Charles Frances

LORENTZIAN 3-MANIFOLDS AND DYNAMICAL SYSTEMS

An invitation to Lorentzian dynamics

The following article aims at presenting classical aspects of dynamical systems preserving a geometric structure, focusing on 3-dimensional Lorentzian dynamics. The reader won't find here any new result, but rather an expository approach of classical ones. Our main goal, in particular, is to introduce part of the techniques and arguments used in [Fraa] to obtain the classification of closed 3-dimensional Lorentzian manifolds admitting a non-compact isometry group. Doing so, we will present a self-contained, and somehow shortened proof of Zeghib's classification [START_REF] Zeghib | Killing fields in compact Lorentz 3-manifolds[END_REF] of non-equicontinuous Lorentzian isometric flows in dimension 3.

1.1. From dynamical systems to geometry. A flow ϕ t X , generated by a vector field X, on a closed manifold M is said to be Anosov when it is non-singular, and there exists a ϕ t X -invariant splitting T M = E -⊕RX ⊕E + such that vectors of E -(resp. of E + ) are exponentially contracted (resp. dilated) under ϕ t X . Precisely, there exist positive constants c and λ such that ||Dϕ t X (u -)|| ≤ ce -λt ||u -|| for every u -∈ E -, t ≥ 0, and ||Dϕ -t X (u + )|| ≤ ce -λt ||u + || for every u + ∈ E + , t ≥ 0. Here, the norms are taken with respect to any auxiliary Riemannian metric on M .

Examples of Anosov flows are provided by the geodesic flows of a closed, negatively curved, Riemannian manifolds (N, h) (the manifold M then corresponds to T 1 N , the unit tangent bundle of N ).

The very definition of Anosov flows already involves a (weak) geometric structure, namely the splitting T M = E -⊕ RX ⊕ E + . A beautiful aspect of the theory is that under certain circumstances, additional and more rigid invariant structures naturally appear. Assume, for simplicity, that our manifold M is 3-dimensional, so that E -and E + are 1-dimensional. Assume moreover that the flow ϕ t X preserves a smooth volume form ω (this is for instance the case if ϕ t X is the geodesic flow of a smooth Riemannian surface). One can then define a metric g on M , given by the relations g(X, X) = 1, g(u -, u + ) = ω(X, u -, u + ) for every (u -, u + ) ∈ E -× E + , and RX is g-orthogonal to E -⊕ E + . Observe that this metric is Lorentzian, namely has signature (-, +, +), and is ϕ t X -invariant by construction. Our volume-preserving condition thus turned our Anosov flow into an isometric Lorentzian flow.

All this nice picture is however damaged by regularity issues. In full generality, the splitting E -⊕ RX ⊕ E + is only Hölder continuous, and so is our Lorentzian metric g. This low regularity is a strong limitation (at least until now) to take advantage of this new geometrical data. But if one assumes that the distributions E -and E + are smooth (as well as the field X), then all the tools of differential geometry can be used to understand the dynamical system (M, ϕ t X ). It seems that Kanai [START_REF] Kanai | Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations[END_REF] was one of the first to apply this kind of ideas in the study of the geodesic flow of negatively curved surfaces. E. Ghys went further and obtained the following beautiful classification of 3-dimensional Anosov flows with smooth distributions E -, E + . Theorem 1.1. [START_REF] Ghys | Flots d'Anosov dont les feuilletages stables sont différentiables[END_REF] Let ϕ t X be a smooth, orientable, Anosov flow on a closed 3-dimensional manifold M . Then ϕ t X is either the suspension of an Anosov diffeomorphism on the 2-torus, or smoothly orbit equivalent to the right diagonal flow, acting on a compact quotient Γ \ SL(2, R).

Actually, Ghys did not use an invariant Lorentzian metric to achieve his classification, but rather another rigid geometric structure often called "contact Lagrangian". This structure comes from the geometrization of second order ODE's, and its study can be traced back to E. Cartan. Theorem 1.1 can also be obtained using Lorentzian geometry, as will be seen in this article.

In arbitrary dimension, when the distributions E + , E -are smooth, and E + ⊕ E -defines a contact distribution, then an Anosov flow ϕ t X preserves a smooth pseudo-Riemannian (not Lorentzian in general) metric, what allowed Y. Benoist, P. Foulon and F. Labourie to extend the initial 3-dimensional classification to all contact Anosov flows on closed manifolds, with smooth stable and unstable distributions (see [START_REF] Benoist | Flots d'Anosov à distributions stable et instable différentiables[END_REF], [START_REF] Benoist | Sur les difféomorphismes d'Anosov affines à feuilletages stable et instable différentiables[END_REF]). A particularly inspiring discussion of that topic can be found in [ There is an identification of SOL as the following subgroup of affine transformations:

SOL      e t 0 0 0 1 0 0 0 e -t   +   u t v   | (t, u, v) ∈ R 3    .
It is easily seen that this group acts freely and transitively on R 3 , preserving the flat Lorentz metric dx 1 dx 3 + dx 2 2 . Let us identify SOL with R 3 via the orbit map g → g.0. If h t denotes the 1-parameter group generated by the element T , then the isometric flow x → x + te 2 corresponds to the right multiplication by h t . To summarize, we have endowed SOL with a complete flat metric, invariant by the group SOL ×{h t }.

If we now pick Γ a lattice in SOL, and look at the right action of h t on Γ\ SOL, we get an isometric flow on the (flat) Lorentzian manifold Γ\ SOL. It is easily checked that Γ\ SOL is diffeomorphic to a hyperbolic torus bundle T 3

A , with A ∈ SL(2, Z) a hyperbolic matrix, and the right action of h t is the suspendion flow of the action of A on T 2 . 1.2.2. Examples derived from geodesic and horocyclic flows on hyperbolic surfaces. The Lie group PSL(2, R) admits a lot of interesting left-invariant Lorentzian metric. The most symmetric one is the anti-de Sitter metric g AdS . It is obtained by left-translating the Killing form of the Lie algebra sl(2, R). The space (PSL(2, R), g AdS ) is a complete Lorentz manifold with constant sectional curvature -1, called anti-de Sitter space AdS 3 . Because the Killing form is Ad-invariant, the metric g AdS is invariant by left and right multiplications of PSL(2, R) on itself. It follows that for any uniform lattice Γ ⊂ PSL(2, R), the metric g AdS induces a Lorentz metric g AdS on the quotient manifold Γ\PSL(2, R), with a non-compact isometry group coming from the right action of PSL(2, R) on Γ\PSL(2, R).

Exponentiating the matrix 0 1 0 0 (resp. 1/2 0 0 -1/2

), one gets a unipotent (resp. R-split) flows {u t } (resp. {h t }) in PSL(2, R). The right action of h t (resp. of u t ) on Γ\PSL(2, R) is an algebraic model for the horocyclic (resp. geodesic) flow on the unit tangent bundle of the hyperbolic surface Γ \ H 2 (when Γ is torsionfree). A remarkable feature is the possibility of deforming in a non-trivial way the previous examples. Consider a representation ρ h : Γ → {h t } (resp. ρ u : Γ → {u t }), and the group Γ

ρ h ⊂ PSL(2, R) × PSL(2, R) (resp. Γ ρu ) defined by {(γ, ρ h (γ)) | γ ∈ Γ} (resp. {(γ, ρ u (γ)) | γ ∈ Γ}).
When the representations ρ h and ρ u are close enough to the trivial representation, the groups Γ ρ h and Γ ρu act properly discontinuously on PSL(2, R), preserving the metric g AdS .

We get in this way (anti-de Sitter) Lorentz manifolds Γ ρ h \ PSL(2, R) and Γ ρu \ PSL(2, R), with a (right) isometric action of {h t } and {u t } respectively. While the topological type of the closed manifolds obtained is the same as that of Γ\ PSL(2, R), the new flow {h t } is no longer conjugated, as a flow, to the geodesic flow. This was noticed by Ghys in [Ghy87, Section 2]. However, there still exists a smooth orbit equivalence between the deformed flow and the initial one [START_REF] Ghys | Flots d'Anosov dont les feuilletages stables sont différentiables[END_REF]Prop 2.4].

The adjoint action of each of the groups {h t } and {u t }, admits invariant Lorentz scalar products on sl(2, R), which are not equal to a multiple of the Killing form. One can left-translate those scalar products and get metrics g u and g h on PSL(2, R) which are respectively PSL(2, R) × {u t } and PSL(2, R) × {h t }-invariant. Actually there are families of such metrics g u and g h which are not pairwise isometric. In the sequel, the metric g AdS and metrics of the form g u or g h , will be refered to as Lorentzian, non-Riemannian, left-invariant metrics on PSL(2, R). Those are the only left-invariant metrics on PSL(2, R), the isometry group of which does not preserve a Riemannian metric. Such metrics induce Lorentzian metrics g h (resp. g u ) on the quotients Γ ρ h \ PSL(2, R) (resp. Γ ρu \ PSL(2, R)) which are invariant by the right action of {h t } (resp. of {u t }).

1.3. Classification results. The previous sections are a motivation for the general study of 3-dimensional Lorentzian manifolds (M 3 , g), admitting an isometric flow ϕ t X which is "dynamically interesting", namely which does not have compact closure in Iso(M, g).

Theorem 1.2. [Zeg96, Theorems 1 and 2] Let (M, g) be a smooth, closed 3-dimensional Lorentz manifold. Let ϕ t X be a flow of Lorentzian isometries which is not relatively compact in Iso(M, g), then up to finite cover, and after replacing eventually ϕ t X by ϕ αt X for α ∈ R * , we are in exactly one of the following situations:

(1) The manifold M is a torus bundle T 3 A , A ∈ SL(2, Z) hyperbolic, and g is a flat metric. The flow ϕ t X is the suspension of the Anosov diffeomorphism A.

(2) The manifold (M, g) is isometric to a quotient Γ ρ h \ PSL(2, R) (resp.

Γ ρu \ PSL(2, R)) endowed with a metric g AdS or g h (resp. g AdS or g u ). The flow ϕ t X coincides with the right action of {h t } (resp. of {u t }).

By "up to finite cover", we mean that we allow taking finite covers and quotients. This statement implies Theorem 1.1 (at least for Anosov flows preserving a smooth volume form, as it is the case for instance for contact flows). Let us observe however, that the proof given in [Zeg96] uses Ghys's result. We will try in the following to provide an alternative proof (which will be independant of Theorem 1.1).

The condition that there exists in Iso(M, g) a 1-parameter group which is not relatively compact in Iso(M, g), amounts to the non-compactness of the identity component Iso o (M, g). One may wonder if Theorem 1.2 concludes the classification of 3-dimensional closed Lorentzian manifolds having a noncompact isometry group.

The answer is negative as the simple following example shows. Let us consider R 3 with the flat metric -dx 2 1 + dx 2 2 + dx 2 3 . This metric is translation invariant, hence induces a metric g f lat on the 3-torus T 3 (obtained as the quotient of R 3 by the action of Z 3 by translations). The isometry group

Iso(T 3 , g f lat ) is O(1, 2) Z T 3 , hence non-compact (recall that O(1, 2) Z is a lattice in O(1, 2))
. We see on this example, that the non-compactness comes from the discrete part of the isometry group. Other 3-dimensional closed Lorentz manifolds display this property (see [Fraa, Section 2]), and some of them (as T 3 ) have a topological type which is not represented by the manifolds of Theorem 1.2. A complete description of the Lorentz 3manifolds (M, g) for which Iso(M, g) is non-compact was recently given in [Fraa].

Theorem 1.3. [Fraa, Theorem A] Let (M, g) be a smooth, closed 3-dimensional Lorentz manifold. Assume that (M, g) is orientable and time-orientable, and that Iso(M, g) is non-compact. Then M is homeomorphic to one of the following spaces:

(1) A quotient Γ\ PSL(2, R), where Γ ⊂ P SL(2, R) is any uniform lattice.

(2) A 3-torus T 3 , or a torus bundle T 3 A , where A ∈ SL(2, Z) can be any hyperbolic or parabolic element.

Conversely, any smooth compact 3-manifold homeomorphic to one of the examples above can be endowed with a smooth Lorentz metric with a noncompact isometry group.

Our aim in this article is to present some ideas involved in the proof of Theorem 1.3, and apply them to give a self-contained proof of Theorem 1.2.

Curvature, Killing fields and the integrability theorem

We consider in this section a 3-dimensional Lorentz manifold (M, g) (actually, the material presented in this section fits into the much more general framework of Cartan geometries, but we stick to the Lorentz case for the sake of simplicity).

Let π : M → M denote the bundle of orthonormal frames on M . This is a principal O(1, 2)-bundle over M , and it is classical (see [START_REF] Kobayashi | Foundations of differential geometry[END_REF][Chap. IV.2 ]) that the Levi-Civita connection associated to g can be interpreted as an Ehresmann connection α on M , namely a O(1, 2)-equivariant 1-form with values in the Lie algebra o(1, 2). Let θ be the soldering form on M , namely the R 3 -valued 1-form on M , which to every ξ ∈ T x M associates the coordinates of the vector π * (ξ) ∈ T x M in the frame x. The sum α + θ is a 1-form ω : T M → o(1, 2) R 3 called the canonical Cartan connection associated to (M, g).

Observe that for every x ∈ M , ω x : T x M → o(1, 2) R 3 is an isomorphism of vector spaces, and the form ω is O(1, 2)-equivariant (where O(1, 2) acts on o(1, 2) R 3 via the adjoint action).

2.1. Killing fields and kill loc -orbits. Recall that a local Killing field on (M, g) is a vector field Z, defined on some open subset U ⊂ M , such that the Lie derivatives L Z g vanishes identically. In other words, the local flow ϕ t Z consists in local isometries for g. Given an open set U , the Killing fields on U define a finite-dimensional Lie algebra kill(U ). Hence, given a point x ∈ M , and a decreasing sequence (U n ) of nested neighborhoods of x, the isomorphism type of the Lie algebras kill(U n ) stabilizes for n large. This define a Lie algebra kill loc (x) that we will call (abusively) the Lie algebra of Local Killing fields around x. For 3dimensional Lorentz manifolds, the dimension of kill loc (x) is at most 6, with equality when g has constant curvature in a neighborhood of x. For analytic structures, the analytic continuation property for Killing fields ensures that the algebraic type of kill loc (x) does not depend on x. In the smooth category, which is the one we consider here, the situation is very different. The dimension, as well as the algebraic type of kill loc (x) may not be constant.

Observe that because local isometries map orthonormal frames to orthonormal frames, any local isometry lifts to a local bundle isomorphism on M , which preserves ω. In the same way, any local Killing field Z defined on U lifts to a vector field defined on Û := π -1 (U ) which satisfies L Z ω = 0. In particular Z commutes with the O(1, 2)-action on M . Conversely, any vector field defined on a connected open subset of M , and such that L Z ω = 0, projects to a local Killing field on M . We will thus indistinctly speak about local Killing fields on M and on M .

The kill loc -orbit of a point x, is the set of points that can be reached from x, by flowing along finitely many successive local Killing fields.

2.2. Curvature. We assume now that the structures considered are of class C ∞ . The notion of Riemannian curvature for g, as well as its higher order covariant derivatives have a counterpart in M . Actually, we will see that it is really fruitful to work on the bundle M instead of the manifold M itself. In all the sequel, we will denote by g the Lie algebra o(1, 2) R 3 .

The curvature of the Cartan connection ω is a 2-form K on M , with values in g, defined as follows. If X and Y are two vector fields on M , the curvature is given by the relation:

K(X, Y ) = dω(X, Y ) + [ω(X), ω(Y )].
Because at each point x of M , the Cartan connection ω establishes an isomorphism between T x M and g, it follows that any k-differential form on M , with values in some vector space W, can be seen as a map from M to Hom(⊗ k g, W). This remark applies for the curvature form K itself, yielding a curvature map κ : M → W 0 , where the vector space W 0 is a sub O(1, 2)-module of Hom(∧ 2 (g/o(1, 2)); g) (the curvature is antisymmetric and vanishes when one argument is tangent to the fibers of M ). 2.2.1. The curvature module. We give here more algebraic informations on the curvature module W 0 in our special case of a 3-dimensional Lorentz manifold.

Let us consider on R 3 the Lorentzian form, with matrix in a basis e, h, f

given by J =   0 0 1 0 1 0 1 0 0   .
We will identify in all what follows the group O(1, 2) with the subgroup of GL(3, R) preserving the bilinear form determined by J. Its Lie algebra is denoted by o(1, 2), and admits the following basis :

E =   0 1 0 0 0 -1 0 0 0   , H =   1 0 0 0 0 0 0 0 -1   , F =   0 0 0 1 0 0 0 -1 0   . Notice the commutation relations [H, E] = E, [H, F ] = -F and [E, F ] = H.
Under the natural identification of g/o(1, 2) with R 3 , our curvature module W 0 is a 6-dimensional Hom(∧ 2 (R 3 ), o(1, 2)) (this constraint on the dimension comes from Bianchi's identities). Choosing e ∧ h, e ∧ f , h ∧ f as a basis for ∧ 2 (R 3 ), and E, H, F as a basis for o(1, 2), an element of Hom(∧ 2 (R 3 ), o(1, 2)) is merely given by a 3×3 matrix. The action of O(1, 2) on Hom(∧ 2 (R 3 ), o(1, 2)) corresponds to the conjugation on matrices.

Scalar matrices are O(1, 2)-invariant, and form a 1-dimensional irreducible submodule (corresponding to constant sectional curvature).

The other irreducible submodule of the curvature module is 5-dimensional, spanned by the matrices:

  0 0 1 0 0 0 0 0 0   ,   0 1 0 0 0 1 0 0 0   ,   1 0 0 0 -2 0 0 0 1   ,   0 0 0 1 0 0 0 1 0   ,   0 0 0 0 0 0 1 0 0   .
2.2.2. Generalized curvature map. We now differentiate the map κ, getting a map Dκ : T M → W 0 . As we already explained it, the connection ω allows to identify Dκ with a map Dκ : M → W 1 , where W 1 = Hom(g, W 0 ). We can then differentiate once again and see D(Dκ) as a map D 2 κ : M → W 2 = Hom(g, W 1 ) Repeating this procedure, we can define the rth-derivative of the curvature D r κ : M → Hom(g, W r ) (with W r defined inductively by W r = Hom(g, W r-1 )). The generalized curvature map of our Lorentz manifold (M, g) is the map κ g = (κ, Dκ, D 2 κ, . . . , D dim(g) κ). The O(1, 2)-module Hom(g, W dim(g) ) will be rather denoted W κ g in the following.

Remark 2.1. Of course, we could have replaced dim(g) by 6 in the expressions above, but we want to emphasize the role of the bound dim(g) when working in the general framework of Cartan geometries.

The integrability theorem.

We are now coming to the most important result of this section. To understand its statement, we first make the trivial remark that given a local Killing field Z defined on some open subset Û ⊂ M , then Dκ g (Z) = 0. The question is about the converse of this property. Given a point x ∈ M , and ξ ∈ T x M such that Dκ g (ξ) = 0, can we ensure that ξ is the evuation at x of a local Killing field Z?

The remarkable fact is that the answer is positive if x stays in a nice dense open subset of M the integrability locus of M . This locus is defined as the subset M int ⊂ M where the rank of the map κ g is locally constant. Observe that M int is a dense open subset in M , which is O(1, 2)-invariant. We will denote by M int its projection on M . The integrability theorem can then be stated as follows:

Theorem 2.2 (Integrability theorem). Let M int ⊂ M the integrability locus of the smooth Lorentz manifold (M, g).

(1) For every x ∈ M int , and every ξ ∈ Ker(D xκ g ), there exists a local Killing field Z around x such that Z(x) = ξ.

(2) The connected components of the fibers of κ g in M int coincide whith the kill loc -orbits in M int .

Some comments about this result are in order. The deepest, and most difficult part, of the theorem is the first point. A first, and weaker version of this statement, can be found in [START_REF] Nomizu | On local and global existence of Killing vector fields[END_REF]. Actually, the intergability condition stated in [START_REF] Nomizu | On local and global existence of Killing vector fields[END_REF] required the vector ξ to be in the kernel of all successive derivatives of the curvature map κ, which makes the conclusions less interesting. It was M. Gromov's insight to see that Killing generators of a finite, sufficiently large order are locally integrable. He stated this integrability result, as well as its amazing consequences for the structure of Is loc -orbits in [START_REF] Gromov | Rigid transformations groups[END_REF][Section 1.6]. Quite recently, those results were recast in the framework of Cartan geometries by K. Melnick in the analytic case (see [START_REF] Melnick | A Frobenius theorem for Cartan geometries, with applications[END_REF]). The reference [START_REF] Pecastaing | On two theorems about local automorphisms of geometric structures[END_REF] gives an alterative approach, allowing to deal with smooth Cartan geometries, leading to the statement of Theorem 2.2. A proof that the integrability property actually holds on the set where the rank of κ g is locally constant (first point of the theorem) follows Pecastaing's ideas and can be found in Annex A of [Frab].

2.3. Enriched structures. We have given the statement of Theorem 2.2 in the case where (M, g) is a (3-dimensional) Lorentz manifold. Actually, this theorem holds for all pseudo-Riemannian manifolds, and also for a much larger class of geometric structures called Cartan geometries. We invite the reader wanting to learn more about this topic, to look at the very comprehensive books [START_REF] Sharpe | Differential geometry[END_REF], and [vS09]. In particular, [START_REF] Čap | Parabolic geometries. I[END_REF]Chap. 4] provides an extensive discussion of examples.

The integrability theorem is actually available for an even greater class of structures, that we call enriched structures, namely a Cartan geometry together with additional tensorial datas (see [Pec16, Section 4] for a detailed account). The example which will be relevant for us is that of a 3-dimensional Lorentz manifold (M, g), together with a Killing field X. The geometric structure considered is then the triple (M, g, X). Given x ∈ M , we can consider local Killing fields for our enriched structure (M, g, X), defined in a neighborhood of x. They are the Killing vector fields for g which moreover commute with X. The Lie algebra of such local fields will be denoted by z X (x). With the same definition as above, we will speak about the z X -orbit of a given point.

Our Killing field X lifts to a vector field (still denoted X) on the bundle M , and provides there a O(1, 2)-equivariant map λ X : M → g defined by λ X (x) = ω(X(x)). Any local vector field Z in z X (x) will lift to a vector field (still denoted Z) in a neighborhood of x ∈ π -1 {x} so that L Z ω = 0 and Z.λ X = 0 (or in other words [Z, X] = 0).

We can now put the curvature κ and the map λ X together, and build a new generalized curvature map

κ g X := (κ, Dκ, . . . , D dim(g) κ, λ X , Dλ X , . . . , D dim(g) λ X
). This defines an integrability locus M int X ⊂ M comprising all point where the rank of κ g X is locally constant. As before this set is O(1, 2)-invariant and projects on a dense open subset M int X ⊂ M . One can then formulate an integrability theorem similar to Theorem 2.3 for the enriched structure (M, g, X).

Theorem 2.3. [Pec16, Th 4.4] Let (M, g, X) be the geometric structure comprising the Lorentz metric g and the Killing field X.

(1) For every x ∈ M int X , and every ξ ∈ Ker(D xκ g X ), there exists a local field Z around x, which is the lift an element of z X (x), and such that ω(Z(x)) = ξ.

(2) The connected components of the fibers of κ g X in M int coincide whith the z X -orbits in M int .

Components of the integrability locus. The integrability locus M int

X is a dense open subset, which can be written as a (possibly infinite) union of connected components. Above each such component M, the generalized curvature map κ g X has constant rank, hence the dimension of z X (x) is constant as x ranges over M. The isomorphism type of z X (x) is thus constant too, so that we will write z X (M) for the Lie algebra isomorphic to z X (x) for all x ∈ M. Observe that when M is not simply connected, monodromy phenomena may appear, so that generally, z X (M) won't admit a realization as a subalebra of vector fields on M.

Basic facts about Lorentzian isometric flows

Before tackling the proof of Theorem 1.2, we recall briefly some elementary facts about isometries of Lorentz manifolds. The first one is a rigidity property of Lorentz isometries. Proposition 3.1. Let (M, g) be a smooth connected Lorentz manifold.

(1) If an isometry f of (M, g) fixes a point x ∈ M , and satisfies

D x f = id, then f is the identity map on M . (2) If (f k ) is a sequence of Iso(M, g), and if x ∈ M is such that f k (x) is bounded in M , and D x f k is bounded, then (f k ) has compact closure in Iso(M, g).
Proof: The fact is classical, and we only outline its proof, which relies essentially on the fact that the exponential map locally linearizes any Lorentz isometry. More precisely, if ξ → exp(x, ξ) denotes the exponential map at x ∈ M , and f ∈ Iso(M, g), the relation:

f (exp(x, ξ)) = exp(f (x), D x f (ξ))
holds for every ξ ∈ T x M in a small neighborhood of 0. In particular, this relation shows that if f (x) = x and D x f = id, then f is the identity on a neighborhood of x. Actually, we also get that the set of points where f (y) = y and D y f = id is open. Since it is clearly closed, we conclude that f is the identity map.

The second point is a refinement of this reasoning. Under the asumption that f k (x) and D x f k are bounded, we may find an extraction such that f k (x) converges to x ∞ , and D x f k converges to a linear, invertible, map

A ∞ : T x M → T x∞ M . Then the relation f (exp(x, ξ)) = exp(f (x), D x f (ξ))
ensures that on a small neighborhood U of x, f k converges for the C ∞ topology to a local isometry f ∞ : U → V . As before, one shows that the set of points in M , having a neighborhood on which f k converges is open and closed in M . As a consequence, one may extract from every subsequence of (f k ) a converging sequence, showing that (f k ) is relatively compact in Iso(M, g). ♦

We now focus on flows of Lorentz isometries on closed 3-dimensional manifolds. The following result was proved in [Zeg96, Prop 4.1]. Lemma 3.2. Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not relatively compact in Iso(M, g). Then

(1) The Lorentz norm g(X, X) is constant on M , and nonnegative.

(2) The field X has no singularities.

(3) If X is lightlike, then any Killing field on M commuting with X is of the form λX, for some λ ∈ R. (4) If X is spacelike, then any lightlike geodesic whose direction u is orthogonal to X is complete.

Proof:

We introduce the (smooth) function ψ = g(X, X), and we assume that it is not constant to zero (if not the first point is proved). We pick a ∈ R\{0} a regular value of ψ. Let Σ be the level set ψ -1 {a}. It is a smooth submanifold of M . We pick x ∈ Σ. By compactness of Σ, we can choose a sequence (t k ) tending to infinity in R, such that y k := ϕ t k X (x) converges to y ∞ in Σ.

We choose U and V small open neighborhoods of x and y ∞ , as well as R = (E 1 , E 2 , E 3 ) a field of orthonormal frames on U ∪ V . If we read D x ϕ t k X in this frame field, we get a sequence A k of matrices in O(1, 2).

We first assume a < 0. Then the identity D x ϕ t k X (X(x)) = X(y k ) yields a timelike vector u such that A k .u converges in R 1,2 . It follows easily that (A k ) is bounded in O(1, 2), and Proposition 3.1 implies that {ϕ t X } is relatively compact in Iso(M, g). This is in contradiction with our hypotheses.

If a > 0 we complete X into a local frame field (X,

E 1 , E 2 ) on U ∪ V , with (E 1 , E 2 ) spanning X ⊥ and satisfying g(E 1 , E 1 ) = g(E 2 , E 2 ) = 0, g(E 1 , E 2 ) = 1. Considering A k as above we have A k (E 1 ) = λ k E 1 and A k (E 2 ) = 1 λ k E 2 , for a sequence λ k → 0. The invariance of ψ under ϕ t X shows that ψ(exp(x, sE 1 )) = exp(y k , λ k sE 1 ). It follows that s → exp(x, sE 1 ) is in- cluded in Σ, hence E 1 (x) ∈ T x Σ. Then E 1 (y k ) ∈ T y k Σ for all k ∈ N, and finally E 1 (y ∞ ) ∈ T y∞ Σ.
One also has ψ(exp(x, sλ k E 2 )) = exp(y k , sE 2 ), what shows that s → exp(y ∞ , sE 2 ) is included in Σ. We reach the conclusion that the three independant vectors E 1 (y ∞ ), E 2 (y ∞ ), X(y ∞ ) belong to T y∞ Σ, contradicting the fact that Σ is a surface.

Observe also that in this last case a > 0, if U and V are shrunk so that all geodesics of direction E 1 (y) or E 2 (y) are defined on (-τ, τ ), τ > 0, then the geodesics of direction E 1 (x) and E 2 (x) will be defined on (-λ -1 k τ, λ -1 k τ ) for every k ∈ N. It follows that those geodesics are complete, proving the last point of the proposition.

We now show that X does not have any singularity. This of course obvious if g(X, X) is constant and positive. Now if g(X, X) is constant equal to 0. Let us assume that x 0 is a singularity of X, and let us look at the action of D x 0 ϕ t X on the 1-sphere corresponding to lightlike directions at x 0 . If this action is nontrivial, then there exist some x on the local lightcone with vertex x 0 (denoted C(x 0 )), such that X(x) is tangent to C(x 0 ) and transverse to the lightlike geodesic through x and x 0 . This forces g(X, X) > 0 at x, a contradiction.

If the action on the lightlike directions at x 0 is trivial, then D x 0 ϕ t X = id, and we get that X is identically 0 by Proposition 3.1. This is again a contradiction. Observe that we have proved the general fact that a nontrivial lightlike Killing field has no singularity.

It remains to prove the third point of the proposition. Let Y be a Kiliing field commuting with X, where X is Killing and lightlike. If Y is colinear to X at each x ∈ M , then there exists a real λ such that Y -λX has a singularity. Since Y -λX is Killing and lightlike, the remarks above imply that Y -λX = 0, and we are done.

It remains to investigate the case where X and Y are linearly independant in a neighborhood of a point x ∈ M . Since ϕ t X preserves the Lorentz volume on M , which is finite by compactness of M , we may assume that x is a recurrent point for ϕ t X . In this case, there exists a local

frame field (E 1 , E 2 , E 3 ) around x, with g(E 1 , E 2 ) = g(E 3 , E 3 ) = 1 and g(E 1 , E 1 ) = g(E 2 , E 2 ) = g(E 1 , E 3 ) = g(E 2 , E 3 ) = 0, such that D x ϕ t X ex- pressed in the frame (E 1 , E 2 , E 3
) is the identity. The second point of Proposition 3.1 then implies that ϕ t X is relatively compact. This is a contradiction. ♦ 4. A Killing field factory 4.1. Recurrence produces Killing fields. We are no proving an essential fact, namely that a non-compact flows ϕ t X produces extra local Killing fields. In the following statement, we use the notation Is X (x) for the isotropy at x, namely the subalgebra of z X (x) comprising all local Killing fields in z X (x) vanishing at x. Proposition 4.1. Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not relatively compact in Iso(M, g).

(1) For every x ∈ M int X , the isotropy algebra Is X (x) is 1-dimensional, and generates a non-compact subgroup of O(T x M ). This subgroup is hyperbolic (resp. parabolic) if X is spacelike (resp. X is lightlike).

(2) On each component M ⊂ M int X , the Lie algebra z X (M) has dimension 3 or 4. In the first case, the z X -orbits on M have dimension 2. In the second case, the component M is a single 3-dimensional z X -orbit.

Proof: The metric g defines a Lorentz volume (attributing volume 1 to any orthonormal frame), which is finite on M and ϕ t X -invariant. Poincaré reccurence theorem ensures that the set of points which are recurrent for ϕ t X has full measure. Let x be such a recurrent point for ϕ t X and choose x ∈ M in the fiber of x. The recurrence hypothesis means that there exist sequences (t k ) and (p k ) in R and O(1, 2) respectively, satisfying |t k | → ∞ and ϕ t k X (x).p -1 k → x. Observe that (p k ) tends to infinity in O(1, 2), because Iso(M, g) acts properly on M (Proposition 3.1). By equivariance of the generalized curvature map, we also have

p k .κ g X (x) → κ g X (x)
. The linear action of O(1, 2) on W κ g X is algebraic, hence its orbits are locally closed. As a consequence, there exists a sequence ( k ) in O(1, 2) with k → id and k .p k .κ g X (x) = κ g X (x). We infer that the stabilizer S(x) of κ g X (x) in O(1, 2) is a non-compact subgroup. This group is algebraic, hence its identity component is non-compact too, and the Lie algebra s(x) is nontrivial. But Theorem 2.3 actually says that elements in s(x) are images under ω of evaluations (at x) of elements of Is X (x). In particular Is X (x) is not reduced to {0}. If Y ∈ Is X (x), then because [X, Y ] = 0, the 1parameter subgroup e tω x(Y (x)) fixes the vector ω x(X (x)). In particular, the horizontal part of ω x(X (x)), which is nonzero by point (2) of Lemma 3.2, is fixed and thus Is X (x) is 1-dimensional. The flow e tω x(Y (x)) is hyperbolic if X(x) is spacelike and parabolic if X(x) is lightlike. The first point of the proposition follows by density of recurrent points in M int X . To prove the second point, we start with x ∈ M int X , and consider a simply connected neighborhood U ⊂ M int X of x. Then, every algebra Is X (y), y ∈ U , is realized as a Lie algebra of Killing fields defined on U . By the first part of our proof, there exists Y a nontrivial Killing field of z X on U , such that Y (x) = 0. The zero locus of Y is a nowhere dense set in U . We can thus pick y ∈ U satisfying Y (y) = 0, and apply again the first point of the proof at y. We get a second nontrivial Killing field Z defined on U , and vanishing at y. Let us consider a lightlike direction u ∈ T y M such that g y (u, Y (y)) = 0, and u is transverse to the zero locus of Z. Let t → γ u (t) be the geodesic passing through y at t = 0 and such that γu (0) = u. Then Clairault's equation ensures that g( γu , Y ) is constant on γ u , hence does not vanish on γ u by our choice of u.

Clairault's equation also ensures that g γu ( γu , Z) = 0, and Z(γ u (t)) = 0 for small, nonzero, values of t. Hence, on some open subset of U , the z Xorbits have dimension ≥ 2, while for every point z ∈ U , the dimension of Is X (z) is ≥ 1 by the first part of the proof. It follows that the dimension of z X (M) is at least 3. Because the dimension of M is 3, the dimension of z X (M) can be 3 (in which case the z X -orbits on M are 2-dimensional) or 4 (in which case M is a single 3-dimensional z X -orbit). ♦ 4.2. Hyperbolic and parabolic components. Proposition 4.1 shows that components of M int X split into two categories. The ones for which the isotropy algebra generates a 1-parameter hyperbolic group will be called hyperbolic components. Those for which the isotropy algebra generates a 1-parameter parabolic group will be called parabolic components. Hyperbolic and parabolic components can themselves be divided into two subgategories. The ones for which z X (M) is 4-dimensional, which are locally homogeneous, and the others for which z X (M) is 3-dimensional. We will call those later components non locally homogeneous. Observe that this is a property about the local action of z X , and there is no obstruction for the metric g itself to be locally homogeneous on a non locally homogeneous component.

4.3.

Existence of closed z X -orbits. One key consequence of the previous results is that components M for which the dimension of z X (M) is minimal, contain closed kill loc -orbits. Proposition 4.2. Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not relatively compact in Iso(M, g). Then every component M ⊂ M int X such that z X (M) has minimal dimension contains a kill loc -orbit which is closed in M .

Proof: The proof follows essentially from two remarks.

The first one is that the stabilizer of κ g X (x) in O(1, 2) is 1-dimensional for every x ∈ M . Indeed, Proposition 4.1 and the density of M int X in M shows that this stabilizer is always at least 1-dimensional. Since there are no 2-dimensional stabilizers in a finite dimensional linear representation of O(1, 2), the only other possibility is that κ g X (x) is stabilized by O o (1, 2). Because the map κ g X has the map λ X as one of its factors, this would imply that λ X (x) is O o (1, 2)-invariant, hence equal to zero. Contradiction with Lemma 3.2.

Because for an algebraic action, orbits can accumulate only on orbits of smaller dimension, we infer that for every x ∈ M , the O(1, 2)-orbit (as well as the O o (1, 2)-orbit) of κ g X (x) is closed in κ g X ( M ). We then call Ω the open subset of M , where the rank of κ g X achieves its maximal value, that we call r max . Theorem 2.3 ensures that Ω ⊂ M int X . Let us denote by Λ the set of points in M where the rank of κ g X is ≤ r max -1. Sard's theorem says that the r max -dimensional Hausdorff measure of the set κ g X ( M ) is zero. It follows that there must be x0 ∈ Ω such that Σ := (κ g X ) -1 (κ g X (x 0 )) is included in Ω. Now Σ is a closed submanifold of M . Theorem 2.3 says that the projection π(Σ) is a submanifold of M int X the connected components of which are z X -orbits. Our first remarks show that the saturation of Σ by the action of O(1, 2) is also closed in M , since it is the inverse image of a O(1, 2)-orbit in W κ g X and such orbits are closed in κ g X ( M ). We infer that π(Σ) is closed in M , and the proposition is proved. ♦

Heisenberg structures and the geometry of parabolic components

5.1. Heisenberg structures on surfaces. We consider the following realisation of the 3-dimensional Heisenberg group as a subgroup of affine transformations of the plane:

1 t 0 1 + x y , x, y, t ∈ R
We will denote by Heis this group. If Σ is a surface, we will call a (Heis, R 2 )-structure on Σ the data of an atlas (U i , ϕ i ), taking values in R 2 , so that transition maps ϕ j • ϕ -1 i are restrictions of elements of Heis. Observe that the action of Heis on R 2 preserves the (degenerate) metric dy 2 , and commutes with the 1-parameter group given by the transformations (x, y) → (x + t, y), t ∈ R. It follows that any (Heis, R 2 )-structure S on Σ defines accordingly a degenerate metric λ on Σ, as well as a flow ψ t . We will say that λ (resp. ψ t ) is the standard metric (resp. the standard flow) associated to the structure S. In the following, we will often use the terminology Heisenberg structure instead of (Heis, R 2 )structure.

Here is an equivalent point of view for Heisenberg structures. Let us consider the Lie algebra h ⊂ o(1, 2) R 3 , spanned by the vectors e, h, E. The corresponding Lie subgroup H ⊂ O(1, 2) R 3 is isomorphic to Heis, and the action of Heis on R 2 is conjugated to that of H on the homogeneous space H/{e tE }. We will retain in the sequel H/{e tE } as a model space for Heisenberg structures. Actually, we will also use the following dual point of view for Heisenberg structure. Let us consider ω H the Maurer-Cartan form on the group H. Then a Heisenber structure on a surface Σ is equivalent to the data of a {e tE }-principal bundle Σ over Σ, as well as a 1-form ω on Σ with values in h, such that the fiber bundle ( Σ, ω) is locally isomorphic to (H, ω H ). We end up this section with the following completeness result.

Proposition 5.1. Let Σ be a closed surface, endowed with an Heisenberg structure S. Then:

(1) The surface Σ is a 2-torus, and the structure is complete, namely is the quotient of R 2 by a discrete subgroup of Heis isomorphic to Z 2 . (2) For such a structure, the standard flow ψ t on Σ is periodic.

Proof: An Heisenberg structure provides an orientation of Σ, and because the standard flow ψ t does not have singularity, the Euler characteristic of Σ is zero. Hence Σ is a 2-torus.

The Heisenberg structure on Σ yields a developing map δ : Σ → R 2 , which is a smooth immersion. Here, Σ stands for the universal cover of Σ. The map δ is ρ-equivariant for a representation ρ : π 1 (Σ) → Heis. The standard flow ψ t is complete on Σ. It follows that the image δ( Σ) is saturated by lines of the form

D u = {(t, u) | t ∈ R}.
Let us call I the projection of δ( Σ) on the second coordinate axis, namely

I = {u ∈ R, D u ⊂ δ( Σ)}.
Then I is invariant by the action of ρ(π 1 (Σ)) on the set of lines of the form D u . It turns out that this action is by translations, and it must also be cocompact since δ is ρ-equivariant and π 1 (Σ) acts cocompactly on Σ. This forces I = R, and δ is onto.

To ckeck that δ is injective, we consider W a vector field on Σ, transverse to the orbits of ψ t , and satisfying λ(W, W ) = 1, where λ stands for the standard metric on Σ defined by the Heisenberg structure. We lift W to Σ to get a (complete) flow φ t W , whose orbits are transverse to those of ψ t . Let us pick x ∈ Σ, and consider the curve γ : s → φ s W .x. The map δ is injective in restriction to γ. Indeed, if we write δ(γ(s)) = (u(s), v(s)), we notice that |v (s)| = 1, since λ(W, W ) = 1. It follows also that v(R) = R, and we conclude that {ψ t (φ s W (x)) | (s, t) ∈ R 2 } is mapped injectively onto R 2 . One deduce easily that δ has the path-lifting property, and δ is a covering between Σ and R 2 , hence a diffeomorphism. We conclude that ρ(π 1 (Σ)) is a discrete subgroup of Heis isomorphic to Z 2 , proving the first point of the proposition.

All discrete copies of Z 2 in Heis intersect the center of Heis nontrivially. This shows that the standard flow ψ t is periodic. ♦ 5.2. Existence of Heisenberg structures on z X -orbits. We now study in more details the geometry of non locally homogeneous components. This section will be devoted to the proof of the next proposition.

Proposition 5.2. Let M be a parabolic component which is not locally homogeneous. Then:

(1) The Lie algebra z X (M) is isomorphic to the 3-dimensional Heisenberg algebra heis.

(2) Any z X -orbit of M is endowed with an invariant Heisenberg structure, for which the restriction of ϕ t X (resp. of the Lorentz product g) coincides with the standard flow (resp. the standard metric).

Let us fix a point x ∈ M. We work in the fiber bundle M (and lift all local Killing fields there). Let Y be a generator of the local isotropy at x. Then we can find x ∈ M in the fiber of x such that ω(Y (x)) = E. Because X commutes with Y , we must have ω(X(x)) = e + αE. Finally, there is a third vector field in z X (x) such that ω(Z(x)) = h + βH + γF .

We call κ 0 the element of Hom(∧ 2 (R 3 ), o(1, 2)) corresponding to the identity matrix, namely κ 0 maps e ∧ h to E, e ∧ f to H and h ∧ f to F . We also call κ 1 the element of Hom(∧ 2 (R 3 ), o(1, 2)) corresponding to the matrix   0 0 1 0 0 0 0 0 0   (see Section 2.2.1).

The two dimensional vector space spanned by κ 0 and κ 1 is the set of fixed points of the action of {e tE } t∈R on the curvature module. We infer that κ(x) is of the form κ = σκ 0 + bκ 1 . In particular, the following identities hold at x:

(1) κ(e ∧ h) = σE, κ(e ∧ f ) = σH, κ(h ∧ f ) = bE + σF.
Notice that σ, b, α,β,γ depend on x and x, but since those points are fixed once for all in the following, there will be considered as constant. Cartan's formula L X ω = ι X dω + d(ι X ω) shows that whenever U, V are two Killing fields on M , the following relation holds:

(2) ω([U, V ]) = K(U, V ) -[ω(U ), ω(V )]
We now write Equation (2) at x, using identities (1), when U and V range over Z, X, Y . For instance, the first equation is (at x):

ω([X, Z]) = (σ + αβ)E + (β -α)e + γh -γαH. Now [X, Z] = 0 hence γ = 0, β = α and σ = -α 2 .
We now write the second equation (still at x):

ω([Y, Z]) = -e + αE = ω(-X + 2αY ).
Next, two Killing fields which coincide at x must be equal (by freeness of the action of isometries on the orthonormal frames). We get the following relation at the level of the Lie algebra z X (x):

[Y, Z] = -X + 2αY.
We see in particular that at each z ∈ M, if U is a local Killing field around z generating the isotropy at z, then ad(U ) is a nilpotent endomorphism of z X (z).

At x, X(x) is lightlike and nonzero and Z(x) is spacelike, orthogonal to X(x). The orthogonal to Z(x) at x is a Lorentzian plane spanned by X(x) and another vector w ∈ T x M . Let us call t → γ(t) the geodesic through x satisfying γ(0) = w. Clairault's equation ensures that the quantities g( γ(t), Z(γ(t))), g( γ(t), X(γ(t))) and g( γ(t), Y (γ(t))) do not depend on t. In particular, for t > 0, both Y (γ(t)) and Z(γ(t)) are orthogonal to γ(t) while X(γ(t)) is not. For t > 0 small enough, Z(γ(t)) is still spacelike, hence nonzero, and γ(t) belongs to M. In particular, the z X -orbit at γ(t) is not 3-dimensional, so that there must exist some real number λ t for which Y (γ(t)) = λ t Z(γ(t)). Observe finally that w is transverse to the curve where Y vanishes, hence for t > 0 small λ t = 0. Using our bracket relation [Y, Z] = -X + 2αY , we compute Trace(ad(Y -λ t Z)) = 2λ t α.

Because we observed that ad(Y -λ t Z) must be nilpotent at γ(t), we get α = 0.

To summarize, we have shown that α = 0 and σ = α 2 = 0 (in particular κ x(e ∧ h) = 0). Moreover, at x:

ω(X) = e, ω(Y ) = E, ω(Z) = h,
and the bracket relations for z X (M) are

[X, Y ] = [X, Z] = 0, [Z, Y ] = X,
showing that z X (M) is isomorphic to heis.

Finally, let us consider Σ the z X -orbit of x. Calling, as in Section 5.1, h the Lie subalgebra spanned by e, h, E in o(1, 2) R 3 , the invariance of ω under local killing flows show that ω(T Σ) ≡ h. This makes Σ into a {e tE }-principal bundle over Σ. The restriction of ω to T Σ yields a Cartan connexion ω : T Σ → h. The curvature of this induced Cartan connection ω is given by κ(e ∧ h), which is identically 0 on Σ since we have σ ≡ 0. It thus follows that ( Σ, ω ) is locally isomorphic, as {e tE }-principal bundle to (H, ω H ) fibering on H/{e tE }, where ω H stands for the Maurer-Cartan form on H. As explained in Section 5.1, this defines an Heisenberg structure on Σ. Because ω = ω |T Σ, and because ω(X(ŷ)) = e for every ŷ ∈ Σ, it follows that the restriction of g to Σ yields the standard metric of the Heisenberg structure, and ϕ t X coincides with the standard flow.

Local homogeneity

The next step is to show that under our assumtion that the flow ϕ t X is not relatively compact, our manifold (M, g) is locally homogeneous. Proposition 6.1. Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not relatively compact in Iso(M, g). Then M consists in a single z X -orbit, hence is locally homogeneous. Proposition 6.1 will be proved if we can show there are no closed 2dimensional z X -orbits. Indeed, Proposition 4.2, together with Proposition 4.1 then imply that all z X -orbits in M int X have dimension 3, hence are open. Proposition 4.2 again shows that one of those z X -orbits must be closed, hence is the entire manifold M . By those remarks, Proposition 6.1 will just be a consequence of Lemmas 6.2 and 6.3 below. Lemma 6.2. Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not relatively compact in Iso(M, g). Then there are no non locally homogeneous hyperbolic components.

Proof: By Proposition 4.1, it is sufficient to prove that for any hyperbolic component M, the z X -orbits are not 2-dimensional. Assume for a contradiction that such M contains a point x having a 2-dimensional z Xorbit, denoted Σ. Let us denote by The Lie algebra Is X (x) is generated by a local Killing field Y around x, vanishing at x, and such that the flow

{D x φ t Y } ⊂ O(T x M
) is a hyperbolic flow. Linearizing Y around x thanks to the exponential map, we see there are two distinct lightlike directions u and v in T x M such that the two geodesics γ u : s → exp(x, su) and γ v : s → exp(x, sv) are left invariant by φ t X . In particular, for s = 0 close to 0, γu (s) and γv (s) are colinear to Y , hence tangent to the z X -orbits at γ u (s) and γ v (s) respectively. By continuity, this property must still hold for s = 0. We infer that u and v are tangent to Σ at x, hence Σ has Lorentz signature. By local homogeneity of Σ, the Gaussian curvature of Σ is constant. The Lie algebra z X (M) should then be isomorphic to the Lie algebra sol (case of curvature 0) or o(1, 2) (case of nonzero constant curvature). This yields a contradiction since the center of both sol and o(1, 2) is trivial (we have implicitely used the fact that a local Killing field on M which vanishes on Σ is trivial, what is easily proved using proposition 3.1). ♦ Lemma 6.3. Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not relatively compact in Iso(M, g). Then there are no non locally homogeneous parabolic components.

Proof:

If such a non locally homogeneous parabolic component M existed, then Proposition 4.2 would provide a kill loc -orbit Σ ⊂ M which is a closed surface. Proposition 5.2 ensures that Σ is endowed with an Heisenberg structure, for which the standard flow ψ t coincides with ϕ t X . Then, Proposition 5.1 shows that the restriction of ϕ t X to Σ is cyclic, namely there exists T > 0 such that ϕ T X restricts to identity on Σ. But one easily checks that a 1-parameter group in o(1, 2) fixing pointwise an hyperplane must be trivial, so that point (1) of Proposition 3.1 ensures that ϕ T X is the identity on M . This contradicts the fact that {ϕ t X } is not relatively compact in Iso(M, g). ♦ 7. Local models and the classification theorem 7.1. Three homogeneous 3-dimensional Lorentzian manifolds.

7.1.1. Lorentz-Heisenberg geometry. We call heis the 3-dimensional Heisenberg Lie algebra, namely the Lie algebra generated by X, Y, Z, with relation [Y, Z] = X. Let Heis be the connected, simply connected, associated Lie group. The Lorentz scalar product defined by < X, Y >= 1, < Z, Z >= 1, and all other products are zero, can be left-translated on Heis to give an homogeneous Lorentz metric g LH called the Lorentz-Heisenberg metric on Heis.

The 1-dimensional group of automorphism I h of Heis, which acts on heis by the matrices   e t 0 0 0 e t 0 0 0 1   , t ∈ R (in the basis Y, Z, X) acts isometrically on (Heis, g LH ). The group G h generated by I h and the left translations coincides with identity component of Iso(Heis, g LH ). Its Lie algebra g h is generated by X, Y, Z, and a fourth element T satisfying [T, Y ] = Y , [T, Z] = -Z and [T, X] = 0. It is isomorphic to the semi-direct product R h heis (the subscript h means that the R-factor integrates into a group of hyperbolic automorphisms of heis). We call X LH the pair (Heis, g LH ), modelling the Lorentz-Heisenberg geometry. Notice that X LH = G h /I h as an homogeneous space.

7.1.2. The geometry X h . We still consider the group G h , and we call I u the Lie subgroup generated by Y + Z. It is easily checked that the adjoint action of I u on g h /I u is unipotent and preserves a Lorentz scalar product. This defines a G h -invariant Lorentz metric on G h /I u (or more accurately, a 2-parameter family of such metrics, since there is such a family of Lorentz scalar products preserved by Ad(I u )). The corresponding Lorentzian homogeneous space G h /I u will be denoted by X h . The identity component of its isometry group is G h . 7.1.3. The geometry X e . We now consider a different 4-dimensional Lie algebra g e , generated by T, X, Y, Z, and defined in the following way. The elements X, Y, Z generate a Lie algebra isomorphic to heis, with bracket relation [Y, Z] = X. The element T normalizes this subalgebra, and the relations are

[T, Y ] = -Z, [T, Z] = Y , [T, X] = 0.
The algebra g e is again a semi-direct product R e heis (subscript e indicates that the R-factor induces a group of elliptic automorphisms on heis). Let I be the 1-dimensional group generated by Y + Z. As in the previous example, it is easily checked that Ad(I) acts on g e /I by unipotent transformations, preserving a Lorentzian scalar product (actually, again a 2-parameters family of such scalar products). This yields a homogeneous Lorentzian manifold G e /I, denoted by X e . The identity component of its isometry group is G e . 7.1.4. Natural foliations, and characteristic flow. We discuss now some remarkable geometric features of the spaces X h and X e .

The groups G h and G e have a derived subgroup H isomorphic to Heis. The center of H is a 1-parameter subgroup {z t }, which coincides with the center of G h (resp. G e ). It follows that z t induces a well defined flow on any manifold M locally modelled on X LH , X h or X e . This flow will be called the characteristic flow of the structure.

The foliation of G h (resp. G e ) the leaves of which are given by the classes gH, induces a G h -invariant (resp. G e -invariant ) codimension one foliation F on X h (resp. X e ). All leaves are lightlike and naturally endowed with an Heisenberg structure (see Section 5.1).

Let α be the degenerate bilinear form on g h (resp. g e ) such that X, Y, Z generate the Kernel of α, and α(T, T ) = 1. Observe that the adjoint action of I h (resp. of I) leaves α invariant, resulting on a G h (resp. G e ) invariant degenerate metric on X h (resp . X e ). It follows that any manifold M locally modelled on X h or X e is naturally endowed with a metric α of signature (+, 0, 0), such that the foliation F integrates the Kernel of α. This yields a transverse Riemannian structure for the foliation F. 7.2. Identifying the Lie algebra z X and the homogeneous models. We are now in position to describe all the possible local homogeneous models, for a Lorentzian manifold (M 3 , g) with a non-relatively compact isometric flow.

Proposition 7.1. Let (M 3 , g) be a smooth, closed, 3-dimensional Lorentz manifold. Let X be a Killing field on M generating a flow ϕ t X which is not relatively compact in Iso(M, g). Then either ϕ t X preserves a metric of constant curvature g 0 on M , or the geometry (M, g) is locally modelled on X LH , X h or X e , and ϕ t X coincides with the characteristic flow. The list of local models is pretty short, and will be shortened even more in the next section.

Obtaining those local models amounts to determine the possibilities for the algebra z X , as well as for the isotropy algebras. It turns out that this can be obtained by very simple algebraic computations, similar to those made in the proof of Proposition 5.2 (we keep the notations of this proof in the following).

Let us start with the computations when the isotropy algebra is hyperbolic (namely M coincides with a hyperbolic component). We pick x ∈ M , x ∈ M in the fiber of X, and T, X, Y, Z generating z X (and Y generating the isotropy). We can choose T, Y, Z and x so that the following relations hold at x.

ω(X) = h + αH, ω(Y ) = H, ω(Z) = e + βE + γF, ω(T ) = f + λE + µF,
for some real numbers α, β, γ, λ, µ.

The curvature κ(x) is invariant under the isotropy, namely the flow {e tH }, hence of the form κ(x) = σκ 0 + bκ 2 (see notation in Section 2.2.1).

In particular, at x, κ(e ∧ h) = (σ + b)E, κ(e ∧ f ) = (σ -2b)H, and

κ(h ∧ f ) = (σ + b)F . Recall the equation (3) ω([U, V ]) = K(U, V ) -[ω(U ), ω(V )]
available for every pair U, V of local Killing fields around x. We first express that X is in the center of z X (all equalities below are available at x). We use these relations to compute

ω([X, T ]) = 0 = κ(h ∧ f ) + [f + λE + µF, h + αH] = λe + (α -µ)f -αλE leading to the extra relations λ = 0, α = µ.
From the simplified relations ω(X) = h + αH, ω(Z) = e + αE and ω(T ) = f + αF , one proceeds writing

ω([Z, T ]) = κ(e ∧ f ) + [f + αF, e + αE] = 2αω(X) + (3σ -α 2 )ω(Y ), ω([Y, Z]) = -[H, e + αE] = -ω(Z) ω([Y, T ]) = -[H, f + αF ] = ω(T ).
Because two Killing fields which coincide at x coincide, we obtain the bracket relations for z X . The possibilities split into three cases.

•

] = Y , [Y , Z] = -(3σ -α 2 )Z, [Y , T ] = 3(3σ -α 2 )T .
We see that Y , Z, T generate a Lie algebra isomorphic to sl(2, R), that acts locally freely on M . The Lie algebra z X is isomorphic to sl(2, R) ⊕ R. We conclude that we have a ϕ t X -invariant (G, X)-structure on M , where X = PSL(2, R), and G = PSL(2, R) × { ht }. Here { ht } is the 1-dimensional subgroup of PSL(2, R) obtained by exponentiating 1 0 0 -1 . Because the group PSL(2, R) × { ht } preserves the bi-invariant metric on PSL(2, R) coming from the Killing form, it follows that ϕ t X preserves a Lorentz metric g 0 of constant curvature -1 on M . Similar computations can be carried out to determine z X in the case where the isotropy is parabolic. Actually, those computations were already detailed in [START_REF] Frances | Variations on gromov's open-dense orbit theorem[END_REF]Section 4.3.3], where all possibilities for local geometries having a 4-dimensional Lie algebra of Killing fields, and a parabolic isotropy, were listed. We restrict here to Lie algebras having a nontrivial center (which are the only candidates for z X ). Again, if we rule out the possibility of (M, g) being of constant curvature, [START_REF] Frances | Variations on gromov's open-dense orbit theorem[END_REF]Prop 4.3] yields three more possibilities for the Lie algebra z X .

• Case 4. The Lie algebra z X is isomorphic to g h (and the isotropy is parabolic). The local model for (M, g) is then X h . • Case 5. The Lie algebra z X is isomorphic to g e . The local model for (M, g) is then X e . • Case 6. The Lie algebra z X is isomorphic to sl(2, R) ⊕ R. This time the isotropy is parabolic, so that the local model for (M, g) is PSL(2, R), endowed with a Lorentz metric invariant under PSL(2, R)× {ũ t }. Here {ũ t } is the 1-dimensional subgroup of PSL(2, R) obtained by exponentiating 0 1 0 0 . Again, there is then a Lorentz metric g 0 of constant curvature -1 on M , which is invariant under ϕ t X .

7.3. Completeness results. The final step toward the classification theorem 1.2 is the global understanding of the manifold M . In this section, X will denote indistinctly X h , X e or X LH . Any Lorentzian manifold (M, g) locally isometric to X is automatically endowed with a (Iso(X), X)-structure (see [START_REF] Thurston | Three-dimensional geometry and topology[END_REF] for more material on this notion). Such a (G, X)-structure on M gives rise to a developing map δ : M → X and a holonomy morphism ρ : Iso( M , g) → Iso(X).

Considering a finite cover of (M, g) we may assume that (M, g) is endowed with a (G h , X h ), (G h , X HL ) or (G e , X e )-structure.

Of crucial importance then, are completeness results for the (G, X)-structures we are considering. For the geometries X e , X h and X LH , this completeness is respectively proved in [DZ10, Prop 7.3] and [DZ10, Section 8.1]. Observe that the proof given in [START_REF] Dumitrescu | Géométries lorentziennes de dimension 3: classification et complétude[END_REF], of the completeness of compact manifolds locally modelled on the Lorentz-Heisenberg geometry uses Theorem 1.2. To avoid any vicious circle, we give below an independent proof (under our standing asumption that ϕ t X is not relatively compact), which follows the arguments of [Ghy87, Thm. 5.7].

Proposition 7.2. Any compact Lorentzian manifold locally modelled on X LH , X h or X e , and for which the characteristic flow is not relatively compact is complete. Namely, the structure is the quotient of the model space by a discrete subgroup of isometries.

Proof:

We first sketch the arguments for structures modelled on X h or X e . In Section 7.1.4, we explained that any M modelled on X h or X e is endowed with a lightlike, codimension 1 foliation F, the leaves of which have a Heisenberg structure. The foliation F integrates the distibution X ⊥ . Replacing M by a double cover, we can pick W a global vector field on M such that g(W, W ) = 1 and g(W, X) = 0. This vector field W is complete, and the proof of Proposition 5.1 works verbatim to show that the Heisenberg structure of each leaf of F is complete. Taking again a double cover of M , we can also pick another vector field W on M which is transverse to F and satisfies α(W , W ) = 1 (here α is the degenerate metric of signature (+, 0, 0) introduced in Section 7.1.4 and defining the transverse Riemannian structure of F. Then, just as in the proof of Proposition 5.1, we get that if F is any lift of a leaf F to the universal cover M , and if ψ t is the flow of the lift of W to M , then ∪ t∈R ψ t ( F ) is mapped diffeomorphically on X e (resp. X h ) by the developing map δ : M → X e (resp. X h ). One easily deduced that δ has the path-lifting property, hence is a diffeomorphism from M to X e (resp. X h ).

Let us now consider the case of Lorentz-Heisenberg geometry. We have again a developing map δ : M → X LH . We recall the basis T, X, Y, Z introduced in Section 7.1.1. Our remark in the study of case 2 (Section 7.2) says that for every g ∈ Heis, the curves t → e tY and t → e tZ are lightlike geodesics for the Lorentz-Heisenberg metric. Those geodesics are orthogonal to the flow generated by the center z t . Our hypothesis that the characteristic flow ϕ t X is not relatively compact ensures that lightlike geodesics orthogonal to X in M are complete (Lemma 3.2). It follows that any piecewise smooth curve of Heis, obtained by successively flowing along the right multiplication under e tY , e tZ and z t = e tX , can be lifted to M through the map δ, with arbitrary initial condition. Let us fix > 0 such that the map f : (r, s, t) → Heis defined by f (r, s, t) = e rY e sZ z t yields a diffeomorphism from (-, ) × (-, ) × (-, ) to its image B . Our previous remark about completeness ensures that whenever x is a point of M , such that δ(x) = g, then gB can be lifted to U containing x, such that δ : U → gB is a diffeomorphism. One easily deduces that δ : M → X LH has the path-lifting property, hence is a diffeomorphism. ♦ Corollary 7.3. (see [Zeg96, Section 14]) For every compact Lorentzian manifold locally modelled on X LH , X h or X e , the characteristic flow is relatively compact.

Proof: Our manifold (M, g) is endowed with a (G, X) structure, where X is X LH , X h or X e , and G is G h or G e . We denote by Γ the image of π 1 (M ) by the holonomy morphism ρ. Because we have shown that the developing map δ is a diffeomorphism, Γ is a discrete subgroup of G. The flow ϕ t X is mapped by ρ to the center {z t } of G.

The algebraic structure is in all the three cases that of a semi-direct product R Heis. Moreover, G is always the identity component of an algebraic group, so that it makes sense to consider Γ, the identity component of the Zariski closure of Γ in G.

• Case 1: Γ ∩ Heis = {1}. In this case Γ projects injectively on R, hence is abelian. The only connected abelian subgroups of G have dimension 1 or 2. If the dimension of Γ is one, then Γ is the exponentiation of some U ∈ g. Observe that U is centralized by an index 2 subgroup of Γ. If U lies in the center of g, then ϕ t X is a periodic flow. If U is not in the center of g, then U gives rise to a Killing field on M which is not proportional to X and commutes with X. Lemma 3.2 says that ϕ t X is relatively compact. If the dimension of Γ is 2, we directly get that γ contains the center {z t }, so that {z t } is relatively compact in Γ/Γ. It follows again that ϕ t X is relatively compact.

• Case 2: Γ ∩ Heis = {1}. In this case Γ = Γ ∩ Heis is a nontrivial, discrete, normal subgroup of Γ. Either Γ is cyclic, in which case it is included in a 1-parameter group of Heis. If this group is the center, then ϕ t X is periodic. If this group is not the center, we again get a Killing field on M which commutes with X, without being proportional to X, so that ϕ t X is relatively compact by Lemma 3.2. Any non-cyclic discrete subgroup of Heis intersects the center nontrivially. hence, if Γ is not cyclic, then ϕ t X is periodic. ♦ 7.4. Structures with constant curvature and classification. All our work so far says that whenever (M, g) is a closed 3-dimensional Lorentz manifold, and ϕ t X is a non relatively compact isometric flow, then either g is flat, or ϕ t X preserves a metric g 0 with constant curvature -1 on M . We are thus reduced to study structures of constant curvature having a non relatively compact flow of isometries.

Thanks to the work of Y. Carrière [START_REF] Carrière | Autour de la conjecture de L. Markus sur les variétés affines[END_REF] (curvature zero) and B. Klingler [START_REF] Klingler | Complétude des variétés lorentziennes à courbure constante[END_REF] (arbitrary curvature), closed Lorentz manifolds of constant curvature are complete, namely are obtained as a quotient of the 1-connected, complete model space of constant curvature by a discrete subgroup Γ of isometries (it follows that the curvature has to be zero or negative). Those completeness results are deeper, and more difficult, than the ones presented above for geometries X e , X h , X LH , and we can't present them here. They will allow us to recover the classification of Theorem 1.2. 7.4.1. Flat structures. The model space in this case is Minkowski space R 1,2 , namely R 3 endowed with the metric dx 1 dx 3 +dx 2 2 . The flow ϕ t X has constant norm (Lemma 3.2), hence its orbits are geodesics. It follows that ϕ t X is mapped by the holonomy morphism to a flow z t of translations, which may be spacelike or lightlike (the timelike case is ruled out by lemma 3.2). We thus may assume, after conjugating in Iso(R 1,2 ), that z t is the flow of lightlike translations T t e 1 , or of spacelike translations T t e 2 . The following result states the completeness of closed flat Lorentz manifolds ([Car89]), as well as a Bieberbach type theorem proved in [START_REF] Fried | Three-dimensional affine crystallographic groups[END_REF] (see also [START_REF] Goldman | The fundamental group of a compact flat Lorentz space form is virtually polycyclic[END_REF]): Theorem 7.4 (Bieberbach's theorem for flat Lorentz manifolds). Let (M, g) be a closed, 3-dimensional, flat Lorentz manifold. There exists a discrete subgroup Γ ⊂ Iso(R 1,2 ) such that (M, g) is isometric to the quotient Γ\R 1,2 . Moreover, there exists a connected 3-dimensional Lie group G ⊂ Iso(R 1,2 ), which is isomorphic to R 3 , Heis or SOL, and which acts simply transitively on R 1,2 , satisfying that Γ 0 = G ∩ Γ has finite index in Γ and is a uniform lattice in G.

Considering a finite cover of (M, g) we may assume Γ = Γ 0 in the previous statement. The centralizer of T t e 1 in Iso(R 1,2 ) is the following 4-dimensional Lie group

G 1 =      1 s -s 2 2 0 1 -s 0 0 1   +   t u v   | (s, t, u, v) ∈ R 4    .
The centralizer of T t e 2 in Iso(R 1,2 ) is the following 4-dimensional Lie group There are no copies of SOL in the group G 1 , and all copies of R 3 and Heis contain the translations along e 1 , namely the 1-parameter group {z t }. We infer that any lattice Γ in such a group meets {z t } non-trivially, implying that ϕ t X is cyclic. We thus rule out this case, and focus on the case where ϕ t X is spacelike, and the centralizer of {z t } is G 2 . There are no copies of Heis in G 2 . The only copy of R 3 is the subgroup of transations, hence any lattice Γ in it will meet {z t } non-trivially. We again get a cyclic flow ϕ t The action of the flow T t e 2 on the quotient Γ\G is (after a possible reparametrization T αt e 2 , α ∈ R * ) the suspension of a hyperbolic toral automorphism. We are in the first case of Theorem 1.2. 7.4.2. Anti-de Sitter structures. We investigate now the case when ϕ t X preserves a Lorentz metric g 0 of curvature -1 on M . Klingler's completeness theorem [START_REF] Klingler | Complétude des variétés lorentziennes à courbure constante[END_REF] ensures that (M, g 0 ) is the quotient of ( PSL(2, R), gAdS (the universal cover of anti-de Sitter space, see Section 1.2.2) by a discrete group of isometries Γ. An important result, known as finiteness of level, says that Γ must intersect the center of PSL(2, R) non trivially. This property was first stated in [START_REF] Kulkarni | 3-dimensional Lorentz space-forms and Seifert fiber spaces[END_REF]. A detailed proof can be found in [Sal99, Theorem 3.3. Denoting by z t the image of ϕ t X by the holonomy morphism, we see that z t must centralize the group Γ. It is thus a flow of the form {id} × {u t } (in which case Γ ρ is of the form Γ ρu , see Section 1.2.2), or of the form {id}×{h t } (in which case Γ ρ is of the form Γ ρ h ).

The manifold (M, g 0 ) is isometric to a quotient Γ ρu \ PSL(2, R) (resp. Γ ρ h \ PSL(2, R)) endowed with a metric g AdS . The flow ϕ t X coincides with the right action of {h t } (resp. of {u t }). Regarding the initial metric g, it must come from a Ad(u t )-invariant (resp. Ad(h t )-invariant) Lorentz scalar product on sl(2, R). It is thus of the form g u or g AdS (resp. g h or g AdS ). This completes the proof of Theorem 1.2.

  DG91, Section 2]. 1.2. Some nice Lorentzian dynamical systems. We now illustrate what was said before on a few paradigmatic examples. 1.2.1. Suspensions of Anosov diffeomorphisms. We call sol the 3-dimensional Lie algebra generated by T, Y, Z, with bracket relations [T, Y ] = Y , [T, Z] = -Z and [Y, Z] = 0. By, SOL, we mean the associated connected, simply connected Lie group.

  ω([Z, X]) = 0 = κ(e ∧ h) + [h + αH, e + βE + γF ] = (α -β)e + γf + (σ + b + αβ)E + αγF which implies γ = 0, α = β, σ + b + α 2 = 0.

  Case 1: α and σ are zero. This implies b = 0 because of the relation σ + b + α 2 = 0, so that the geometry is flat. The only nontrivial bracket relations are [Y, Z] = -Z and [Y, T ] = T . The Lie algebra z X is isomorphic to sol ⊕R. • Case 2: α = 0 and α 2 = 3σ. Then we put X = 2αX, and we have [Z, T ] = X . It follows that Z, T, X generate a subalgebra isomorphic to heis, that acts locally freely on M , and z X is isomorphic to g h . Our manifold (M, g) is locally modelled on the Lorentz-Heisenberg geometry X LH . Let us make a further observation. Our computations led to the relations ω(Z) = e + αE and ω(T ) = f + αF . Because [E, e] = 0 = [F, f ], we get that the curves t → e tY and t → e tZ are (lightlike) geodesics for the Lorentz-Heisenberg metric. • case 3: α = 0 and α 2 = 3σ. Then we can define Y = 2αX + (3σ -α 2 )Y , and the bracket relations become [Z, T

  X in this case, what contradicts our hypothesis of non-compactness. The only remaining case is when G < G 2 is isomorphic to SOL. It is thus a group of the form β ∈ R * .

  2.3].It follows that up to finite cover, (M, g) is actually isometric to a quotient of (PSL(2, R), g AdS ) by a discrete subgroup Γ ⊂ PSL(2, R) × PSL(2, R). The structure of the group Γ is well understood. Up to conjugacy, there exists Γ 0 a uniform lattice in PSL(2, R), and a representation ρ: Γ 0 → PSL(2, R) such that Γ = Γ ρ = {(γ, ρ(γ)) ∈ PSL(2, R) × PSL(2, R) | γ ∈ Γ 0 }.This was established in [KR85, Theorem 5.2] when Γ is torsion-free. For a group with torsion, the adapted proof can be found in [Tho14, Lemma 4.3.1].