Actions of semisimple Lie groups preserving a degenerate Riemannian metric

We prove a rigidity of the lightcone in Minkowski space. It is essentially the unique space endowed with a degenerate Riemannian metric, of lightlike type, and supporting an isometric non-proper action of a semi-simple group.


Introduction
Our subject of study here is lightlike metrics on smooth manifolds. First, a lightlike scalar product on a vector space E is a symmetric bilinear form b which is positive but non-definite, and has exactly a 1-dimensional kernel. If E has dimension 1 + n, then, in some linear coordinates (x 0 , x 1 , . . . , x n ), the associated quadratic form q can be written q = (x 1 ) 2 + . . . + (x n ) 2 . Now, a lightlike metric h on a manifold M is a smooth tensor which is a lightlike scalar product on the tangent space of each point. 1.0.1. Characteristic foliation. The Kernel of h is a 1-dimensional sub-bundle N ⊂ T M , and thus determines a 1-dimensional foliation N , called the characteristic (or null, normal, radical, isotropic...) foliation of h. By definition, any null curve (i.e. a curve with everywhere isotropic speed) of (M, h) trough x is contained in the null leaf N x . The (abstract) normal bundle of N , i.e. the quotient T M/N is a Riemannian vector bundle. Conversely, a lightlike metric consists in giving a 1-dimensional foliation together with a Riemannian metric on its normal bundle.
1.1. Major motivations. Lightlike geometry appears naturally in a lot of geometric situations. We list now some natural examples motivating their study.
1.1.1. Submanifolds of Lorentz manifolds. Let M be a submanifold in a Lorentz manifold (V, g). The metric g is non-degenerate with signature − + . . . +. However, for a given x ∈ M , the restriction h x of g to T x M has not necessarily the same signature. Two easy stable situations are those where h x is everywhere of Riemannian type (M is spacelike), or h x is everywhere of Lorentzian type (M is timelike). In both cases, all the submanifold theory valid in the Riemannian context generalizes: shape operator, Gauss and Codazzi equations... The delicate situation is when h x is degenerate for any x. Because the ambient metric has Lorentz signature, h x is then lightlike as defined above. Unfortunately, by opposition to the previous cases, these lightlike submanifolds are generally "to poor" to generate a coherent extrinsic local metric differential geometry. Let us give examples of interesting lightlike submanifolds: • Horizons of domains of dependence and black holes. Unfortunately, they have an essential disadvantage: their lower smoothness. One can believe that smooth horizons are sufficiently rigid to be classifiable (see for instance [6,20,13]).
• Characteristic hypersurfaces of the wave equation. There is a nice interpretation of lightlike hypersurfaces in terms of propagation of waves: a hypersurface is degenerate iff it is characteristic for the wave equation (on the ambient Lorentz space) [12]. These hypersurfaces enjoy the nice property that their null curves are geodesic in the ambient space (this is not true for submanifolds of higher codimension). However, no deeper study of their extrinsic geometry seems to be available in the literature.
• Lightlike geodesic hypersurfaces. They are characterized by the fact that their lightlike metrics are basic (see the example 1.2.1). They inherit a connection from the ambient space. See [8,9,22,23], for their use in Lorentz dynamics.
• Degenerate orbits of Lorentz isometric actions. Let G be a Lie group acting isometrically on a Lorentz manifold (V, g). Then, any orbit which is lightlike at a point is lightlike everywhere, hence yields an embedded lightlike submanifold in V . The problem of understanding these lightlike orbits, and more generally degenerate invariant submanifolds, is essential when studying such isometric actions.
• Terminology. We believe that the choose of the word "lightlike" here is widely justified from the relationship between lightlike submanifolds and fields on one hand, and geometric as well as physical optics in general Relativity on the other hand (see for instance [21]). We also think this terminology is naturally adapted to our situation here, but less for the general situation of "singular pseudo-Riemannian" metrics (compare with [10,17]).
1.1.2. From submanifolds to intrinsic lightlike geometry. In the last example given above, when restricting the action of the Lie group G to a lightlike orbit, we are led to study the isometric action of G on a lightlike submanifold in a Lorentzian manifold. In fact, one realizes that the submanifold structure is irrelevant in this problem, and the pertinent framework is that of isometric actions on abstract lightlike manifolds.
The main difficulty when dealing with this intrinsic formulation is that we loose the rigidity of the ambient action: as we will see below, the isometry group of a lightlike manifold can be infinitely dimensional.

1.2.
Two fundamental examples. We give now two important examples of lightlike geometries, which are in some sense antagonistic.
We will denote its (linear) orthogonal group by O(0, n) (this is somehow natural since reminiscent of the notation O(1, n)). We have: It is naturally isomorphic to the affine similarity group R×Euc n = R.O(n)⋉ R n (here Euc n = O(n) ⋉ R n is the group of rigid motions of the Euclidian space of dimension n).
Let us now see R 1+n as a lightlike manifold. The group of its affine isometric transformations is O(0, n) ⋉ R 1+n .
• Contrary to the non-degenerate case, there is here a huge group (infinitely dimensional) of non-affine isometries. Take any where ψ 2 ∈ Euc n , and ψ 1 : R n+1 → R is a smooth function with ∂ψ 1 ∂x 0 = 0 (in order to get a diffeomorphism).
More generally, let (L, g) be a Riemannian manifold, and M = R × L endowed with the lightlike metric 0 ⊕ g, that is the null foliation is given by the R-factor, and the metric does not depend on the coordinate along it. Then, we have also here an infinitely dimensional group of isometric transformations given by: ψ : (t, l) ∈ R × L → (ψ 1 (t, l), ψ 2 (l)), where ψ 2 is an isometry of L, e.g. ψ 2 the identity map, and ∂ψ 1 ∂t = 0 . Conversely, assume that the lightlike metric (M, h) is such that there exists a non-singular vector field X tangent to the characteristic foliation, which flow preserves h (equivalently, the Lie derivative L X h = 0). Then, locally, there is a metric splitting M = R × L as above. Observe in fact that any vector field collinear to X will preserve h too, in other words any vector field orienting the characteristic foliation N preserves h. Let us call the lightlike metric basic in this case (they can also be naturally called, locally product, or stationary). This terminology is justified by the fact that h is the pull-back by the projection map M → L of the Riemannian metric on the basis L.
• Recall the classical notion from the geometric theory foliations: a 1dimensional foliation N on a manifold M is transversally Riemannian (one then says N is a transversally Riemannian flow), if it is the characteristic foliation of some lightlike metric h on M , which is preserved by (local) vector fields tangent to N . Therefore this data is strictly equivalent to giving a locally basic lightlike metric on M . Of course, the usual classical definition does not involve lightlike metrics.
There is a well developed theory of transversally Riemannian foliations, with sharp conclusions in the 1-dimensional case [7,19]...
The isometry group of a basic lightlike metric contains at least all flows tangent to N which form an infinitely dimensional group (surely not so beautiful). However, these metrics are somehow tame, since, at least locally, the metric is encoded in an associated Riemannian one. Moreover, it t was proved by D. Kupeli [17] (and reproduced in many other places) that some kinf of Levi-Civita connection exists exactly if the lightlike metric is basic. The connection is never unique, and so enrichment of the structure is always in order. Actually, the most useful additional structure is that of a screen, mostly developed in [10], which allows to develop "calculus", and get sometimes invariant quantities (see for instance [3]). Nevertheless, there is generally no distinguished screen left invariant by the isometry group, so that this notion will not be helpful for us.

1.2.2.
The example of the Lightcone in Minkowski space. We are going now to consider an opposite situation, where the isometry group is "big", though remaining finitely dimensional. Let M in 1,n be the Minkowski space of dimension 1 + n, that is R 1+n endowed with the form q = −x 2 0 + x 2 1 + . . . x 2 n . The isotropic (positive) cone, or lightcone, Co n is the set {q(x) = 0, x 0 > 0}. The metric induced by q on Co n is lightlike. The group O + (1, n) (subgroup of O(1, n) preserving the positive cone) acts isometrically on Co n . This action is in fact transitive so that Co n = O + (1, n)/Euc n−1 becomes a lightlike homogenous space, with isotropy group Euc n−1 = O(n − 1) ⋉ R n−1 , the group of rigid motions of the Euclidean space of dimension n − 1.
A key observation is: • For n = 2, there is no rigidity at all, even globally, since to any diffeomophism of the circle corresponds an isometry of Co 2 .
This theorem, which will be proved in §2, shows in particular that for n ≥ 3, Co n is a homogeneous lightlike manifold with isometry group O(1, n).
Remark that for the sake of simplicity, we will often use the notation O(1, n) to mean its identity component SO 0 (1, n), and generally any finite index subgroups of O(1, n). Actually, to be precise, we can say that our geometric descriptions of objects are always given up to a finite cover.
It seems likely that being homogeneous and having a maximal isotropy O(0, n − 1) characterizes the flat case, i.e. R 0,n−1 , and having a maximal unimodular isotropy, i.e. Euc n−1 , characterizes the lightcone. In some sense the lightcone is the maximally symmetric non-flat lightlike space, analogous to spaces of constant non-zero curvature in the pseudo-Riemannian case.
1.3. Statement of results. The present article contains in particular detailed proofs of the results announced in [5]. Before giving the statements, let us recall that two (lightlike) metrics h and h ′ on a manifold M are said to be homothetic if h = λh ′ , for a real λ > 0. A Lie group acts locally faithfully on M if the kernel of the action is a discrete subgroup.
One motivation of the present work was Theorem 1.6 of [9], that we state here as follows: 9]). Let G be a connected group with finite center, locally isomorphic to O(1, n) or O(2, n), n ≥ 3. If G acts isometrically on a Lorentz manifold, and has a degenerate orbit with non-compact stabilizer, then G is locally isomorphic to O(1, n), and the orbit is homothetic to the lightcone Co n .
Here, we prove an intrinsic version of this result: Theorem 1.3. Let G be a non-compact semi-simple Lie group with finite center acting locally faithfully, isometrically and non-properly on a lightlike manifold (M, h). Assume that G has no factor locally isomorphic to SL(2, R). Then, looking if necessary at a finite cover of G: • G has an orbit which is homothetic, up to a finite cover, to a metric product Co n × N , where N is a Riemannian H ′ -homogeneous manifold. The action of H × H ′ on Co n × N is the product action.
Using this theorem and working a little bit more, we can also handle the case where some factors of G are locally isometric to SL(2, R), when the action is transitive. The following result can be thought as a converse to Theorem 1. The non-properness assumption is essential in the previous theorems. If one removes it, "everything becomes possible". Indeed, consider a Lie group L and a lightlike scalar product on its Lie algebra l. Translating it on L by left multiplication yields a lightlike metric on L, with isometry group containing L, acting by left translations.
It is quite surprising that kind of global rigidity theorems can be proved in the framework of lightlike metrics, which are not rigid geometric structures (see §1.2.1). Here, it is, in some sense, the algebraic assumption of semisimplicity which makes the situation rigid. However, since any Lie algebra is a semi-direct product of a semi-simple and a solvable one, it is natural to start looking to actions of semi-simple Lie groups.
When the manifold M is compact, only one simple Lie group can act isometrically, as shows the: 1.4. The mixed signature case: sub-Lorentz metrics. This notion will naturally modelize the situation of general submanifolds in Lorentz submanifolds. A sub-Lorentz metric g on M is a symmetric covariant 2 -tensor, which is at each point, a scalar product of either Lorentz, Euclidean, or lightlike type. The point is that we allow the type to vary over M . So, if (L, h) is a Lorentz manifold, and M a submanifold of L, then the restriction on h on M is a sub-Lorentz metric (this fact raises the inverse problem, i.e. isometric embedding of sub-Lorentz metrics in Lorentz manifolds). We think it is worthwhile to investigate the geometry of these natural and rich structures (see for instance [18] for a research of normal forms of these metrics in dimension 2).
We restrict our investigation here to an adaptation of our lightlike results to this sub-Lorentz situation.
In the case of Minkowski space, the isometry group is not semi-simple.
The Lorentz and lightlike dynamics are unified in the following statement: Theorem 1.6. Let G be a semi-simple group with finite center , no compact factor and no local factor isomorphic to SL(2, R), acting isometrically nonproperly on a sub-Lorentz manifold M . Then, up to a finite cover, G has a factor G ′ isomorphic to O(1, n) or O(2, n) and having some orbit homothetic to dS n , AdS n or Co n .

Preliminaries
Thus, the different rigidity phenomena are just consequences of classical analogous rigidity results for conformal transformations on the sphere.

SL(2, R)-homogeneous spaces.
Understanding these spaces is worthwhile in our context, since one can take advantage of restricting the G-action to small simpler groups, e.g. SL(2, R) or a finite cover, which always exist in semi-simple Lie groups.

Notations.
Let SL(2, R) be the Lie group of 2 × 2-matrices with determinant 1. It is known that any one parameter subgroup of SL(2, R) is conjugate to one of the following: The corresponding derivatives of A + and N at the identity are Together with Z = 0 0 1 0 , X and Y span the Lie algebra sl(2, R) and satisfy the bracket relations: As usual, we denote by A (resp. K), the subgroup generated by Let Af f (R) be the subgroup of upper triangular matrices: and aff(R) its Lie algebra.
Non-connected 1-dimensional subgroups of Af f (R) can be constructed as follows. Let Γ 0 be a cyclic subgroup of A generated by an element γ ∈ A.
The semi-direct product Γ 0 ⋉N is a closed subgroup. Conversely, any closed 1-dimensional non-connected subgroup of Af f (R) is obtained like this.
The "classical" classification of the SL(2, R)-homogenous spaces, allows one to recognize the lightlike ones.
with an affine flat connection, together with a lightlike metric.
It is locally an Anti de Sitter space, i.e. a Lorentz manifold with negative constant curvature. Proof. The proof of the first part is standard; we just give details in the lightlike case.
Let Σ be be an SL(2, R)-homogeneous space of dimension ≥ 2 i.e Σ ∼ = SL(2, R)/ H , where H is the stabilizer of some p ∈ Σ and conjugated, as showed above, to one of the following subgroups: K, N, Γ 0 N and Γ. Let h be the Lie algebra of H. Considering the isotropy representation . This is just because the Killing form on sl(2, R) has Lorentz signature. If moreover ρ H (Γ) is conjugated to a subgroup of O(0, 2), then ρ H (Γ) has to be finite. Since the Kernel of the adjoint representation of SL(2, R) is finite, we get that Γ is finite. Therefore the unique lightlike SL(2, R)-homogeneous space of dimension ≥ 2 with non-compact isotropy is R 2 \ {0}.
In order to check that the lightlike metric has the form αdθ 2 (for some α ∈ R * + ), one argues as follows. At p = (1, 0), the vector X is the unique nontrivial eigenspace of ρ N , and thus the orbit of p by the flow φ t X must coincide with the null leaf N (1,0) , which is therefore a radial half-line. The other null leaves are also radial, since they are images of N (1,0) by the SL(2, R)-action. By homogeneity, the metric must have the form αdθ 2 .
For a latter use, let us state the following fact, which follows directly from the previous description of the lightlike surface R 2 \ {0}.
, then Y vanishes at p and X is isotropic at p.

Generalities on semi-simple groups; notations.
[See for instance [14]] Let G be a semi-simple group acting isometrically on (M, h). This means that we have a smooth homomorphism ρ : G → Dif f ∞ (M ), such that for every g ∈ G, ρ(g) acts as an isometry for h (i.e ρ(g) * h = h). Let g be the Lie algebra of G. For any X in g , we will generally use the notation φ t X instead of ρ(exp(tX)). By a slight abuse of language, we will also denote by X the vector field of M generated by the flow φ t X . We get, for every p ∈ M , a homomorphism λ p : g → T p M , defined by λ p (X) = X p . The flow φ t X stabilizes p iff X p = 0, and we denote by g p the Lie algebra of the stabilizer of p.
We say that X ∈ g is lightlike at p ∈ M (or isotropic) (resp. spacelike) if h p (X p , X p ) = 0 (resp. h p (X p , X p ) > 0).
We denote by s p the subspace of all vectors of g which are isotropic at p ∈ M .
Let O be a lightlike G-orbit of some p ∈ M , that is O ∼ = G/G p , where G p is the stabilizer of p. The tangent space T p O is identified by λ p to the quotient g/g p . In fact the isotropy representation on T p O is equivalent to the adjoint representation Ad of G p on g/g p . In particular G p is mapped, up to conjugacy, to a subgroup of O(0, n).
Similarly, the Euclidian space T p O/N p is identified to g/s p , where Ad : G p −→ GL(g/s p ) preserves a positive inner product on g/s p , so that G p acts on g/s p by orthogonal matrices. In particular, if we consider the tangent representation: ad : g p −→ End(g/s p ), the Lie subalgebra g p acts by skew symmetric matrices on g/s p . We will use the same notation for the elements of the quotients g/g p and g/s p and their representatives in the Lie algebra g.
We fix once for all a Cartan involution Θ on the Lie algebra g. This yields a Cartan decomposition g = k ⊕ p, k (resp. p) being the eigenspace of Θ associated with the eigenvalue +1 (resp. −1).
We choose a, a maximal abelian subalgebra of p, and m the centralizer of a in k. This choice yields a rootspace decomposition of g, namely there is a finite family Σ + = {α 1 , ..., α s } of nonzero elements of a * , such that g = ⊕ α∈Σ + g −α ⊕ g 0 ⊕ ⊕ α∈Σ + g α . For every X ∈ a, ad(X)(Y ) = α(X)Y , as soon as Y ∈ g α . The Lie subalgebra g 0 is in the kernel of ad(X), for every X ∈ a and splits as a sum: g 0 = a ⊕ m.
The positive Weyl chamber a + ⊂ a, contains those X ∈ a, such that α(X) ≥ 0, for all α ∈ Σ + . Its image by the exponential map is denoted by A + . Let The stable subalgebra ( for a) W s = ⊕ α∈Σ − g α , and the unstable one W u = ⊕ α∈Σ + g α are both nilpotent subalgebras of g, mapped diffeomorphically by the exponential map of g onto two subgroups N + ⊂ G, and N − ⊂ G.
Given X ∈ a, its stable algebra is W s X = ⊕ α(X)<0 g α , and its unstable algebra is W u X = ⊕ α(X)>0 g α . Let us prove now a lemma which will be useful in the sequel: Lemma 2.4. The subalgebra W s X has the following properties: (1) Since Y ∈ g p , ad Y acts on g/s p by a skew symetric endomorphism, which is moreover nilpotent since Y ∈ W s X . Hence ad Y acts by the null endomorphism on g/s p , which means that ad Y maps g to s p .
(2) ad Y acts as a nilpotent endomorphism of g/g p (identified with the tangent space), and has s p /g p (identified to the isotropic direction) as a 1-dimensional eigenspace. By nilpotency, the action on it is trivial, i.e ad Y maps s p into g p .
Finally, recall that a semi-simple Lie group of finite center admits a Cartan decomposition G = KAK, where K is a maximal compact subgroup of G.

Non-proper actions.
(See for instance [15] for a recent survey about these notions).
there is a sequence of transformations g k ∈ G, such that, both (p k ) and (q k ) = (g k (p k )) lie in a compact subset of M , but (g k ) tends to ∞ in G, i.e. leaves any compact of G.
-The sequence (g k ) is called a "return sequence" for (p k ).
-In the sequel, we will sometimes assume that (p k ) and (q k ) converge to p and q in M .
One says that the group G acts non-properly if it admits a non-escaping sequence.
A nice criterion for actions of semi-simple Lie groups of finite center to be non-proper, is the next: Lemma 2.6. Let G be a non-compact semi-simple group with finite center. Then G acts non-properly iff any Cartan subgroup A acts non-properly.
Proof. G admits a Cartan decomposition KAK, where K is compact. Let (p k ) be a non-escaping sequence of the G-action, and (g k ) its return sequence. Write g k = l k a k r k ∈ KAK. Then, p ′ k = r k (p k ) is a non-escaping sequence for the A-action, with associated return sequence (a k ). Obviously (a k ) goes to infinity in A since (g k ) goes to infinity in G.

A Key fact on the stable space
Here we state a crucial ingredient for the proofs of all our theorems. In all what follows, G is a non-compact semi-simple Lie group with finite center acting locally faithfully, non-properly and isometrically on a lightlike manifold (M, h). The main result of this section is: Proposition 3.1. If no factor of G is locally isomorphic to SL(2, R), there exists a Cartan subalgebra a 0 such that for some X 0 ∈ a 0 and p 0 ∈ M , both X 0 and its stable algebra W s X 0 are isotropic at p 0 .
3.1. Starting fact. The non-properness of the action of G leads to a fundamental fact, already observed in [16] for Lorentzian metrics, which is the existence of p ∈ M and X ∈ a such that W s X is isotropic at p. Let us recall its proof. (1) If the flow of X ∈ a acts non-properly, and p k → p is a non-escaping sequence for the action of φ t X , then the stable space W s X is isotropic at p.
(2) More generally, if p k → p is a non-escaping sequence for the Aaction, then there exists X ∈ a such that W s X is isotropic at p ∈ M . Proof.
(1) Denote φ t X = exptX the flow of X, and let (t k ) be a return time sequence for (p k ), i.e. φ t k X is a return sequence of p k , what means that q k = φ t k X (p k ) stay in a compact subset of M .
. Assume that α(X) < 0, then: On the left hand side, passing to the limit yields h p (Y p , Y p ).
On the right hand side, since (q k ) lie in a compact set, h q k (Y q k , Y q k ) is bounded. Therefore, since α(X) < 0, this right hand term tends to 0, yielding h p (Y p , Y p ) = 0. This proves that W s X is isotropic at p.
(2) Let (X k ) be a sequence in a such that exp(X k ) is a return sequence for (p k ). Let . be a Euclidian norm on a and, considering if necessary a subsequence, assume ( X k X k ) converges to some X ∈ a. As above, one proves that that W s X is isotropic at p.

Remark 3.3. This result is nothing but a generalization of the linear (punctual) easy fact: If a matrix
A preserves a lightlike scalar product, then its corresponding stable and unstable spaces are isotropic. In our particular case, if X ∈ a ∩ g p (i.e. X stabilizes p), then both of W s X and W u X are isotropic at p.

3.2.
Proof of Proposition 3.1. The proof follows from several observations. The simplest one is that for lightlike metrics (in contrast with the Lorentz case), the isotropic direction is unique on each tangent space T p M . Furthermore, it coincides with the non-trivial eigenspace (if any) of any infinitesimal isometry fixing p. The hypothesis made in Proposition 3.1, that G has no factor locally isomorphic to SL(2, R) will be only used in Lemma 3.7.

Lemma 3.4. For any p ∈ M , the subspace of isotropic vectors s p is a Lie subalgebra of g.
Proof. Let X, Y ∈ s p , and let φ t X be the isometric flow generated by X on M , then: ) − Y p ] since X, Y are isotropic at p, their integral curves at p are supported by the null leaf N p , and thus Y φ t X (p) is isotropic. Since φ −t X is an isometry, dφ −t X (Y φ t X (p) ) is also isotropic. Lemma 3.5. G stabilizes no p ∈ M .
Proof. Suppose by contradiction that G stabilizes p ∈ M , then G acts on T p M by: ρ : g → d p g ∈ GL(T p M ). Since G preserves the lightlike scalar product h p , it is mapped by ρ into a subgroup of O(0, n). Thus, at the level of Lie algebras, we get a homomorphism dρ : g → o(0, n). Now, we prove: Sublemma 3.6. Any homomorphism from g to o(0, n) is trivial.
Proof. Without loss of generality, we can suppose that g is simple. Let λ be a homomorphism from g to o(0, n), π the projection from o(0, n) to o(n). Consider the homomorphism λ • π : g −→ o(n). Since g is simple and noncompact, it has no non-trivial homomorphism into the Lie algebra of a compact group; this implies that λ • π is trivial. So, g is mapped by λ into the kernel g 0 of π, that is the algebra of matrices of the form Since g 0 is solvable and g is simple, we conclude that λ is trivial.
As a corollary, the ρ-image of any connected compact subgroup K ⊂ G is trivial. However such K preserves a Riemannian metric. But on a connected manifold M , a Riemannian isometry which fixes a point and has a trivial derivative at this point, is the identity on M . This is easily seen since in the neighbourhood of any fixed point, a Riemannian isometry is linearized thanks to the exponential map. Hence K acts trivially on M , and therefore G does not act faithfully, which contradicts our hypothesis and completes the proof of our lemma. Proof. Assume by contradiction that, a ∩ g p = {0}, and let us take X = 0 in this intersection. Apply Remark 3.3 to X to get that the subspaces W s X and W u X are both isotropic at p. It is a general fact that n, the Lie subalgebra generated by W s X and W u X , is an ideal of g (see for instance [16]), and it is in particular a factor of g. It acts on the 1-manifold N p . This action is faithful, otherwise its kernel s would be the Lie algebra of a semi-simple group S ⊂ G, which would have fixed points on M , in contradiction with Lemma 3.5. Now, the only the semi-simple algebra acting faithfully on a 1-manifold is sl(2, R). This contradicts our hypothesis that g has no such factor. (1) If H is linear, then it is isomorphic to either SL(2, R) or P SL(2, R).
(2) If H is a subgroup of a Lie group G with finite center, then it is a finite covering of P SL(2, R).
Proof. The point is that all the representations of the Lie algebra sl(2, R) integrate to actions of the group SL(2, R) itself (and not merely its universal cover). Indeed, the classical classification asserts that all the irreducible representations are isomorphic to symmetric powers of the standard representation, or equivalently to representations on spaces of homogeneous polynomials of a given degree, in two variables x and y (see for instance [14]). Clearly, SL(2, R) acts on these polynomials, and P SL(2, R) acts iff the degree is even. For the last point, observe that the adjoint representation of G has finite Kernel.
End of the proof of Proposition 3.1. From Proposition 3.2, there exist X ∈ a and p ∈ M , such that W s X is isotropic at p. Since g has no local factor isomorphic to sl(2, R), we have dim W s X > 1 (otherwise the subalgebra a ⊕ Σ α(X)≥0 g α is supplementary to W s X and would have codimension 1, and therefore g acts on a 1-manifold). For a lightlike metric, an isotropic space has dimension at most 1, so that the evaluation of W s X at p has at most dimension 1, and thus W s X contains at least a non-zero vector Y 0 vanishing at p.
By the Jacobson-Morozov Theorem (see [14]), the nilpotent element Y 0 belongs to some subalgebra h isomorphic to sl(2, R), i.e. generated by an Let H ⊂ G be the group associated to h. From the lemma above and the fact that G has finite center, H is a finite covering of P SL(2, R). Let us call Σ the H-orbit of p. Since Y 0 vanishes at p, but not X 0 (by Lemma 3.7), Σ is a lightlike surface, homothetic up to finite cover to (R 2 \ {0}, dθ 2 ). Also, Y 0 ∈ g p implies that H acts non-properly on Σ. The group exp(RX 0 ) is a Cartan subgroup of H, and by Lemma 2.6, exp(RX 0 ) also acts nonproperly. Thus, we can find (q k ) a sequence of Σ converging to p 0 ∈ Σ, and a sequence of return times (t k ), such that h k .q k converges in Σ, where h k = exp(t k X 0 ). Because in any finite-dimensional representation of sl(2, R), any R-split element is mapped on some R-split element, the Cartan subalgebra RX 0 is contained in a Cartan subalgebra of the ambient algebra g, say a 0 . Now, we apply the first part of Proposition 3.2 to X 0 and a 0 , and deduce that W s X 0 is isotropic at p 0 (where W s X 0 is defined relatively to a 0 ). In particular, Y 0 is isotropic at p 0 , and Fact 2.3 then ensures that X 0 is also isotropic at p 0 .  More precisely, any p ∈ M , for which there exists X such that W s X is isotropic at p, has a lightlike orbit on which the action is non-proper.
Proof. We have already seen at the end of the proof of Proposition 3.1, that W s X has dimension > 1. If it is isotropic at p, then it contains elements Y ∈ W s X ∩ g p vanishing at p. But Y is a nilpotent element in g, and in particular Ad(exp(tY )) is non-compact, proving that the stabilizer of p is non-compact. Therefore the action of G on the G-orbit O of p is non-proper.
Let us show that O is lightlike. Lemma 3.5 shows that O can not be reduced to p. Also, O can not be 1-dimensional. Indeed, a factor of G should then act faithfully on O, and we already saw that such a factor would be locally isomorphic to SL(2, R).
Suppose now by contradiction that O is Riemannian. Then any vector which is isotropic at p must vanish there, in particular W s X ⊂ g p . Consider Y ∈ W s X , and its infinitesimal action on the tangent space of the orbit at p. This action is just ad(Y ) : g/g p −→ g/g p . If O is supposed to be Riemannian, it is at the same time skew symmetric and nilpotent, hence trivial (on g/g p ), which means: ad(Y ) (g) ⊂ g p , for all Y ∈ W s X . Now, let us pick Y 0 ∈ W s X , and use the Jacobson-Morozov theorem to get an sl(2, R)triple (Z 0 , X 0 , Y 0 ). Then ad(Y 0 )(Z 0 ) = X 0 , so that X 0 ∈ g p . But we saw that X 0 is in a Cartan subalgebra of g, yielding a contradiction with Lemma 3.7.

4.2.
Proof in the simple case. We now give the proof of Theorem 1.3, assuming that the group G is simple with finite center, and the action is transitive and non-proper. The general case of semi-simple groups will be handled in the next section. The proof will be achieved in several steps.
Step 1: There exist p ∈ M and X in some Cartan subalgebra a, such that W s X ⊂ g p .
Proof. Proposition 3.2 says that for some p ∈ M , both X and W s X are isotropic at p. For any Y in W s X , the Lie algebra generated by X and Y is isomorphic to the Lie algebra aff(R) and acts on the null leaf N p . Up to isomorphism, there are exactly two actions of af f (R) on a connected 1-manifold: (1) The usual affine action of af f (R) on the line. Here, a conjugate of X vanishes somewhere. (2) The non-faithful action, for which Y acts trivially.
We can not be in the first case without contradicting Lemma 3.7, so that only possibility (2) occurs, and thus W s X ⊂ g p .
Step 2: The R-rank of g equals 1.
Proof. Suppose the R-rank of g > 1. Let α be a root such that α(X) > 0 and β an adjacent root in the Dynkin diagram, according to the choice of a basis Φ of positive simple roots where, γ ∈ Φ =⇒ γ(X) ≥ 0. (See [14]).
By definition, α + β is also a root and (α + β)(X) > 0, that is, g −α and g −(α+β) are different and contained in W s X . Let T α and T α+β be the vectors of a dual to α and α + β respectively. They are linearly independent.
Moreover, T α ∈ [g α , g −α ] ⊂ ad g (W s X ), and similarly for T α+β , By the first step and Lemma 2.4, T α and T α+β are isotropic at p. Hence, there is a non-trivial linear combination of them which vanishes at p. This contradicts Proposition 3.7 claiming that a ∩ g p = 0. Therefore, g has rank 1.

Remark 4.2. It is exactly here that we need G to be simple!
Step 3: The Lie algebra g is isomorphic to o(1, n) Proof. Suppose that g is not isomorphic to o(1, n), then we have two roots α, 2α, such that α(X) > 0, Let us continue the proof assuming the claim. Consider a non-zero Y ∈ [g +2α , g −α ] ⊂ g +α . By Lemma 2.4, Y is isotropic at p. Let Θ be the Cartan involution, then ΘY ∈ W s X , and hence belongs to g p , by Step 1. Lemma 2.4 then implies that [Y, ΘY ] ∈ g p , in particular a ∩ g p = 0, which contradicts Lemma 3.7.
Proof of the claim. First, the rank 1 simple Lie groups of non-compact type are known to be the isometry groups of symmetric spaces of negative curvature. They are the real, complex and quaternionic hyperbolic spaces, together with the hyperbolic Cayley plane. A direct computation can be performed and yields the claim. Let us give another synthetic proof. By contradiction, the sum l = g 0 +g −α +g −2α +g +2α would be a subalgebra of g. For the sake of simplicity, let us work with groups instead of algebras. Let L be the group associated to our last subalgebral. Clearly, L is not compact. The point is that there is a dichotomy for non-compact connected isometry subgroups of negatively curved symmetric spaces. If they have a non-trivial solvable radical, then they fix a point at infinity, and thus are contained in a parabolic group and in particular have a compact simple (Levi)-part (see [11]). If not the group is semi-simple. It is clear that our L contains a non-compact semi-simple, and therefore by the dichotomy, it is semi-simple. But, once semi-simple, L will have a "symmetric " root decomposition, i.e. the negative of a root is a root, too. Thus, there must exist a non-trivial root space corresponding to α, which contradicts the true definition of l.
Step 4: The full isotropic subalgebra is s p = a ⊕ m ⊕ g −α Proof. Recall that m is the Lie algebra of the centralizer of a in the maximal compact K. Since m ⊂ [g α , g −α ], Lemma 2.4 implies that it is isotropic at p.
On the other hand, if Y ∈ g + α is isotropic at p, Lemma 2.4 implies that the semi-simple element [Y, ΘY ] ∈ [g −α , g α ] ⊂ [s p , W s p ∩ g p ] is in the the stabilizer subalgebra of p, which contradicts Lemma 3.7. Therefore, the isotropic subalgebra is exactly s p = a ⊕ m ⊕ g α .
Step 5: The full stabilizer subalgebra is g p = m ⊕ g α .
Proof. Let Z ∈ m, then it is isotropic at p. Suppose by contradiction that Z / ∈ g p . Then, there exists an element Z + λX ∈ g p , λ ∈ R * . We let it act on the normal space of the null leaf.
The action of X on g/s p is identified to its action on g +α , by the previous step. In particular, the X-action has non-zero real eigenvalues.
The action of m (and in particular Z) on g/s p has purely imaginary eigenvalues, since m is contained the Lie algebra of a maximal compact group.
On one hand, since X and Z commute (by definition of m), the action of Z + λX on g/s p must have eigenvalues with non-trivial real part.
On the other hand, Z+λX ∈ g p and acts as a skew symmetric endomorphism on g/s p , and thus has only purely imaginary eigenvalues: contradiction. This shows that m ⊂ g p , but since a ∩ g p = 0, and g p ⊂ s p , which was calculated in the previous space, we get the equality g p = m ⊕ g − α.
End: Since g is isomorphic to o(1, n), and the Lie algebra of the stabilizer g p is isomorphic to the Lie algebra of the group of Euclidian motions Euc n , we conclude that M is a covering of the Lightcone in Minkowski space, which completes the proof of Theorem 1.3 when G is simple.

4.3.
End of the proof. Thanks to Lemma 4.1, the complete proof of Theoremsimple.transitive reduces to the study of non-proper transitive actions of semi-simple groups with no factor locally isomorphic to SL(2, R). The work made above will be useful thanks to the following reduction lemma: Proof. Write g = h 1 ⊕ ... ⊕ h s , where the h i 's are the simple factors of g, and call X i the projection of X on h i . If W s X i ,h i denotes the stable space of X i relatively to h i , it is straightforward to check that X s X = W s X 1 ,h 1 ⊕...⊕W s Xs,hs . In particular, if W s X is isotropic at p and h is a simple factor on which X has a non-trivial projection X ′ , then W s X ′ ,h is non-trivial and isotropic at p. We infer from Corollary 4.1 that the H-orbit of p is lightlike and the action of H on it is non-proper.
By this lemma, there is a simple factor H of G having a lightlike non-proper orbit H.p. It follows from the previous section that H is locally isomorphic to O(1, n), n ≥ 3, and H.p is homothetic to Co n . There is a semi-simple group H ′ such that G is a finite quotient of H × H ′ . This product still acts locally faithfully on M , so that we will assume G = H × H ′ in the following. Consider O = G.p the G-orbit containing this H.p. The remaining part of Theorem 1.3 is the geometric description of O: it is a direct metric product Hp × H ′ p (up to a finite cover). This is the containt of the following lemma, which will be also useful when dealing with groups having factors locally isomorphic to SL(2, R).  O(1, 2)). Then h ′ is the identity map of Co d (resp. Co 1 ).
Proof. We begin with the case of Co 1 . An isometry of Co 1 is just a diffeomorphism of S 1 . If such a diffeomorphism commutes with the projective action of O(1, 2) on S 1 , it must fix, in particular, all the fixed points of parabolic elements in O(1, 2). But the set of these fixed points is precisely S 1 , so that we are done. Now, in higher dimension, we saw that writing Co d as R × S d−1 with the metric 0 ⊕ g S d−1 , the isometry h ′ is of the form: (t, x) → (t − µ(x), φ(x)).
Here φ is a conformal transformation of S d−1 satisfying φ * g S d−1 = e 2µ g S d−1 . Now, if h ′ commutes with the action of O(1, d), it must leave invariant any line of fixed points of parabolic elements in O(1, d). This yields φ(x) = x, and finally µ(x) = 0.
The lemma implies that h ′ acts identically on H.p, a contradiction with the fact that it acts non-trivially on N p .
2) Let S be the isotropy group of p in H. Since H and H ′ commute, S acts trivially on H ′ .p: s(h ′ .p) = h ′ s.p = h ′ .p. In particular, S will act trivially on T p (H ′ .p) ∩ T p (H.p). But T p (N p ) is the only subspace of T p (H p ) on which the action is trivial, and since we have already seen that T p (H ′ .p) must be transverse to T p (N p ), we get T p (H ′ .p) ∩ T p (H.p) = {0}. Now, there is a S-invariant splitting T p M = T p (H ′ .p) ⊕ T p (N p ) ⊕ E, where E is a Riemannian subspace of T p (H.p), on which S acts irreducibly by the standard action of O(n−1) on R n−1 . Let us call F the orthogonal of T p (H.p) in T p M . This space is transverse to E, so that F is the graph of a linear map A : T p (H ′ .p) ⊕ T p (N p ) → E. This map A intertwines the trivial action of S on T p (H ′ .p) ⊕ T p (N p ) with the irreducible one on E, so that A = 0, and F = T p (H ′ .p) ⊕ T p (N p ). As a consequence, the sum T p (H ′ .p) ⊕ T p (H.p) is orthogonal for the metric h p , and by homogeneity of M , this remains true at every point of M .

4.4.
Proof of corollary 1.4. Here, we assume that G is semi-simple, noncompact, with finite center. The group G acts transitively and non-properly on a lightlike manifold (M, h). Looking at a finite cover of G if necessary, we assume that G = H 1 × ... × H s , where each H i is a simple group with finite center. For p ∈ M , and every i = 1, ..., s, we call G i p the projection of the isotropy group G p on H i , and H i p the intersection G p ∩ H i . Each H i p is a normal subgroup of G p . Since G p is non-compact, some G i p has noncompact closure; for example i = 1. Performing a Cartan decomposition of a sequence (g k ) in G p tending to infinity, and using Proposition 3.2, we get X 1 in a Cartan subalgebra of h 1 , and some p ′ ∈ M such that W X 1 is isotropic at p ′ . If H 1 is not locally isomorphic to SL(2, R), we get that W X 1 has dimension > 1, and thus H 1 .p ′ is lightlike and carries a non-proper action of H 1 . By the previous study, H 1 is isomorphic to O(1, n), n ≥ 3, and H 1 .p ′ is homothetic to Co n . We can then apply Proposition 4.5 to conclude.
We are left with the case where H 1 is a finite cover of P SL(2, R), and G 1 p does not have compact closure. We claim that the orbit H 1 .p can not have dimension 3. Indeed, let < > p the pullback of h p in the Lie algebra h 1 . Let g ∈ G p , and g j the projection of g on H j . Since D p g leaves h p invariant, and the H j 's commute, we get that < > p is Ad(g 1 )-invariant. But < > p is either Riemannian, or lightlike. In both cases, we saw ( see the proof of Proposition 2.1) that the subgroup S ⊂ H 1 such that Ad(S) preserves < > p is compact, contradicting the fact that G 1 p does not have compact closure. Now, if H 1 .p is of dimension 2 and Riemannian, H 1 p is a maximal compact subgroup K ⊂ H 1 . Now, since H 1 p is normal in G p , we get that K is normal in G 1 p , what yields G 1 p = K, and a new contradiction. We conclude that H 1 .p is either 1-dimensional and lightlike, or 2-dimensional and lightlike. It follows from Proposition 2.1 that H 1 .p is homothetic to a cone Co 1 or Co 2 . We then get the conclusion thanks to Proposition 4.5.

Proof of theorem 1.5
Here, we assume that G is simple with finite center, and acts locally faithfully by isometries of a compact lightlike manifold (M, h).
We first assume, by contradiction, that G is not locally isomorphic to P SL(2, R). By compactness , every sequence of M is non-escaping. It follows from Proposition 3.2 that for every X in a Cartan subalgebra of g, W s X is isotropic at every p ∈ M . Thus, using the last point of Lemma 4.1, and the conclusions of Corollary 1.4, we get that G is locally isomorphic to O(1, n), and any G-orbit is homothetic to Co n , n ≥ 3. Let us call K a maximal compact subgroup of G, and let K 0 the stabilizer in K of a given point p 0 ∈ M . As we saw it in the proof of Lemma 3.5, the compact group K 0 preserves a Riemannian metric on M . Since any Riemannian isometry can be linearized around any fixed point thanks to the exponential map, it is not difficult to prove that the set of fixed points of K 0 is a closed submanifold of M , that we call M 0 . We know explicitely the action of K on Co n , and observe that every orbit of K is of Riemannian type. Let S(k/k 0 ) denote the set of euclidean scalar products on k/k 0 . There is a continuous map µ : M 0 → S(k/k 0 ) defined in the following way: if X and Y are two vectors of k, and X and Y are their projections on k/k 0 , then µ(p)(X, Y ) = h p (X(p), Y (p)). Now, on G.p 0 , there is a 1-parameter flow of homotheties h t , which transforms h |G.p 0 into e 2t h |G.p 0 , and commutes with the action of K (in particular, it leaves M 0 ∩ G.p 0 invariant). From this, it follows that µ(h t .p 0 ) = e 2t µ(p 0 ). Now, by compactness of M 0 , there is a sequence (t k ) tending to +∞ such that h t k .p 0 tends to p ∞ ∈ M 0 . We should get by continuity of µ: lim k→+∞ e 2t k µ(p 0 ) = µ(p ∞ ), which yields the desired contradiction.
It remains to understand what happens if G has finite center, and is locally isomorphic to P SL(2, R). Let us fix X, Y, Z a standard basis of g: [Y, Z] = X,[X, Y ] = 2Y , and [X, Z] = −2Z. It follows from Proposition 3.2 that Y and Z are isotropic at every p ∈ M . As a consequence, at any p ∈ M , a nontrivial linear combination of Y and Z has to vanish, so that all the orbits of G are lightlike and have dimension at most 2. If there is a 2-dimensional orbit G.p 0 , Proposition 2.1 ensures that it is homothetic to R 2 \{0} endowed with the metric dθ 2 (namely Co 2 ). We get a contradiction exactly as above, using the action of a maximal compact group and the homothetic flow on Co 2 (here k 0 = 0).
We conclude that every G orbit is 1-dimensional and lightlike. Since G has finite center, these orbits are finite coverings of the circle, hence closed.
6. Proof of Theorem 1.6 Let us first summarize results on Lorentz dynamics in the following statement, fully proved in [9], but early partially proved for instance in [1,2,4,16].
Theorem 6.1. Let G be a semi-simple group with finite center, no compact factor and no local factor isomorphic to SL(2, R), acting isometrically nonproperly on a Lorentz manifold M . Then, up to a finite cover, G has a factor G ′ isomorphic to O(1, n) or O(2, n) and having some orbit homothetic to dS n or AdS n .
Most developments along the article, in particular Proposition 3.2, do not explicitly involve the lightlike nature of the ambient metric, and apply equally to the Lorentz case, and by the way to the general sub-Lorentz case. This allows one to find a non-proper G-orbit O, i.e. with a stabilizer algebra containing nilpotent elements (see the end of proof of Proposition 3.1). One checks easily that O can not be Riemannian. If O is Lorentz, then, apply Theorem 6.1 (in the homogeneous case), and if it is lightlike, then apply Theorem 1.3.
6.0.1. Some remaining questions. The results of [9] are stronger than the statement of Theorem 6.1, since they contain a detailed geometric description of the Lorentz manifold M (a warped product structure...). This is the missing part of Theorem 1.3 in the lightlike non-homogeneous case and Theorem 1.6 in the sub-Lorentz case. In particular, in this last sub-Lorentz situation, it remains to see whether the manifold is or not pure, i.e. everywhere lightlike, or everywhere Lorentz?