∈ L⊥
γ(t)and U(0) = e, we have
0≡ ∇S
˙γ(t)[U(t)] = h∇g
˙γ(t)U(t)i
which implies that
∇g
˙γ(t)U(t) = f(t)V(t)
with a function f: [0,1] →R. Since V(t) is parallel, the vector field
U(t)−Zt
0
f(s)ds ·V(t)
is parallel along γwith respect to ∇gand equals ein t= 0. Hence,
Pg
γ(e)=U(1) −Z1
0
f(s)ds ·V(1)= [U(1)] = PS
γ([e]).
This proves statement (6) and the proposition.
Since G0
p:= prSO(n)(Hol0
p(M, g)) ⊂SO(Sp), as a subgroup of SO(n), is compact, it acts com-
pletely reducible on
Sp=V0⊕V1⊕...⊕Vk
with V0trivial and Viirreducible for i= 1,...,k, and moreover, using the Bianchi-identity, it can
be shown (see [6] or [23]) that
G0
p=G1×...×Gk
is a direct product of subgroups Giacting irreducibly on Viand trivial on Vjfor j6=i. Furthermore,
in [23] we have shown that G0
pacts as a Riemannian holonomy representation, i.e. G0
pis trivial or
a product of the groups from the Berger list, i.e. of
(7) SO(n),U(m),SU(m),Sp(k),Sp(k)·Sp(1),Spin(7),and G2,
and of isotropy groups of Riemannian symmetric spaces. In section 7 we will use this in order to
prove Theorem 2.
4. Holonomy groups and coverings
In the following we will consider manifolds that are given as a quotient by a group of diffeo-
morphisms. We recall the following facts (see for example [29, Chapter 7]): Let Γ be a group of
diffeomorphisms of a smooth manifold M. We say that Γ is properly discontinuous, if
(PD1) each p∈Mhas a neighborhood U, such that γ(U)∩U=∅for all γ∈Γ\ {1}and
(PD2) two points pand q, which are not in the same orbit under Γ, have neighborhoods Upand
Uqsuch that γ(Up)∩Uq=∅for all γ∈Γ.
Clearly a properly discontinuous group acts freely on M. If Γ is a properly discontinous group of
diffeomorphisms of M, the quotient space M/Γ is a smooth manifold and the projection π:M→
M/Γ is a smooth covering map [29, Chapter 7, Proposition 7.7]). If Γ is a properly discontinuous
group of isometries of a semi-Riemannian manifold M, then there is a unique metric on M/Γ, such
that π:M→M/Γ is a semi-Riemannian covering ([29, Chapter 7, Corollary 7.12]).
10 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
Remark 2. Let ΓΩ
λ,λ∈R\ {0}, be the group of diffeomorphisms of an appropriate domain
Ω⊂R2generated by
ϕλ(v, u) := (eλv, e−λu),(v, u)∈Ω.
For Ω0:= R2\ {(0,0}, the group ΓΩ0
λfails to satisfy (PD2) for the points p= (1,0), q= (0,1).
For the halfspace Ω := {(v , u)∈R2|u > 0}, the group ΓΩ
λis properly discontinuous whereas the
groups ΓΩ
λ1,λ2⊂Diff(Ω), generated by ϕλ1and ϕλ1for λ1,λ2linearly independent over Q, fail to
satisfy (PD1) since there is a sequence of integers (kn, ln)∈Z×Zsuch that 0 6=knλ1+lnλ2
n→∞
−→ 0.
In the following section we will have to check that certain group actions on manifolds define
quotients that are manifolds again. The following simple observation is useful. Its proof is straight-
forward.
Definition 1. Let M1and M2be two smooth connected manifolds and Γ ⊂Diff(M1)×Diff (M2)
a group of diffeomorphisms of M1×M2. We denote by Γi:= proji(Γ) ⊂Diff(Mi) the projections
and for σ∈Γ2by Γσ⊂Γ1the σ-section Γσ:= {γ∈Γ1|(γ, σ)∈Γ}. We say that Γ is of quotient
type, if
(i) Γ2is properly discontinuous,
(ii) ΓIdM2⊂Γ1satisfies (PD1) and
(iii) Γσ⊂Γ1satisfies (PD2) for all σ∈Γ2,
(or if the same is true exchanging the role of M1and M2).
Clearly, (iii) is satisfied if the section sets Γσ⊂Γ1are finite for all σ∈Γ2.
Lemma 2. Let Γ⊂Diff(M1)×Diff(M2)be a group of quotient type. Then Γis properly discon-
tinuous.
The fundamental relation (2) between the fundamental group and the quotient of the full holo-
nomy by the restricted holonomy group is a special case of the following fact when applied to the
universal covering. In the following when referring to semi-Riemannian manifold, for brevity of
notation we do not mention explicitly the metrics. Furthermore, e
Peγdenotes the parallel transport
along a curve eγin f
Mand Pγalong a curve γin M, both with respect to the Levi-Civita connections
of the semi-Riemannian metrics on f
Mand M, respectively.
Proposition 3. Let f
Mbe a connected semi-Riemannian manifold and Γa properly discontinuous
group of isometries of f
Minducing the semi-Riemannian covering π:f
M→M:= f
M/Γ. Then, for
any points p∈Mand epin the fibre π−1(p)we have:
(i) The holonomy group Holep(f
M)injects homomorphically into Holp(M)via
ι:e
Peγ7→ Pπ◦eγ,
for eγa loop at ep, and the image is a normal subgroup.
(ii) The following map is a surjective group homomorphism,
Φ : Γ →Holp(M)/Holep(f
M)
σ7→ [Pγ],
where γis a loop at pthat, when lifted to a curve eγstarting at ep, ends at σ−1(ep).
(iii) Let γbe a loop in Mat p∈M. Then, using the identification of TpMwith Tepf
Mby dπep, the
parallel transport along γis given by
(8) Pγ=dσσ−1(ep)◦˜
P˜γ,
where eγis the lift of γstarting at epand eγ(1) = σ−1(ep)with σ∈Γ. In particular,
(9) φ(σ) := dσσ−1(ep)◦˜
P˜γ= (dσ−1|p)−1◦˜
P˜γ
is a representative of Φ(σ)∈Holp(M)/Holep(f
M).
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 11
Proof. (i) Clearly, ιis a group homomorphisms: If eγand e
δare loops at ep∈f
M, we have3:
ι(e
Peγ·e
Pe
δ) = ι(e
Peγ∗e
δ) = Pπ◦(eγ∗e
δ)=Pπ◦eγ∗π◦e
δ=ι(e
Peγ)·ι(e
Pe
δ).
That ιis injective follows from the fact that πis a local isometry, which implies that
dπ−1
ep◦Pπ◦eγ◦dπep=e
Peγ
(see for example [29, p. 91]). To show that the image of ιis a normal subgroup, we proceed in the
same way as for the restricted holonomy group. Let γbe a loop at p∈Mthat lifts to a loop at ep
and δa loop at p=π(ep). Denote by e
δthe lift of δthat starts at epand by eγthe lift of γstarting
at e
δ(1) ∈π−1(p). Note that eγis a loop at e
δ(1) and we have
Pγ·Pδ=Pδ·Pδ−1∗γ∗δ=Pδ·Pπ◦(e
δ−1∗eγ∗e
δ).
Since e
δ−1∗eγ∗e
δis a loop at ep, the image of ιis normal.
(ii) First we have to verify that the map Φ is well defined. For two curves eγand e
δstarting at
ep∈f
Mand ending at σ−1(ep) for a σ∈Γ we can write
e
Pe
δ=e
Peγ·e
Peγ−1∗e
δ
which shows that
Pπ◦e
δ=Pπ◦eγ·Pπ◦(eγ−1∗e
δ).
Noting that eγ−1∗e
δis a loop at epshows that the image of σunder Φ does not depend on the chosen
curve eγ.
Next we show that Φ is a group homomorphism. Let σ1and σ2two elements of Γ, and γ1,γ2two
loops at p∈Mthat lift to curves starting at epand ending at σ−1
1(ep) and σ−1
2(ep), respectively. Now,
consider the curve eγ:= (σ−1
2◦eγ1)∗eγ2in f
M.eγstarts in epand ends in σ−1
2(σ−1
1(ep)) = (σ1·σ2)−1(ep)
and projects to γ1∗γ2. Hence, we have
Φ(σ1)·Φ(σ2) = [Pγ1∗γ2] = [Pπ◦eγ] = Φ(σ1·σ2).
Finally, Φ is surjective since f
Mis connected.
(iii) Since πis a local isometry, we have Pγ=dπ˜γ(1) ◦˜
P˜γ◦(dπ˜γ(0))−1. Then the statement
follows using the identification TpM≃Tepf
Mby dπepand dπσ−1(ep)=dπep◦dσσ−1(ep).
5. Construction of Lorentzian manifolds with disconnected holonomy
The following proposition shows that special classes of non-connected subgroups of the stabilizer
StabO(1,n+1)(L) can be realized as holonomy group of Lorentzian manifolds.
Proposition 4. Let G⊂O(n)be the holonomy group of an n-dimensional Riemannian manifold.
Then the subgroups
G⋉ Rn,(R+×G)⋉ Rn,(R∗×G)⋉ Rn,(Z2×G)⋉ Rn
in StabO(1,n+1)(L)can be realized as holonomy groups of Lorentzian manifolds.
Proof. The first part of the proof follows [21] or [4], where it was given for the restricted holonomy
group. We fix a Riemannian manifold (N, h) whose holonomy group is given by the not necessarily
connected group G. Furthermore we denote by (v, u) coordinates on R2and fix an open domain
Ω in R2, for example Ω = R2. Also we chose a smooth function f∈C∞(Ω ×N) with the property
that
06= det Hessh
p(f(v0, u0,·)) for some p∈Nand (v0, u0)∈Ω.(10)
3Note, that by γ∗δwe denote the joint path that first runs through δand then through γ.
12 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
Then we define a Lorentzian manifold (M, g ) by
(11) M= Ω ×N , gf,h = 2dvdu + 2fdu2+h.
Computing the Levi-Civita connection ∇gof gwe obtain as the only non-vanishing terms
∇g
XY=∇h
XY,
∇g
∂uX=∇g
X∂u=df(X)∂v,
∇g
∂u∂u=∂u(f)∂v−gradh(f),
∇g
∂u∂v=∇g
∂v∂u=∂v(f)∂v
for X, Y ∈Γ(T N ), ∇hand gradhwith respect to the Riemannian metric hon N. This allows
us to compute the parallel transport from a point q= (v0, u0, p) of a vector X0∈TpNalong a
curve δ= (v, u, γ) : [0,1] →Mfor γa curve in Nand δ(0) = q. Indeed, if X: [0,1] →T N is the
vector field along γthat is the parallel transport of X0∈TpNwith respect to ∇hand the function
ϕ: [0,1] →Rsatisfies the ODE
˙ϕ+ϕ·˙u·∂vf◦δ+ ˙u·df (X)◦δ= 0,
ϕ(0) = 0,
then the vector field,
ϕ·(∂v◦δ) + X(12)
is the parallel transport of X0along δ. This shows that for the curve δwe have that
prTpN◦ Pg
δ|TpN=Ph
γ.
Hence, the O(n) projection of Holq(M , g) is given as G= Holp(N, h).
Furthermore, when computing the curvature Rgof gwe get
Rg(∂u, X)Y= Hessh(f)(X, Y )∂v.
Taking this at points (v0, u0, p) where det Hessh
p(f(v0, u0,·)) 6= 0, this shows that the holonomy
algebra, and hence both, the restricted and the full holonomy group contain Rn.
Next we look at the R-component of the holonomy group. Clearly, when ∂vf= 0, the vector
field ∂vis parallel and what we have shown so far implies that Hol(M, g) = G⋉ Rn. Otherwise,
again by computing the curvature term,
Rg(X, ∂u)∂v=X(∂v(f))∂v
and by choosing fsuch that ∂vfis not constant on N, we conclude that the holonomy algebra
contains Rand hence the restricted holonomy contains R+. Therefore, as (M, g) is clearly time-
orientable, the full holonomy group is equal (R+×G)⋉ Rn.
Using this result, we construct non-time-orientable Lorentzian manifolds with holonomy group
(R∗×G)⋉ Rnas well as with holonomy group (Z2×G)⋉ Rn. We set Ω := R2\{(0,0)}, and choose
a function f∈C∞(Ω ×N) which satisfies f(−v, −u, p) = f(v, u, p) and condition (10). Then, for
the Lorentzian manifold as defined in (11), the map
(v, u, p)7→ σ(v, u, p) := (−v, −u, p)
is an isometry of (M , gf,h) and generates the group Z2in the isometry group of (M , gf,h). This
group acts properly discontinous on Mand hence M/Z2becomes a smooth Lorentzian manifold,
which is not time-orientable. By Proposition 3, its holonomy is given as (R∗×G)⋉ Rnif ∂vfis not
constant on N, and as (Z2×G)⋉Rnif fis chosen such that ∂vf= 0. Indeed, if δis a loop in M/Z2,
starting and ending in π(0,1, p), with a non-closed lift e
δ, then e
δis given by e
δ(t) = (v(t), u(t), γ(t)),
starting at (0,1, p) and ending at (0,−1, p) = σ−1(0,1, p), where γis a loop in Nbased at p. Then,
by the above formulae, the parallel transport to (0,−1, p) of the vector ∂vand a vector X0tangent
to Nat (0,1, p) along e
δis given as a∂vfor a positive number a, which is 1 when ∂vf= 0, and by
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 13
Ph
γ(X0) + b∂v, respectively. Hence, as dσ(0,−1,p)(∂v) = −∂vand dσ(0,−1,p)|T N = Id, formula (9) in
Proposition 3 gives the result.
Remark 3. Lorentzian metrics of the form (11) are sometimes called pf-waves, for “plane-fronted
waves” , or Np-waves, “N-fronted with parallel rays” (for example in [12] and [16]). Special classes
of pf-waves are
a) pp-waves, “plane fronted with parallel rays”, for which his flat,
b) plane waves, for which his flat and fis a quadratic polynomial in the coordinates on N
with u-dependent coefficients, and
c) Cahen-Wallach spaces, [11, 10], which are Lorentzian symmetric spaces. Here his flat and
fis a quadratic polynomial in the coordinates on Nwith constant coefficients.
Note that the construction for manifolds with holonomy (Z2×G)⋉ Rncannot be generalised
directly to discrete subgroups of Lorentz boosts of R1,1: As we observed in remark 2, a group of
Lorentz boosts Γ generated by diag(eλ,e−λ)⊂SO(1,1) will not act properly on R1,1\ {0}and the
quotient will not be Hausdorff. On the other hand, taking Ω smaller by removing a closed ball
around the origin in R1,1, Ω would no longer be invariant under Γ. We will avoid these difficulties
in the following construction by which we will obtain Lorentzian manifolds with disconnected
holonomy group contained in the stabiliser of a null line from the following data:
1) A simply connected Riemannian manifold (N, h) together with a function fon Mwith
the property (10).
2) a properly discontinuous subgroup Γ of isometries of the Lorentzian manifold
(f
M:= Ω ×N , gf,h := 2dvdu + 2f du2+h).
Theorem 3. Let (N, h)be a Riemanian manifold, Ωan open domain in R2with global coordinates
(v, u),fa smooth function on Ω×Nwith ∂vf= 0 and satisfying property (10). Then every
isometry σof the Lorentzian manifold
f
M= Ω ×N, gf,h = 2dvdu + 2fdu2+h
can be written as
(13) σ(v, u, p) = aσv+τσ(v, u, p),u
aσ
+bσ, νσ(v, u, p),
with certain aσ∈R∗,bσ∈R,τσ∈C∞(f
M)and νσ∈C∞(f
M , N), such that
dτσ(∂v) = dνσ(∂v) = 0,
and for each (v, u)∈Ω,νσ(v, u, .)is an isometry of the Riemannian manifold (N, h).
Furthermore, let Γbe a properly discontinuous group of isometries of (f
M , gf,h)and let π:
f
M→M:= f
M/Γbe the corresponding covering of the Lorentzian manifold M. Then, at a point
π(eq)∈Mfor eq= (v0, u0, p)∈f
M, the map Φ : Γ →Holq(M)/Holeq(f
M)of Proposition 3, with the
image written in the decomposition
R·∂v(eq)⊕TpN⊕R·(∂u−f ∂v)(eq)
of Teqf
M≃Tπ(eq)M, is given by the representative
ˆ
φ(σ) =
aσ0 0
0 (dν0
σ−1|p)−1◦Ph
σ0
0 0 a−1
σ
∈Φ(σ),
where
i) ν0
σ−1:= νσ−1(v0, u0, .)the isometry of Ndefined by σ−1at (v0, u0)∈Ω,
ii) Ph
σdenotes the parallel transport with respect to halong some curve γ: [0,1] →Nwith
γ(0) = pand γ(1) = νσ−1(v0, u0, p).
14 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
In particular, the full holonomy group of the Lorentzian manifold Mis given as
(14) Holπ(eq)(M) = nˆ
φ(σ)σ∈Γo·Holp(N, h)⋉ Rn.
Proof. The construction of gf,h and the conditions on fimply by Proposition 4 that the holonomy
group of ( f
M , gf,h) is given by the group Holp(N, h)⋉ Rn. Furthermore, the vector field ∂vis
parallel and so is its push forward σ∗∂vby an isometry σ. Indeed, for an isometry σwe have
0 = σ∗(e
∇X∂v) = e
∇(σ∗X)(σ∗∂v)
for all vector fields Xon f
M. By the conditions on f,f
Mis indecomposable, which implies that
σ∗∂vis a constant multiple of ∂v, as otherwise ∂vand σ∗∂vwould span a non-degenerate parallel
distribution. Hence, we write the isometry σin components according to f
M= Ω ×Nand using
global coordinates (v , u) on Ω as σ= (v◦σ, u ◦σ, νσ). Then, when ∂v(v◦σ) denotes the directional
derivative ∂v(v◦σ)(p) = d(v◦σ)p(∂v(p)), we have
σ∗∂v=∂v(v◦σ)◦σ−1·∂v
and that ∂v(v◦σ) is constant, i.e., there is a constant aσ∈R∗and a function τσof uand p∈N
such that v◦σ(v, u, p) = aσv+τσ(u, p). Also, because of
dσ(∂v) = aσ∂v+d(u◦σ)(∂v)∂u+dνσ(∂v)
we get that u◦σand νσdo not depend on v. We also note that σ∗Xis still orthogonal to ∂vfor
X∈Γ(T N ). This implies that dσ(X)∈R·∂v⊕T N and thus that d(u◦σ)(X) = 0 which shows
that u◦σis a function only of the coordinate u. We also note that
1 = g(∂v, ∂u) = σ∗g(∂v, ∂u) = aσ
d
du (u◦σ),
which shows that u◦σ=1
aσu+bσwith a constant bσ∈R. Finally
h(X, Y ) = g(X, Y ) = σ∗g(X, Y ) = ν∗
σh(X, Y )
for all X, Y ∈T N shows that νσ(v, u, .) is an isometry of (N , h) This proves formula (13).
In order to prove the result about the holonomy group, consider a curve
e
δ(t) = (v(t) := v◦e
δ(t), u(t) := u◦e
δ(t), γ(t))
in f
Mwith e
δ(0) = eq= (v0, u0, p) and e
δ(1) = σ−1(eq), γa curve in Nwith γ(0) = pand γ(1) =
νσ−1(eq). Then the formulae in the proof of of Proposition 4 imply that the parallel transport along
e
δ(t) = (v(t), u(t), γ(t)) is given in the decomposition
R·∂v⊕T N ⊕R·(∂u−f ∂v)
as
e
Pe
δ=
1∗ ∗
0Ph
γ∗
001
.
Secondly, we have seen that the differential of σat σ−1(eq) is given as
dσσ−1(eq)=
aσ∗ ∗
0dNνσ|σ−1(eq)∗
0 0 a−1
σ
=
aσ∗ ∗
0 (dν0
σ−1|p)−1∗
0 0 a−1
σ
.
Having this, we can apply formula (8) in Proposition 3 directly. For a loop δ: [0,1] →f
M/Γ
at π(eq), the lift is a curve e
δin f
Msuch that e
δ(0) = eqand e
δ(1) = σ−1(eq) for a σ∈Γ. Hence, by
formula (8) in Proposition 3 the parallel transport along δis given by
Pδ=
aσ∗ ∗
0 (dν0
σ−1|p)−1◦Ph
γ∗
0 0 a−1
σ
.
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 15
The same argument as in the proof of Proposition 1 shows, that we can use the matrix
ˆ
φ(σ) =
aσ0 0
0 (dν0
σ−1|p)−1◦Ph
γ0
0 0 a−1
σ
as representative of the class Φ(σ)∈H olq(M)/H oleq(f
M). This proves the second statement.
The examples in the next section will be based on this Theorem. For most of them we will
apply the following
Corollary 1. Let (N, h)be an n-dimensional Riemannian manifold, Γa properly discontinuous
group of isometries of (N, h), and faΓ-invariant function on Nsatisfying property (10). Fix a
not necessarily finite set of generators (γ1, γ2,...)of Γ. Corresponding to these generators of Γfix
a sequence of integers m= (m1, m2,...)and of real numbers λ:= (λ1, λ2,...).
(1) Consider the Lorentzian metric
gf,h = 2dvdu + 2fdu2+h
on f
M:= R2×N. Suppose that the group Γmgenerated in the isometry group of (f
M , gf,h)
by the isometries
σmi(v, u, p) := ((−1)miv, (−1)miu, γi(p))
for i= 1,2,... is of quotient type or restrict to f
M0:= (R2\ {(0,0)})×Notherwise.
Then Γmis properly discontinuous and the holonomy group of the Lorentzian manifold
(M:= f
M(0)/Γm, gf,h)at π(v, u, p)is given as
LΓm·(Holp(N)⋉ Rn),
where LΓmis the group that is generated in O(1, n + 1) by the linear maps
(−1)mi0 0
0φ(γi) 0
0 0 (−1)mi
∈Z2×φ(Γ),
where φ(γi)is the representative for the class Φ(γi)∈Holπ(p)(N/Γ)/Holp(N)as described
in of Proposition 3. In particular, for appropriate choices of m,(M, g f,h)is not time-
orientable, admits a parallel null line bundle but no parallel vector field.
(2) Set Ω := {(v, u)∈R2|u > 0}and consider the Lorentzian metric
gf/u2,h = 2dvdu +2
u2fdu2+h
on f
M:= Ω×N. Suppose that the group Γλgenerated in the isometry group of (M, gf /u2,h )
by the isometries
σλi(v, u, p) := (eλiv, e−λiu, γi(p))
for i= 1,2,... is of quotient type. Then Γλis properly discontinuous and the holonomy
group of the Lorentzian manifold (M:= f
M/Γλ, gf /u2,h)at π(v, u, p)is given as
LΓλ·(Holp(N)⋉ Rn),
where LΓλis the group that is generated in O(1, n + 1) by the linear maps
eλi0 0
0φ(γi) 0
0 0 e−λi
∈R+×φ(Γ).
In particular, (M, gf,h)is time-orientable, admits a parallel null line bundle, but, for ap-
propriate choices of λ, admits no parallel vector field.
16 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
Proof. Since Γmand Γλare of quotient type, by Lemma 2 they are properly discontinuous as well.
By the constructions of the Lorentzian metrics, they also act isometrically. Then the formula for
the holonomy group of the Lorentzian quotient follows from formula (14) in Theorem 3.
Remark 4. Obviously, the constructions presented in this section always give non-compact ex-
amples. But in simple situations they can be modified in order to obtain compact Lorentzian
manifolds by replacing Ω by a torus S1×S1with a suitable metric. Start with a compact Rie-
mannian manifold (N, h) with holonomy Gand a function fon S1×S1×N, even in the first two
entries and satisfying condition (10). Then the Lorentzian metric on M=S1×S1×Ngiven by
gf,h = 2dϕdθ +f dθ2+h,
where ϕand θare standard angle coordinates on S1×S1, has holonomy G⋉ Rnor (R+⋉G)⋉ Rn.
Then we can consider the same involution on Mas in Proposition 4, in coordinates
(ϕ, θ, p)7→ (ϕ+π, θ +π, p),
to obtain compact Lorentzian manifolds with holonomy (Z2×G)⋉ Rnor (R∗×G)⋉ Rn.
This can be generalised to the situation when we have a cocompact properly discontinuous
group ΓNof isometries of (N , h) being generated by γ1, γ2,.... We choose the function fto be
ΓN-invariant and independent of θand ϕ, fix natural numbers m= (m1, m2,...) and consider the
group Γmof isometries of (M, g f,h) which is generated by
(15) (ϕ, θ, p)7→ (ϕ+miπ, θ +miπ, γi(p)),
for i= 1,2,.... Then Γmis of quotient type and the holonomy of M/Γmis contained in the group
(Z2×Hol(N/ΓN)) ⋉ Rnbut might have a coupling between the Z2and the Hol(N/ΓN) part.
6. Examples with disconnected holonomy
In the following we will consider quotients of certain Lorentzian manifolds by a discrete group
of isometries. These examples will illustrate some of the characteristic features of the holonomy
groups that can be obtained by the construction given in Theorem 3.
6.1. Flat manifolds. We will start off with the flat case and give an example of a flat Lorentzian
manifold with indecomposable, non irreducible full holonomy (see Remark 1). Let R1,n+1 be the
Minkowski space and E(1, n + 1) = O(1, n + 1) ⋉ R1,n+1 its isometry group. Any isometry γhas
the form γ(x) = Aγx+vγ, where Aγ∈O(1, n + 1) and vγ∈Rn+2. For a discrete subgroup
Γ⊂E(1, n + 1) we denote by
LΓ:= {Aγ|γ∈Γ} ⊂ O(1, n + 1)
its linear part. Then, by Proposition 3, the full holonomy group of a flat space-time R1,n+1/Γ is
given by
Hol(R1,n+1/Γ) = LΓ.
In many cases, the holonomy group of R1,n+1/Γ stays trivial or acts decomposable. For example,
if R1,n+1/Γ is a complete homogeneous flat Lorentzian space, then Γ is a group of translations,
hence its holonomy group is trivial (see [33]). The same is true for non-complete flat homoge-
neous Lorentzian spaces (see [13] for the result). Looking at the affine classification of compact
3-dimensional flat space-times which are proper in the sense that their holonomy is contained in
SO0(1,2) in [33, Sec. 3.6], one finds all with trivial, with decomposable as well as with indecom-
posable holonomy group. Typically, an indecomposable holonomy group appears, if Γ contains an
element γ, such that Aγhas a lightlike eigenvector. We show this with a 4-dimensional example.
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 17
For a fixed θ∈Rwe consider the matrix in Aθ∈SO(1,3):
Aθ:=
1−cos θsin θ−1
2
0 cos θ−sin θ1
0 sin θcos θ0
0 0 0 1
.
where we used the basis (ℓ, e1, e2, ℓ∗) as in the first section. Then, by k7→ (Aθ)kwe obtain an
immersion of the finite group Zminto O(1,3) if θis a rational multiple of π, and an immersion of
the group Zinto O(1,3) if θis an irrational multiple of π. If we denote the image of this immersion
by H, the connected component of His given by the identity and thus acts decomposable in a
trivial way, but the group Hdoes not admit a non-degenerate invariant subspace, and hence it acts
indecomposable. The group Hcan be easily realized as the holonomy group of a flat space-time.
For that we consider coordinates (v, u, x, y ) on R4and the flat Minkowski metric
g0= 2dudv +dx2+dy2.
For θ∈R, let Γ ⊂E(1,3) be the discrete group of isometries generated by
φθ(v, u, x, y) = Aθ(v, u, x, y) + (0,1,0,0).
Γ acts freely and properly. Hence, the quotient R1,3/Γ is a flat Lorentzian manifold with trivial
restricted holonomy and full holonomy H.
6.2. pp-waves. The next examples will be quotients of Lorentzian manifolds which are usually
referred to as pp-waves (see Remark 3). We will define them as follows: Let f
Mbe an open set in
Rn+2 on which we fix global coordinates as (v, u, x1,...,xn). Let fbe a smooth function on f
M
that does not depend on the vcoordinate, i.e. ∂vf= 0. A pp-wave metric on f
Mis a Lorentzian
metric gfwhich, in these coordinates, is given as
gf= 2dvdu + 2f du2+
n
X
i=1
(dxi)2.(16)
The vector fields
∂v, ∂i, ∂u−f∂v
(17)
form a global frame in which the metric gfis given as
gf=
001
01n0
100
.
Here we used the obvious notation ∂v:= ∂
∂v ,∂u:= ∂
∂u and ∂i:= ∂
∂xi,i= 1,...,n. Here and in the
following, all matrices are written with respect to the basis of the tangent spaces given by these
vector fields at the corresponding points.
Proposition 5. The holonomy group of an (n+2)-dimensional pp-wave is abelian and if the matrix
∂i∂jfis non-degenerate at a point, the holonomy group is given by
Rn≃
1yt1
2yty
01n−y
0 0 1
y∈Rn
.
Conversely, any Lorentzian manifold with this holonomy group is locally a pp-wave.
Proof. pp-waves are a special case of the manifolds considered in Proposition 4. Hence the first
statement follows from the proof of Proposition 4. For the converse statement see [21].
18 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
The formula (12) in the proof of Proposition 4 shows in addition that for a curve δ(t) =
(v(t), u(t), γ(t)) in a pp-wave, the parallel transport along δis given by matrices of the form
Pδ=
1∗ ∗
01n∗
001
.(18)
6.3. Lorentzian symmetric spaces. As a first class of manifolds that are covered by a pp-wave,
we consider Lorentzian symmetric spaces. Let (M, g) be an indecomposable Lorentzian symmetric
space of dimension n+ 2 ≥3. Then its transvection group G(M) is either solvable or semi-
simple [11]. In the latter case, (M, g ) is a space of constant sectional curvature κ6= 0, hence its
holonomy group acts irreducibly. The case of solvable transvection group was described by Cahen
and Wallach in [11, 10], see also [28]. The simply-connected models are given by the following
special pp-waves: Let λ= (λ1,...,λn) be an n-tupel of real numbers λj∈R\{0}. Then the
pp-waves Mλ:= (Rn+2, gλ), where
gλ:= 2dv du +
n
X
j=1
λj(xj)2du2+
n
X
j=1
(dxj)2,
are symmetric. If λπ= (λπ(1) ,...,λπ(n)) is a permutation of λand c > 0, then Mλis isometric to
Mcλπ. By Proposition 5 the holonomy group of Mλis abelian and isomorphic to Rn.
Any indecomposable solvable Lorentzian symmetric space (M , g) of dimension n+ 2 ≥3 is
isometric to a quotient Mλ/Γ, where λ∈(R\{0})nand Γ is a discrete subgroup of the centralizer
ZI(Mλ)(G(Mλ)) of the transvection group G(Mλ) in the isometry group I(Mλ) of Mλ. For the
centralizer Zλ:= ZI(Mλ)(G(Mλ)) the following is known (see [9]):
1) If there is a positive λior if there are two numbers λi, λjsuch that λi
λj6∈ {q2|q∈Q}, then
Zλ={tα|α∈R}, where tαis the translation tα(v, x, u) = (v+α, u, x).
2) If λi=−k2
iand ki
kj∈Qfor all i, j ∈ {1,...,n}, then Zλis generated by {tα|α∈R}and
the isometry
ϕβ(v, u, x) = (v, u +β π, (−1)βk1x1,...,(−1)βknxn),
where β:= min{r∈R+|rki∈Zfor all i∈ {1,...,n}}.
Clearly, any discrete subgroup in Zλis generated by a translation tαin the v-component and
a power of ϕβ. Hence, denote by Γm,α the discrete group of isometries generated by a translation
tαand by the isometry ϕm
βwith α∈Rand m∈N0(m= 0 in the first case). Γm,α is obviously
properly discontinuous. Consequently, any indecomposable solvable Lorentzian symmetric space
Mis isometric to Mλ/Γm,α for an appropriate mand α.
Proposition 6. Let Mλ/Γm,α be an indecomposable solvable Lorentzian symmetric space. Then
the full holonomy group is given by
Hol(Mλ/Γm,α)≃Rnif meven,
Z2⋉ Rn={Id, Sm}⋉ Rnif modd,
where in case of modd Smis the reflection Sm=diag((−1)mβk1,...,(−1)mβ kn)∈O(n). In
particular, any indecomposable solvable Lorentzian symmetric space is time-orientable and admits
a global parallel lightlike vector field.
Proof. In order to determine the holonomy group of Mwe apply formula (14) of Theorem 3. As
the parallel transport in Mλis of the form (18), we only have to compute the differentials of the
isometries in Γm,α. But these differentials are clearly given by the identity or by
dϕm
β=
1∗ ∗
0Sm∗
0 0 1
.
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 19
This proves the Proposition.
6.4. Quotients of pp-waves. Now, by applying Theorem 3 and Corollary 1 we construct two
types of quotients of pp-waves for which the full holonomy group has also elements in the dilatation
part R∗of StabO(1,n+1)(L). We fix linear independent vectors (a1,...,ap) in Rnand consider the
discrete group of translations of Rn,
Γ(a1,...,ap):= {ta|a=
p
X
i=1
miai, mi∈Z},
which is generated by the translations by the ai. Furthermore, we fix a Γ(a1,...,ap)-invariant function
f∈C∞(Rn) satsifying the property (10).
In the first example we consider the metric
egf= 2dvdu + 2f(x)du2+
n
X
i=1
(dxi)2
on f
M=R2×Rn=Rn+2. Then, for ta∈Γ with a=Pp
i=1 miaiwe define a isometry ϕaof ( f
M , gf)
by
ϕa(v, u, x) := ((−1)Pimiv, (−1)Pimiu, x +a).
and apply Theorem 3 to the properly discontinuous group
Γ := {ϕa|ta∈Γ(a1,...,ap)} ≃ Zp.
Then, by Corollary 1, the full holonomy group of the Lorentzian manifold M=Rn+2/Γ with
the Lorentzian metric induced by egfis given by
±1yt∗
01n∗
0 0 ±1
y∈Rn
≃Z2⋉ Rn,
whereas the restricted holonomy is given by Rn. In particular, Mis not time-oriented and admits
a parallel lightlike line, but no parallel lightlike vector field.
In order to construct an example which is time-orientable we involve Lorentz boosts on R1,1
in this construction and consider now the pp-wave metric defined by 1
u2f(x) on the half-space
f
M:= Ω ×Rnwith Ω = {(v, u)∈R2|u > 0} ⊂ Rn+2 , i.e.,
(19) ˜gf /u2= 2dvdu +2
u2f(x)du2+ (dx1)2+...+ (dxn)2.
Let λ1,...,λp∈Rbe linearly independent over Qand define for ta∈Γ, a=Pimiai, the
isometries
ψa(v, u, x) := (ePimiλiv, e−Pimiλiu, x +a).
on (f
M , egf /u2). The group Γ := {ψa|ta∈Γ(a1,...,ap)} ≃ Zpis of quotient type, hence properly
discontinuous, and ˜gf/u2induces a Lorentzian metric on the quotient M:= f
M/Γ. Since λ1,...,λp
are independent over Q, the epimorphism F:π1(M)։Holp(L,∇L) is injective. Hence, the full
holonomy group of the Lorentzian manifold M=f
M/Γ is given by
ePjkjλjyt∗
0 1 ∗
0 0 e−Pjkjλj
y∈Rn, kj∈Z
≃Zp⋉ Rn,
whereas the restricted holonomy group is given by Rn. In particular, the manifold Mis time-
orientable admitting a parallel null line, but no parallel lightlike vector field.
20 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
6.5. Coupled holonomy. The aim of this section is to modify the examples of the previous
section in a way so that the holonomy group features a coupling between the linear part in O(n)
and the scaling component in R∗in the sense that some group elements act simultaneously on R
and on Rn. The first examples are direct consequences of Corollary 1.
First we produce 4-dimensional examples. The first class of examples is based on 2-dimensional
flat Riemannian space-forms which are either diffeomorphic to a M¨obius strip or to a Kleinian
bottle (for both cases, see [33, Chap. 2.2.5]). In case of the M¨obius strip, this manifold is given
by R2/Γ1, where Γ1is infinite cyclic generated by γ= (B, tb)∈O(2) ⋉ R2, with a reflection B
fixing the non-zero vector b∈R2. In case of the Kleinian bottle this manifold is given by R2/Γ2,
where Γ2is generated by γ1= (B, tb) and γ2= (I , ta) in O(2) ⋉ R2, and Bis again a reflection
with B(b) = b6= 0 and B(a) = −a6= 0. For the construction of pp-waves, we fix two Γi-invariant
functions fion R2with a non-degenerate Hessian.
Consider the pp-wave metrics
egfi:= 2dvdu + 2fi(x1, x2)du2+ (dx1)2+ (dx2)2
on f
M:= R4and
egfi/u2:= 2dvdu +2
u2fi(x1, x2)du2+ (dx1)2+ (dx2)2
on f
M:= {(v, u, x1, x2)|u > 0}, both with holonomy R2. We fix numbers m, m1, m2∈Nand
λ, λ1, λ2in Rand define the groups of isometries Γ1
m, Γ1
λ, Γ2
m1,m2and Γ2
λ1,λ2as in Corollary 1, with
respect to these numbers and the fixed generators γof Γ1and γ1and γ2of Γ2. These groups are
of quotient type. Then, by Corollary 1 we obtain the following holonomy groups for the quotients
Hol(f
M/Γ1
m, gf1) =
(−1)km wt∗
0Bk∗
0 0 (−1)km
w∈R2, k ∈Z
,
Hol(f
M/Γ2
m1m2, gf2) =
(−1)k1m1+k2m2wt∗
0Bk1∗
0 0 (−1)k1m1+k2m2
w∈R2, k1, k2∈Z
,
Hol(f
M/Γ1
λ, gf1/u2) =
ekλ wt∗
0Bk∗
0 0 e−kλ
w∈R2, k ∈Z
,
Hol(f
M/Γ2
λ1λ2, gf2/u2) =
ek1λ1+k2λ2wt∗
0Bk1∗
0 0 e−k1λ1−k2λ2
w∈R2, k1, k2∈Z
.
Thus, by different choices of the numbers m, m1, m2,and λ, λ1, λ2we can realize the groups Z2⋉R2,
(Z2⊕Z2)⋉ R2,Z ⋉ R2, and (Z⊕Z)⋉ R2, where the integer parts are immersed into (R∗×
O(2)) coupling both factors, as holonomy groups of 4-dimensional not time-orientable Lorentzian
manifolds.
Now we start the construction with a group of isometries of R2that is not properly discontinuous.
Fix an angle θand consider the group Γθgenerated by the rotation Dθby θ. This group is
isomorphic to Zor Zpdepending on whether θis a rational multiple of πor not. We fix a
rotational invariant function fon R2with property (10) and consider again the pp-wave metric
gf/u2= 2dvdu +2
u2f+ (dx1)2+ (dx2)2
on f
M= Ω ×R2with Ω = {(v, u)∈R2|u > 0}. Then, for λ, c ∈R, the group Γ generated by
ϕ(v, u, x) := (eλv+c, e−λu, Dθ(x))
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 21
acts on f
Mby isometries of gf/u2and is properly discontinuous. Indeed, since we restrict the action
to {u > 0}the group of diffeomorphisms on Ω generated by
(v, u)7→ (eλv+c, e−λu)
is of quotient type and therefore properly discontinuous. Using this, we obtain that Γ is of quotient
type as well and Lemma 2 applies again. Hence, f
M/Γ is a Lorentzian manifold with holonomy
Z ⋉ Rnif λ6= 0, i.e.,
Hol(f
M/Γ, gf /u2) =
ekλ wt∗
0Dk
θ∗
0 0 e−kλ
w∈R2, k ∈Z
.
In a similar way we can construct non-time-orientable Lorentzian manifolds by taking Ω = R2\
{(0,0)}and ϕ(v, u, x) := (−v, −u, Dθ). Here the holonomy Z ⋉ Rnor Zp⋉ Rnacts as Z2in the
R∗part.
Remark 5. Note that, if in the prevous example we assume θ6=π, the orthogonal part
prO(2)(Hol( f
M/Γ, gf /u2)) ≃Γθcannot be realised as holonomy group of a complete Riemann-
ian manifold. If there was a complete 2-dimensional Riemannian manifold M2with holonomy Γθ,
then M2would be a flat Euclidean space-form. But the holonomy groups of flat 2-dimensional
space-forms are known to be trivial (for the plane, the cylinder and the torus) or generated by a
reflection (for the M¨obius strip and the Kleinian bottle) (see [33], Chap. 2.2.5). We do not know
if this is also true if we drop the assumption of completeness.
This is in contrast to the situation when considering only the connected component of the
holonomy group. Here it was proven in [23] that prSO(n)(Hol0) is always the connected holonomy
of a Riemannian manifold, and, by results in Riemannian holonomy theory (an overview can be
found in [19]), can be realised as holonomy of a complete Riemannian manifold.
In the same way, 5-dimensional examples can be constructed starting with flat 3-dimensional
Riemannian space-forms. These are classified (see [33, Chap. 3.3.5]) and one can easily single
out those which are generated by discrete groups Γ with non-trivial linear part LΓ. Then, similar
constructions as above give various examples where the R∗and the O(3) part of the holonomy
is coupled. We show this only for the 3-manifolds of type Sθ
1in the list of 3-dimensional flat
space-forms. Let θ∈(0,2π) and let us consider the discrete group Γ ⊂E(3) generated by
(A, ta)∈O(3) ⋉ R3, where a∈R3is an eigenvector of Aand A|a⊥=Dθis the rotation by the
angle θ. We fix a function f∈C∞(R3) which is Γ-invariant and has a non-degenerate Hessian in
a certain point and consider the pp-wave metric gfand gf /u2on the appropriate f
M, where we
choose the coordinates in such a way, that ais a positive multiple of ∂1. Let m∈Nand λ∈Rbe
fixed numbers and denote by Γmand Γλthe properly discontinuous group of isometries on f
Mas
in Corollary 1. Then the holonomy group of the quotients is given by
Hol(f
M/Γm, gf) =
(−1)km s wt∗
0 1 0 ∗
0 0 Dkθ ∗
0 0 0 (−1)km
k∈Z, s ∈R, w ∈R2
,
Hol(f
M/Γλ, gf /u2) =
ekλ s wt∗
0 1 0 ∗
0 0 Dkθ ∗
0 0 0 e−kλ
k∈Z, s ∈R, w ∈R2
.
Again, by appropriate choice of λand θwe can realise the groups Z ⋉ R3and Z2q⋉ R3as holonomy
group of a 5-dimensional Lorentzian manifold, where Zand Z2qare immersed into R∗×O(3)
coupling both factors.
22 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
6.6. An infinitely generated holonomy group. Again by applying Corollary 1 we will give an
example of a 4-dimensional Lorentzian manifold with infinitely generated holonomy group arising
as a quotient of a pp-wave. Consider R2with the flat metric h0= (dx1)2+ (dx2)2and restrict h0
to the manifold N:= R2\Z2. The fundamental group Γ := π1Nof Nis a free group infinitely
generated by the loops going around the holes, but the holonomy group of (N, h0) is trivial. Fix
a function fon Nsatisfying (10). The universal cover of Nis R2and Nis the quotient of R2by
π1N. It is equipped with the Γ-invariant pull-back hof the flat metric h0on N. Also the function
fpulls back to a Γ-invariant function on R2. We apply Corollary 1 to (R2, h) and Γ.
Set Ω := {(v, u)∈R2|u > 0}and define a Lorentzian metric on f
M:= Ω ×R2by the usual
procedure
egf/u2,h = 2dvdu +2
u2fdu +h.
Now we fix generators (γ1, γ2,... ) of Γ and real numbers λ:= (λ1, λ2,... ) which are linearly
independent over Qand define the following isometries of ( f
M , gf/u2,h ):
ϕi(v, u, x) = (eλiv, e−λiu, γi(x)).
Let Γλbe the group of isometries of ( f
M , egf /u2,h) that is generated by the ϕifor i= 1,2,... .
Since the fundamental group Γ = π1Nacts as group of deck transformations of the universal
covering π:R2→N, it is properly discontinuous. Moreover, since Γ is a free group, the sections
(Γλ)σ⊂(Γλ)1consist only of one element for all σ∈Γ. Hence, Γλis of quotient type and by
Lemma 2 properly discontinuous, and the quotient M=f
M/Γλis a smooth Lorentzian manifold
with the induced Lorentzian metric gf/u2,h. Then Corollary 1 shows that the holonomy group of
(M, gf /u2,h) is generated by the following matrices
eλiwt∗
012∗
0 0 e−λi
∈O(1,3),
with w∈R2and λione of the fixed real numbers. Since these were chosen linearly independent
over Q, the quotient Holp(M)/Hol(f
M) = Holp(M)/Hol0(M) and also the holonomy group of the
induced connection on the null line bundle Lis infinitely generated.
6.7. Examples with curved screen bundle. Using constructions in Riemannian geometry one
obtains Lorentzian manifolds with parallel null line, curved screen bundle and disconnected holo-
nomy with a coupling between the R∗and the O(n) part.
The first construction is based on Riemannian manifolds that go back to Hitchin [18] and
McInnes [24, 25] and which are described in detail in [27]. Let Nbe the complete intersection of
m+ 1 quadrics in CP2p+1, defined as the common zero set of quadratic polynomials with strictly
positive real coefficients, and endowed with the Riemannian metric hfrom CP2p+1. Then (N, h) is
a simply connected K¨ahler manifold. On Nwe have an involution defined by complex conjugation,
σ([z0,...,z2p+1]) := ([z0, . . . , z2p+1]),
which is an anti-holomorphic isometry of (N , h). Then, if Γ denotes the group of isometries gener-
ated by σ, in [27, Corollary 9] it is proven that (N/Γ, h) is a compact 2p-dimensional Riemannian
manifold with holonomy Z2⋉SU(p). In addition, (N/Γ, h) is Ricci flat and admits a parallel spinor
field. Note that Φ(σ) is represented by the conjugation on the tangent space with respect to the
complex structure given by the K¨ahler structure. We denote this involution by σ∗.
If we fix a Γ-invariant function fon Nand, for m∈Zand λ∈R, perform both constructions
as in Corollary 1, we obtain (Z2⋉SU(p)) ⋉ R2pand (Z ⋉ SU(p)) ⋉ R2pas holonomy groups, with
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 23
Z2and Zacting simultaneously on Rand SU(p), i.e.,
Hol(f
M/Γm, gf ,h) =
(−1)km 0 0
0σk
∗0
0 0 (−1)km
k∈Z,
·(SU(p)⋉ R2p),
Hol(f
M/Γλ, gf /u2,h) =
ekλ 0 0
0σk
∗0
0 0 e−kλ
k∈Z
·(SU(p)⋉ R2p).
Note that both manifolds no longer admit parallel spinors because they do not admit a parallel
vector field. We will consider the full holonomy groups of Lorentzian manifolds with parallel spinors
in the next section.
In [32] Wilking constructed a remarkable example of a compact Riemannian manifold with non-
compact holonomy group. The manifold is given as a quotient of a 5-dimensional solvable Lie
group S:= R4⋊ R with a left-invariant Riemannian metric by a cocompact subgroup Γ := Λ ⋊ Z
where Λ is a certain lattice of R4. Then S/Γ has restricted holonomy SO(3) ⊂SO(5), acting
trivially on a 2-dimensional subspace of TpS, and SO(3) ×Zas full holonomy group. Again, by
the constructions in Corollary 1 we can produce 7-Lorentzian manifolds with parallel null line and
full holonomy (Z×SO(3)) ⋉ R5where Zacts on Rand SO(3) simultaneously, either as Z2or as Z.
In the same way as described in Remark 4 we can even construct compact Lorentzian manifolds
of the form (S1×S1)/Z2×S/Γ with indecomposable holonomy (Z2×Z×SO(3)) ⋉ R5or of the
form (S1×S1×S)/Γm, where Γmis generated as in formula (15), and with holonomy contained
in (Z2×Z×SO(3)) ⋉ R5, but possible with non-trivial Z2⊂R∗-part which is now coupled to
Z⊂O(5).
7. Full holonomy groups of Lorentzian manifolds with parallel null spinor
A Lorentzian spin manifold (M , g) of dimension n+2 >2 is a time- and space-oriented Lorentzian
manifold with a spin structure e
O(M, g)→O(M, g ) that is a reduction with respect to the double
cover Λ : Spin0(1, n + 1) →SO0(1, n + 1), where O(M, g ) is the bundle of time- and space-oriented
frames over Morthonormal for g. This allows us to write the tangent bundle as
T M = O(M, g)×SO0(1,n+1) R1,n+1 =e
O(M, g)×Spin0(1,n+1) R1,n+1,
and defines the spinor bundle
Σ = e
O(M, g)×Spin0(1,n+1) ∆1,n+1,
where ∆1,n+1 is the spinor module. The spinor bundle is equipped with a metric of neutral
signature, a Clifford multiplication
·:T M ×Σ→Σ,
and a covariant derivative ∇Σ. All these structures are compatible with each other (for details see
for example [3]).
Every section in the spinor bundle ϕ∈Γ(Σ) defines a causal vector field Vϕ∈Γ(T M ) via
g(Vϕ, Y ) = −hY·ϕ, ϕi.
Both, ϕand Vϕhave the same zero set, and the spinor field is of zero length, a null spinor, if Vϕ
is of zero length, i.e. a null vector field. If the spinor bundle admits a section ϕwith ∇Σϕ= 0, for
short, a parallel spinor, then Vϕis a parallel vector field as well.
Since a Lorentzian spin manifold is assumed to be time- and space-oriented, its full holonomy
H:= Holp(M, g ) is contained in the connected component SO0(1, n + 1) of O(1, n + 1), but is not
necessarily connected itself. Using Proposition 1 and results in [23] and [30] we can summarise
what we know so far about the holonomy group in this situation:
24 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
Proposition 7. Let (M, g)be a Lorentzian spin manifold with special holonomy. If (M , g)admits
a parallel spinor, then (M , g)admits a parallel null vector field and its full holonomy group His
given as
H=G⋉ Rn,
with G⊂SO(n), and the connected component G0of Gis trivial or given as a direct product of
some of the following groups
SU(m),Sp(k),G2,Spin(7).
Proof. The parallel spinor on (M , g) induces a parallel vector field, which, as H= Hol(M, g ) acts
indecomposably, has to be null. Hence, the full holonomy group fixes a null vector,
Holp(M, g )⊂StabSO0(TpM)(Vϕ(p)) ≃SO(n)⋉ Rn,
and the restricted holonomy must be of type 2 or 4. Using the fact that for the coupled type 4
the group G0has a centre, in [23] it was shown, that the existence of such a parallel spinor implies
that the restricted holonomy must be of uncoupled type 2,
H0= G0⋉ Rn,
where G0is a Riemannian holonomy group which is isomorphic to a subgroup in Spin(n) admitting
a fixed spinor. Based on Berger’s list [7], these groups were determined by Wang [30]. Using
Proposition 1 we obtain that H=G⋉ Rnwith G⊂SO(n).
We will now generalize this result to the full holonomy group.
Proposition 8. Let (M, g)be a Lorentzian manifold of dimension (n+ 2) >2which is time- and
space-orientable with indecomposable restricted holonomy group H0⊂(R+×SO(n)) ⋉ Rn. Then
we have the following two implications:
1) If (M, g)admits a spin structure with a parallel spinor field, then the full holonomy group H
is given as H=G⋉Rnwith G⊂SO(n)and there exists a homomorphism Φ : G→Spin(n)
(a) with λ◦Φ = IdGfor λ: Spin(n)→SO(n)the twofold cover, and
(b) there is a spinor win the spinor module ∆nsuch that Φ(G)w=w.
2) If the full holonomy group is H=G⋉ Rnwith G⊂SO(n)and there is a homomorphism
Φ : G→Spin(n)with (a) and (b), then (M, g )has a spin structure with a parallel spinor.
Proof. Let (M , g) be time- and space-oriented and H= Holp(M, g )⊂SO0(1, n + 1) be its full
holonomy group. The proof relies on the following three observations:
1) If (M, g) admits a spin structure with a parallel spinor field, then there is a homomorphism
Ψ : H→Spin0(1, n + 1) with Λ ◦Ψ = IdHand a spinor v∈∆1,n+1 such that Ψ(H)v=v.
2) If there is a homomorphism Ψ : H→Spin0(1, n + 1) with Λ ◦Ψ = IdHand a spinor
v∈∆1,n+1 with Ψ(H)v=v, then (M, g ) has a spin structure with a parallel spinor on
(M, g).
3) If H=G⋉ Rn, then there is a homomorphism Ψ : H→Spin0(1, n + 1) with Λ ◦φ= IdH
and a spinor v∈∆1,n+1 such that Ψ(H)v=vif and only if there is a homomorphims
Φ : G→Spin(n) with (1a) and (1b).
The first observation was made by Wang in [31]. Indeed, if (M, g ) has a parallel spinor field, the
holonomy group ˜
Hof the spin connection fixes a spinor v∈∆1,n+1 and maps onto H, hence it
does not contain −1∈Spin0(1, n + 1). Therefore, Λ|˜
H:˜
H→His an isomorphism and we can
define Ψ := (Λ|˜
H)−1:H→Spin0(1, n + 1).
The second observation was made by Semmelmann and Moroianu in [27, Lemma 5] for Rie-
mannian signature but their proof works in any signature. Indeed, if His the holonomy bundle
through a frame in O(M , g) and Ψ : H→Spin0(1, n + 1) is a homomorphism with Λ ◦Ψ = Id, we
can define the spin structure by e
O(M, g) := H ×HSpin0(1, n + 1) which projects canonically onto
O(M, g ) = H ×HSO0(1, n + 1).
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 25
We have to proof the third observation. First assume that Ψ : H=G⋉ Rn→Spin0(1, n + 1)
is given. Since Λ ◦Ψ = IdH, the restriction of Ψ to Gmaps into Λ−1(SO(n)) = Spin(n)⊂
Spin0(1, n + 1). Hence we can define
Φ := Ψ|G:G→Spin(n).
Since λ= Λ|Spin(n), we also get that Φ ◦λ= IdG. Now, Rn⊂His a connected closed Abelian
subgroup. When realising Spin0(1, n +1) in the Clifford algebra Cl(1, n +1), the image of Rnunder
Ψ is given by
Ψ(Rn) = {1 + ℓ·x|x∈Rn} ⊂ Spin0(1, n + 1) ⊂ Cl(1, n + 1),
where (ℓ, e1,...,en, ℓ∗) is a basis as in (4) and Rn= span(e1,...,en). On the other hand, if
Φ : G→SO(n) is given, we define
Ψ : H=G⋉ Rn→Spin0(1, n + 1)
g·x7→ Φ(g)·(1 + ℓ·x).
One can check that Ψ ◦Λ = IdH.
It remains to verify that there is fixed spinor v∈∆1,n+1 if, and only if, there is a fixed spinor
w∈∆n. To this end we consider the Clifford algebra Cl(1,1) of the 2-dimensional space Rℓ⊕Rℓ∗
with the induced signature (1,1)-scalar product, fix a basis (u1, u2) in ∆1,1satisfying ℓ·u1=√2u2,
ℓ·u2= 0, ℓ∗·u1= 0, ℓ∗·u2=−√2u1and assign to a spinor v∈∆1,n+1 two spinors v1, v2∈∆n
by identifying
∆1,n+1 ≃∆n⊗∆1,1
v7→ v1⊗u1+v2⊗u2.
Then a computation shows that (g·a)(v) = vfor all g∈Ψ(G) and a∈Ψ(Rn) if, and only if,
gv2−√2(g·x)v1=v2,
gv1=v1
for all g∈Ψ(G) and all x∈Rn. The first equation for g=1implies that x·v1= 0 for all x∈Rn
and hence, v1= 0. Thus, v∈∆1,n+1 is fixed under Ψ(H) if, and only if, v=v2⊗u2and v2is
fixed under Ψ(G) = Φ(G). This shows observation (7) and completes the proof.
Corollary 2. Let H=G⋉ Rn⊂SO0(1, n + 1) with G⊂SO(n). Then the following vector spaces
have the same dimension:
i) spinors in ∆nfixed under G,
ii) spinors in ∆1,n+1 fixed under H,
iii) parallel spinors fields on a Lorentzian manifold with holonomy group H=G⋉ Rn.
Finally, what is needed to complete a proof of Theorem 2 is a classification of subgroups of
SO(n) that fix a spinor in ∆nand have connected component SU(m), Sp(k), G2and Spin(7). This
result can be obtained from [24] and [31].
Theorem 4 (McInnes [24] and Wang [31]).Let G⊂SO(n)be Lie group with connected component
G0equal to SU(m),Sp(k),G2or Spin(7) with a non-vanishing fixed spinor in ∆n. Then Gis equal
to one of the groups in the following table, in which Nis the dimension of spinors fixed under G,
26 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
G0n G N conditions
SU(m) 2mSU(m)2
SU(m)⋊ Z21mdivisible by 4
Sp(k)k+ 1
Sp(k) 4kSp(k)×Zd(k+ 1)/d d > 1, d odd and divides k+ 1
Sp(k)·Z2d2k
2d+ 1 keven, 1< d ≤2d
Sp(k)·Q4dk
2dif k
2odd keven, 1< d ≤2d,
k
2d+ 1 if k
2even
Sp(k)·B4dsee ref. [31] keven and conditions in [31]
Sp(k)·Γ1keven
Spin78Spin71
G27G21
Here
(1) Q4dis the double cover of the dihedral group D2dof order 2d,
(2) Sp(k)·B4dfor d= 6,12,30, and B4dis the double cover in Sp(1) of the polyhedral groups
P2din SO(3), i.e. the tetrahedral group P12, the octahedral group P24, and the icosahedral
group P60, and
(3) Γis an infinite subgroup of U(1) ⋊ Z2.
Steps in the proof. Since Gis contained in the normaliser of G0in O(n), first we need a list of
normalisers of the possible G0’s. They can be found in [8, 10.114] with a correction made in [24]
for the SU(m)-case. The cases in which G0is equal to G2or Spin(7) are trivial, as both groups are
equal to their own normaliser in O(n). We are left with G0being SU(m) or Sp(k). Their normaliser
in O(n) is given as U(m)⋊ Z2, where Z2acts by complex conjugations, and as Sp(k)·S p(1).
First assume that G/G0is finite. In [24], McInnes classified the possible holonomy groups of
compact Ricci flat Riemannian manifolds. Since their fundamental group is finite, as the first,
purely algebraic step in McInnes’ proof, possible subgroups Gin SU(m)⋊ Z2and Sp(k)·S p(1)
with finite quotients G/SU(m) and G/Sp(k) are listed. For SU(m) they are of the form
Zmr ·SU(m) or (Zmr ·SU(m)) ⋊ Z2,
with a positive integer rand with Zmr ∈U(1). For Sp(k) the list is longer:
(i) Zr·Sp(k), with rodd,
(ii) Z2r·Sp(k), with reven,
(iii) Q4d·Sp(k), where Q4dis the double cover of the dihedral group D2dof order 2d,
(iv) B4d·Sp(k) for d= 6,12,30, and B4dis the double cover in Sp(1) of the polyhedral groups
P2din SO(3), i.e. the tetrahedral group P12, the octahedral group P24 , and the icosahedral
group P60.
Using geometric arguments McInnes shortened this list to obtain all possible holonomy groups of
compact Ricci flat Riemannian manifolds, but since we cannot apply geometric arguments for our
purpose, we cannot use this shorter list. Instead we use results by Wang in [31], where the full
holonomy groups of Riemannian manifolds — compact and non-compact — with parallel spinors
are classified. Although in the compact case Wang can start from the shorter list obtained by
ON THE FULL HOLONOMY GROUP OF SPECIAL LORENTZIAN MANIFOLDS 27
McInnes, for the non-compact case in [31, Proof of Theorem 4.1] only algebraic arguments can be
used and the full list above has to be checked for the existence of fixed spinors. In the SU(m) case
Wang shows that r= 0, that is, only SU(m) itself and SU(m)⋊ Z2remains. In the Sp(m) case
Wang obtains the list in the table in the Theorem.
To conclude this section we consider the question whether there exist Lorentzian manifolds with
the holonomy groups in Theorem 2 and special causality properties. In Proposition 4 we proved
that starting with a Riemannian manifold (N, gN) with full holonomy group Gand a function
f∈C∞(R×N) such that det(HessN(f))|p6= 0 at some point p∈R×N, we obtain a Lorentzian
manifold M:= R2×Nwith the metric
gf,h = 2dvdu + 2fdu2+gN
(20)
with full holonomy G⋉ Rn. Proposition 8 and Corollary 2 show, that in case of a spin manifold
(N, gN), the Lorentzian manifold (M, g f,h) is spin as well and the dimension of the spaces of parallel
spinor fields on (M, gf,h) and (N, gN) are the same. Moreover, for Lorentzian manifolds of type
(M, gf,h ) various causality properties are known (see for example [12] and [16]). Let us quote here
the following two results.
1) If (N, gN) is a complete Riemannian manifold, the function fdoes not depend on uand
is at most quadratic at spacial infinity, i.e., there exist x0∈Nand real constants r, c > 0
such that
f(x)≤c·dN(x0, x)2for all x∈Nwith dN(x0, x)≥r,
then (M, gf,h ) is geodesically complete. Here dNis the distance function of (N, gN).
2) If (N, gN) is a complete Riemannian manifold and the function −fis spacial subquadratic,
i.e., there exist x0∈Nand continious functions p, c1, c2∈C(R,[0,∞)) with p(u)<2 such
that
−f(u, x)≤c1(u)dN(x0, x)p(u)+c2(u) for all (u, x)∈R×N,
then (M, gf,h ) is globally hyperbolic.
Of course, both conditions for fcan be realized in addition to detHessNf(u0, x0)6= 0. Hence,
each of the groups in Theorem 2 can be realized as holonomy group of a Lorentzian manifold,
and in addition, if the group Gin Theorem 4 is the holonomy group of a complete Riemannian
manifold, then Hcan be realized by a geodesically complete as well as by a globally hyperbolic
Lorentzian manifold.
If one is interested in globally hyperbolic manifolds with complete or even compact space-like
Cauchy surfaces, another construction based on Lorentzian cylinders is useful, which from the spin
geometric point of view first was studied by B¨ar, Gauduchon and Moroianu in [2] and further
developed in the context of special holonomy by the first author and M¨uller in [5]. Formulated for
our situation the result is:
Proposition 9 ([5]).Let (N, gN)be an n-dimensional irreducible Riemannian spin manifold of
dimension nwith parallel spinors, (F, gF)the warped product (F=R×N, gF=ds2+e−4sgN)
over (N, gN),C:T F →T F a Codazzi tensor on (F, gF)with only positive eigenvalues and a∈R
a positive constant. Then the Lorentzian manifold (M, gC)given by
M:= (−a, ∞)×R×N, gC:= −dt2+C+ 2(t+a)IdT F ∗gF
(21)
has full holonomy
Hol(0,0,p)(M, g C) = C−1◦Holp(N, gN)◦C)⋉ Rn.
Moreover, if (N , gN)is complete, then the Lorentzian manifold (M, gC)is globally hyperbolic and
the space-like slices ({t} × F, gC
t= (C+ 2(t+a)IdT F )∗gF)are complete Cauchy surfaces.
Proof. The proof in [5], Theorem 3, states the result for the reduced holonomy groups. Using
Proposition 1 in addition, we obtain the result for the full holonomy group.
28 HELGA BAUM, KORDIAN L ¨
ARZ, AND THOMAS LEISTNER
Explicit examples for Codazzi tensors Con the warped product (R×N, ds2+e−4sgN) are given
in [5]. Take for example a bounded, strictly increasing function f∈C∞(R) with f(0) = 0 and
f(s)< λ for all s∈R. Then Cf:R∂s⊕T N →R∂s⊕T N given by
Cf:= e2sf′(s) 0
0 2e2s(λ−f(s))IdT N
is a Codazzi tensor on (F, gF) and the metric (21) is given by
gCf=−dt2+e2sf′(s) + 2a+ 2t2ds2+ 4e−2st+e−2sa+λ−f(s)2gN.
These two constructions reduce the problem of finding, for each Gin the table in Theorem 1, a
Lorentzian manifold with holonomy G⋉ Rnto the Riemannian case. First, one has to ensure the
existence of Riemannian manifolds with holonomy group G. Then, for geodesically complete or
globally hyperbolic Lorentzian metrics, one needs complete Riemannian manifolds with holonomy
group G. For connected holonomy groups we can built on the deep existence results for complete
and even compact Riemannian manifolds with special holonomy obtained by several authors (for
an overview see [19]). Based on the examples with connected holonomy groups, Moroianu and
Semmelmann in [27] constructed Riemannian manifolds with parallel spinor for each of the non-
connected groups Gin the table in Theorem 2. For SU(m)⋊ Z2they construct a compact manifold,
and for the remaining groups the metrics are obtained by removing points from compact spaces or
by cone constructions, thus these metrics are not complete. This yields the following conclusion.
Corollary 3. For each of the groups Gin Theorem 2 there exist Lorentzian manifolds with ho-
lonomy G⋉ Rnand parallel spinors. Moreover, for the connected groups Gand for SU(m)⋊ Z2,
there exist geodesically complete as well as globally hyperbolic Lorentzian manifolds with complete
spacelike Cauchy surfaces and holonomy G⋉ Rn.
It would be interesting to know, if the groups S p(m)×Zd, and S p(m)·Γ in Theorem 4 can be
realized as holonomy group of a complete Riemannian manifold.
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(Baum & L¨arz) Humboldt-Universit¨
at Berlin, Institut f¨
ur Mathematik, Rudower Chaussee 25, 12489
Berlin, Germany
E-mail address:baum@math.hu-berlin.de & laerz@math.hu-berlin.de
(Leistner) School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
E-mail address:thomas.leistner@adelaide.edu.au
