Homogeneous Lorentzian Manifolds

Homogeneous Lorentzian manifolds

D.V. Alekseevsky

Edinburgh University

Vienna, July 2011 We discuss the problem of classiﬁcation of homogeneous Lorentzian manifolds M = G/H with ( mostly proper ) action of an isometry group G and connected stability subgroup H.

We recall the classical result by Nadine Kowal- sky about nonproper homogeneous Lorentzian manifolds M = G/H ( such that the stabilizer H is non compact) and its recent generaliza- tions by M. Deﬀaf, K. Melnick and A. Zeghib.

1 We give a necessary and suﬃcient condition that a proper homogeneous manifold M = G/H admits an invariant Lorentz metric.

A homogeneous manifold M = G/H with a compact stabilizer H is called minimal admissible if it admits an invariant Lorentzian metric, but any homogeneous manifold M˜ = G/H˜ with a bigger (connected) stability group H˜ ⊃ H does not admits such a metric. We classify minimal admissible compact homogeneous manifolds M = G/H.

2 We show that any minimal admissible manifold M = G/H of a simple compact Lie group G is the total space of the canonical homogeneous S1-bundle over the minimal orbit F = Ad Gh = G/ZG(h) of the adjoint representation of G. The explicit form of invariant Lorentzian metrics is also given.

3 We reduce the classiﬁcation of minimal admissible manifold M = G/H of a simple noncompact Lie group G to description of minimal orbits of the isotropy representation of the associated non-compact symmetric space S = G/K and give a list of such manifolds of dimension ≤ 11. We discuss also the problem of description of nonproper homogeneous Lorentzian manifolds of non semisimple Lie group G.

4 Proper and nonproper homogeneous Lorentzian manifolds

An action of a Lie group G on a manifold M is called proper if the map

G × M → M × M, (g, x) 7→ (gx, x) is proper, or, equivalently, G preserves a com- plete Riemannian metric on M. In this case G-manifold M is called proper. The orbit space M/G of a proper G-manifold is a metric space and has a structure of a stratiﬁed manifold.

5 For nonproper G-manifold, the topology of the orbit space is very bad and can be non- Hausdorff, see e.g. the action of the Lorentz group on the Minkowski space. On the other hand, generically the isometry group of a compact Lorentzian manifold is compact and, hence, acts properly. M. Gro- mov states the problem of description of all compact Lorentzian manifolds which admits a noncompact (= nonproper) isometry group It is a special case of his more general problem of classification of geometric structures of finite order on compact manifold with a noncompact group of automorphisms.

6 G.D’Ambra proved that the isometry group of analytic simply connected compact Lorentzian manifold is compact. In a noncompact case, a surprising result was proven by N. Kowalsky: The only simple Lie groups which act as nonproper isometry groups of a Lorentzian manifolds are SO1,n and SO2,n. This implies that the only nonproper homogeneous Lorentz G- manifolds with a simple group G are the spaces of constant curvature (De Sitter and anti De Sitter spaces).

7 This result was recently generalized to the case of a semisimple group in the important paper by M. Deﬀaf,K. Melnick and A. Zeghib (2008). In particular, they show that the only nonproper homogeneous Lorentzian manifold of a semisimple Lie group is a (lo- cal) product of a homogeneous Riemannian manifold and (anti) De Sitter space.

8 Proper homogeneous Lorentzian manifolds

Let M = G/H be a homogeneous manifold with an eﬀective action of G. Then the fol- lowing conditions are equivalent: a) M = G/H is proper, that is the action of G on M is proper; b) the stabilizer H is compact. c) M admits an invariant Riemannian metric (which is deﬁned by an H-invariant Eucliden metric go in ToM, o = eH ∈ M

An H-invariant metric go can be constructed as the center of the ball of minimal radius in 2 ∗ S (To M) which contains the orbit j(H)g1 of some Euclidean metric g1.

9 Criterion for existence of an invariant Lorentzian metric on a proper manifold M = G/H

Proposition 1 A proper homogeneous manifold M = G/H admits an invariant Lorentzian metric iff the isotropy group j(H) preserves an 1-dimensional subspace L = Rv, If 1-form η defines the hyperplane L⊥ = ker η, then an invariant Lorentzian metric is defined by the τ(H)-invariant Lorentz scalar product of the form

g = g0 − λη ⊗ η where λ > 0 is suﬃciently big number, and g0 is a j(H)-invariant Euclidean metric. Any invariant Lorentzian metric can be obtained by this construction.

10 Remark 1 The invariant subspace L = Rv defines an invariant field of directions in M. If the group H is connected the above condition means that j(H) has an invariant vector v which defines an invariant vector field. This vector field is timelike vector field with respect to the described Lorentzian metric.

We will always assume that the stability subgroup H is connected (and compact ).

Deﬁnition 1 A compact connected subgroup H of a Lie group G is called admissible if the isotropy representation τ(H) ⊂ GL(ToG/H) has an invariant vector. A maximal subgroup with this property is called maximal admissible.

11 Observation

A closed subgroup of an admissible subgroup H ⊂ G is admissible.

This reduces the problem of classiﬁcation of admissible subgroups H (hence homogeneous Lorentzian manifolds M = G/H with given isometry group G) to the description of maximal admissible subgroups of a given Lie group G.

12 Deﬁnition 2 We say that a homogeneous Lorentzian manifold M = G/H is a minimal Lorentzian G-manifold if H is a maximal admissible subgroup.

Then any other homogeneous Lorentzian G- manifold has the form M0 = G/H0 where H0 ⊂ H and it is the total space of the ﬁbre bundle M0 = G/H0 → G/H over a minimal Lorentzian G-manifold with the compact ﬁbre H/H0.

13 Homogeneous Lorentzian manifolds of simple compact Lie group

Let G be a compact simple Lie group. The adjoint orbit F = Ad Gt ' G/ZG(t) of G is called to be minimal, if the stability subgroup ZG(t) ( which is the centralizer of an element t ∈ g ) is not contained properly in the centralizer of other non-zero element t0 ∈ g. Recall that the centralizer ZG(t) is connected. The orbit F if minimal if and only if ZG(t) has 1-dimensional center T 1 = {exp λt} and 1 can be written as ZG(t) = H · T where H is a semisimple normal subgroup.

14 Minimal adjoint orbits corresponds to simple roots α of the Lie algebra g. Moreover, the Dynkin diagram of the semisimple group H is obtained from the Dynkin diagram of g by deleting the vertex α. We will denote the minimal orbit associated with a simple root α by Fα.

15 List of all such semisimple subgroups H of simple Lie groups G:

G = SUn,H = SUp × SUq, p = 1, 2, ··· , n − 1; n G = SOn,H = SUp × SOq, p = 1, 2, ··· , [2]; G = Spn,H = Spp × Spq, p = 1, 2, ··· , n − 1; s l G = G2,H = SU2,SU2 s l s l G = F4,H = Sp3,SU3 · SU2,SU2 · SU3, Spin7; G = E6,H = Spin10,SU2 · SU5,SU3 · SU3 · SU2,SU6; G = E7,H = E6,SU2 · Spin10,SU3 · SU5, SU4 · SU3 · SU2,SU6 · SU2, Spin12,SU7. G = E8,H = E7,SU2 · E6,SU3 · Spin10,SU4 · SU5, SU5 · SU3 · SU2,SU7 · SU2, Spin14,SU8.

16 1 Let Fα = G/H · T be a minimal orbit associated with a simple root α. Then

1 π : Mα = G/H → Fα = G/H · T is a principal ﬁbration with the structure group T 1. Denote by

1 θ : TMα → R = Lie(T ) the G-invariant principal connection,s.t. θ(t) = 1, θ(p) = 0 where

g = (h + Rt) + p is the reductive decomposition associated with 1 the orbit Fα = G/H · T . We say that π is the canonical T 1 bundle with connection over the orbit Fα.

17 The tangent space ToFα ' p as an Ad (H·T 1)- module is decomposed into mutually non equivalent irreducible submodules

p = p1 + ··· + pm. (1) The number m is the Dynkin number m(α) = mi of the corresponding simple root α = αi that is the coordinate mi over αi in the de- P composition µ = j mjαj of the maximal root µ with respect to the simple roots α1, ··· , αr. Any invariant Riemannian metric gF in F at the point o = e(H · T 1) is given by

go = λ1b1 + ··· + λmbm.

18 Theorem Any minimal admissible manifold of a simple compact Lie group G is the total space Mα = G/H of the canonical ﬁbration 1 over a minimal orbit F = Fα = G/Hα · T . Moreover, assume that M = G/Hα is not the total space of the sphere bundle of a compact rank one symmetric space that is

n S(S ) = SOn+1/SOn−1,

7 7 6 Spin7/SU3 = S(S ) = S × S ,

3 1 3 2 S(S ) = SU2 × SU2/T = S × S ;

n S(CP ) = SUn+1/SUn,

n S(H ) = Spn+1/Sp1 × Spn−2,

2 S(OP ) = F4/Spin7.

19 Then any invariant Lorentz metric g on M is given by 2 ∗ g = −λθ + π gF where θ is the principal connection, In particular, the metric g depends on m(α)+1 positive parameters, where m(α) is the Dynkin mark.

20 Homogeneous Lorentzian manifolds of a simple noncompact Lie group

Case when the group G has inﬁnite center

Let G be a simple Lie group with inﬁnite center and S = G/K · R associated non-compact irreducible Hermitian symmetric space with the symmetric decomposition

g = (k + Rt) + p where Rt is the 1-dimensional centralizer of the Lie algebra k of the maximal compact subgroup K of G and ad t|p is j(K · R)-invariant complex structure in the tangent space p = ToS.

21 Proposition 2 Let G be a simple non-compact Lie group with inﬁnite center and S = G/K ·R the associated Hermitian symmetric space. Then minimal admissible G-manifold is M = G/K. An invariant Lorentzian metrics on M is given by the scalar product

2 g = −λθ + gp in m = Rt + p, where λ > 0, θ is the 1-form dual to the vector t and p is the invariant Euclidean scalar product in p. In particular,

π : M = G/K → S = G/K · R is a pseudo-Riemannian submersion.

22 Duality Let G is a simple noncompact Lie group with ﬁnite center, S = G/K the associated noncompact symmetric space and Sˆ = G/Kˆ the dual compact symmetric space. If

g = h + p is a symmetric decomposition of S. Then

ˆg = h + ip is the symmetric decomposition of Sˆ.

23 This implies the natural bijection between (maximal) admissible subgroups H ⊂ K of the dual Lie groups G and Gˆ.

Proposition 3 There exists a natural one- to-one correspondence between proper homogeneous Lorentzian G-manifolds M = G/H of a simple noncompact Lie group G and homogeneous Lorentzian manifolds Mˆ = G/Hˆ of the dual compact Lie group Gˆ such that the stabilizer H belongs to the subgroup K ⊂ Gˆ.

24 Homogeneous Lorentzian manifolds of class I and class II

Let M = G/H, H ⊂ K be an admissible homogeneous space of a noncompact simple Lie group G with the reductive decomposition g = h+(n+p). Then the space mH = nH +pH of j(H)-invariant vectors is not zero.

Deﬁnition 3 We say that the admissible homogeneous manifold M = G/H belongs to the class I if nH 6= 0 and belongs to the class II if pH 6= 0.

25 Proposition 4 An admissible G-manifold M = G/H of a simple noncompact Lie group G belongs to the class I if it admits an invariant Lorentzian metric such that π : M = G/H → S = G/K is a pseudo-Riemannian submersion with totally geodesic Lorentzian fibres over the noncompact Riemannian symmetric space S = G/K. In particular, the invariant time-like vector filed generate a compact group S1. An admissible manifold M = G/H belongs to the class II if it admits an invariant Lorentzian metric with a time-like invariant vector field, which generates a noncompact 1-parameter subgroup R.

26 Remark It is possible that a minimal admissible G-manifold belongs to the class I and the class II at the same time.

Let K ⊂ GL(V ) be a linear Lie group. Recall that by the (connected) stabilizer Kv of a vector v ∈ V we understand the connected component of the subgroup which preserves v.

Deﬁnition 4 Let K ⊂ GL(V ) be a linear Lie group. The orbit Kv of a vector v 6= 0 is called a minimal orbit is the the ( connected) stabilizer Kv does not contained properly in 0 the (connected ) stabilizer Kw of any other non-zero vector w. Then the stabilizer Kv is called a maximal stabilizer.

27 Proposition 5 Let M = G/H be a minimal admissible homogeneous G-manifold of a simple noncompact Lie group G.

i) If M belongs to the class I, then H is a maximal admissible subgroup of a maximal compact subgroup K ⊃ H of G.

1. If M belongs to the class II, then H = Kv is a maximal (connected) stabilizer of the isotropy representation of the Riemannian symmetric space S = G/K.

28 Let g = k + p be the symmetric decomposition of a symmetric space S = G/K. For any nonzero vector v ∈ p we denote by kv the stability subalgebra of the isotropy representation j(k) and by Kv ⊂ K corresponding connected stability subgroup.

Deﬁnition 5 The subalgebra kv ⊂ k (resp., corresponding subgroup Kv ⊂ K) is called a maximal stability subalgebra (resp.,maximal stability subgroup) if it does not contained properly in any other stability subalgebra (resp., stability subgroup) of the isotropy representation of G/K.

29 Proposition 6 Let S = G/K be a symmetric space of noncompact type and H ⊂ K a maximal admissible subgroup of K such that the admissible manifold G/H belongs to the class II. Then H = Hv is a maximal stability subgroup of K. Conversely, any maximal stability subgroup Kv of K is admissible and deﬁnes an admissible manifold M = G/Kv of the class II.

30 Let g = k + p be the symmetric decomposition of a symmetric space S = G/K. For any nonzero vector v ∈ p we denote by kv the stability subalgebra of the isotropy representation τ(k) and by Kv ⊂ K corresponding connected stability subgroup.

Deﬁnition 6 The subalgebra kv (resp., corresponding subgroup Kv) is called a maximal stability subalgebra (resp.,stability subgroup) if it does not contained properly in another stability subalgebra (resp., stability subgroup).

31 Proposition 7 Let S = G/K be a symmetric space of non compact type. Then maximal admissible subgroups of G are exactly maximal stability subgroups Kv of the isotropy representation of the symmetric space S.

Since the dual symmetric spaces have the same isotropy representation we have

Corollary 1 Let Sˆ = G/Kˆ be the dual compact symmetric space. Then there exists a natural 1-1 correspondence between (minimal ) proper homogeneous Lorentzian G-manifolds and homogeneous Lorentzian manifolds of the group Gˆ whose stabilizer is contained in K. These dual Lorentzian manifolds has the form M = G/H, Mˆ = G/Hˆ where H a closed subgroup H of some maximal stability subgroup Kv and carries an invariant Lorentzian metric which is deﬁned by an H-invariant Euclidean metric in the tangent space and an invariant vector v.The manifolds M, Mˆ have the same isotropy representation.

32 So classiﬁcation of proper homogeneous type II Lorentzian manifolds of a semisimple group G reduces to description of maximal stability subgroups Kv of the isotropy representation of the associated symmetric space S = G/K.

It is suﬃcient to describe such subgroup for simple Lie groups.

As a simple example, consider the case when G has real rank 1 (that is the symmetric space G/K has rank one). Then the stability subgroups Kv is unique ( up to a conjugation), hence, maximal.

33 List of minimal homogeneous Lorentz G-manifolds M = G/H for real rank one groups:

n S = H = SO1,n/SOn,M = SOn+1/SOn−1 n S = CH = SU1,n/Un,M = SU1,n/Un−1 n S = HH = Sp1,n/Sp1 × Spn,M = Sp1,n/SO3 · Spn−1 2 S = OP = F4/Spin9,M = F4/Spin7.

34 We have reduced the classiﬁcation of proper homogeneous Lorentzian manifolds of a non compact semisimple Lie group to description of maximal stability subgroups of the isotropy representation of irreducible symmetric spaces. Stability subgroups of most irreducible symmetric spaces are know. We will not state the result for all cases and only consider one example.

35 Case of the group G = SLn(R) Let S = SLn(R/SOn) be the symmetric space of positive deﬁned matrices. The isotropy representation is the standard 2 n representation of K = SOn in S R . The stability subgroups are

SOp1 × · · · × SOps. Maximal admissible subgroups are SOp × SOq, p + q = n.

We get the minimal homogeneous Lorentzian manifolds

Mp,q = SLn/SOp × SOq.

36 A matrix v with two eigenvalues, e.g., v = diag(λId p, Id q) deﬁnes an invariant vector ﬁeld on Mp,q which allows to construct an invariant Lorentzian metric. Any proper homogeneous Lorentzian manifold of the group SLn has the form

M = SLn/H where H is a closed subgroup of SOp × SOq. Such manifold is ﬁbred over the space

G/K = SLn(R)/SOp × SOq.

37 Theorem 1 All minimal admissible class II manifolds Md = G/H of dimension d ≤ 11 where G is a simple noncompact Lie group are described in the Table I. There are also indicated the maximal compact subgroup K of G and the space m = ToG/K of its isotropy representation, the dimension m of the symmetric space G/K and the ﬁbre K/H of the natural G-equivariant ﬁbration M = G/H → S = G/K over the symmetric space S = G/K.

38 Table I.

d Md K m m K/H 2 1 3 SL2(R) SO2 R 2 S 3 2 5 SO1,3/SO2 SO3 R 3 S 2 3 2 7 SL3(R)/SO2 SO3 S0(R ) 5 S 2 3 7 SU1,2/U1 U2 C 4 S 4 3 7 SO1,4/SO3 SO4 R 4 S 5 4 9 SO1,5/SO4 SO5 R 5 S 3 5 11 SU1,3/U2 U3 C 6 S 6 5 11 SO1,6/SO5 SO6 R 6 S diag 2 2 3 11 G2/SU2 SU2 × SU2 C ⊗ C 8 S