Rym Smaï

Enveloping space of a globally hyperbolic conformally flat spacetime

We prove that any simply-connected globally hyperbolic conformally flat spacetime V can be conformally embedded in a bigger conformally flat spacetime, called enveloping space of V , containing all the conformally flat Cauchy-extensions of V , in particular its C 0 -maximal extension. As a result, we establish a new proof of the existence and the uniqueness of the C 0 -maximal extension of a globally hyperbolic conformally flat spacetime. Furthermore, this approach allows us to prove that C 0 -maximal extensions respect inclusion.

 Definitions 2 and 4]. Spacetimes of constant curvature are examples of rigid categories.

1

to Einstein equations constitute a rigid category and it turns out that the arguments of Choquet-Bruhat and Geroch could be adapted to any other rigid category.

 Sections 3.2 & 3.3]). Her proof is mainly based on Zorn lemma and so does not give any description of the C 0 -maximal extension. In this paper, we propose a new approach which allows us to give a constructive proof of the existence and the uniqueness of the C 0 -maximal extension. Indeed, given a simply-connected GH conformally flat spacetime M , we construct a bigger conformally flat spacetime E(M ) in which M and all its conformally flat Cauchy-extensions embeds conformally. The C 0 -maximal extension of M turns out to be the Cauchy development of a Cauchy hypersurface of M in E(M ). The images in E(M ) of the previous embeddings satisfy the nice property of being causally convex. A subset U of a spacetime is causally convex if any causal curve joining two points of U is contained in U . While convexity is a metric notion, causal convexity is a conformal notion. Let us add that causal convexity is a strong property in a GH spacetime: it is a classical fact that any causally convex open subset of a GH spacetime is GH. Theorem 1. Let M be a simply-connected globally hyperbolic conformally flat spacetime. There exists a conformally flat spacetime E(M ) with the following properties: 1. E(M ) fibers trivially over a conformally flat Riemannian manifold B diffeomorphic to any Cauchy hypersurface of M ; 2. M embeds conformally in E(M ) as a causally convex open subset; 3. all the conformally flat Cauchy-extensions of M embeds conformally in E(M ) as causally convex open subsets. In particular, the C 0 -maximal extension of M is the Cauchy development of a Cauchy hypersurface of M in E(M ). Such a spacetime E(M ) is called an enveloping space of M .

This result still holds for the larger class of developable GH conformally flat spacetimes (see Definition 11). In Section 4.3, we describe causally convex open subsets of an enveloping space E(M ) then, in Section 7, we characterize those which are C 0 -maximal.

A consequence of Theorem 1 is the following result: Corollary 1. Any globally hyperbolic conformally flat spacetime admits a C 0 -maximal extension, unique up to conformal diffeomorphism.

Globally hyperbolic spacetimes 2.1 Preliminaries on spacetimes

The aim of this preliminary section is to introduce the concept of causality in a Lorentzian manifold and briefly recall some basic causal notions as causal curves, future and past of points, lightcones, achronal and acausal subsets, etc.

Introduction

The notion of maximal extension of a globally hyperbolic spacetime arises from the resolution of Einstein equations in general relativity. This physical theory suggests that our universe is modelized by a Lorentzian manifold (M, g) of dimension 4 where the metric g satisfies some PDEs, the so-called Einstein equations. A way to solve them is to require that M is homeomorphic to S × R where S is a Riemannian manifold. This allows to define a Cauchy problem where the initial data is the Riemannian manifold (S, h) equipped with a (2, 0)-tensor II. A solution is a Lorentzian metric g on S × R such that the restriction of g to S × {0} is h and II is the shape operator of this hypersurface. It turns out that a necessary condition to have such a solution is that h and II satisfy the constraint equations. Conversely, Choquet-Bruhat and Geroch [12, Section 2] proved that when the constraint equations are satisfied, a local solution exists. Two natural questions arise: is it possible to extend this solution to a maximal one? If yes, is it unique up to isometry? Choquet-Bruhat and Geroch answered positively to both questions (see [START_REF] Geroch | Global aspects of the cauchy problem in general relativity[END_REF]Theorem 3]).

The solutions of the Cauchy problem associated to Einstein equations turn out to be globally hyperbolic (abbrev. GH) (see Definition 2). More generally, Geroch [START_REF] Paul | The domain of dependence[END_REF] proved that any GH spacetime admits an embedded Riemannian hypersurface which intersects every inextensible causal curve exactly once, called Cauchy hypersurface. It turns out that all smooth Cauchy hypersurfaces of a GH spacetime are diffeomorphic.

There is a natural partial ordering on GH spacetimes: given two GH spacetimes, M and N , we say that N is a Cauchy-extension of M if there exists an isometric embedding from M to N sending every Cauchy hypersurface of M on a Cauchy hypersurface of N . Such an embedding is called an isometric Cauchy-embedding. We can ask again in this general setting the questions of the existence and the uniqueness, up to isometry, of a maximal extension. The answer to both questions is yes within a rigid category 1 of spacetimes. Actually, the spacetimes which are solution of a Cauchy problem associated Now we ask the following question. Let V be a globally hyperbolic conformally flat spacetime and let U be a causally convex open subset of V . Does the C 0 -maximal extension of U embed conformally in the C 0 -maximal extension of V ?

The C 0 -maximal extensions of U and V are a priori abstract objects which depends on the Cauchy hypersurfaces of U and V respectively. These last ones are completely independant so the question above is not tautological. We prove in Section 6 that the answer is yes, in other words, that C 0 -maximal extensions preserve inclusion. Theorem 2. Let V be a globally hyperbolic conformally flat spacetime and let U be a causally convex open subset of V . Then, the C 0 -maximal extension of U is conformally equivalent to a causally convex open subset of the C 0 -maximal extension of V .

Overview of the paper

In Section 2, we introduce the premiminary material on causality of spacetimes. We focus in particular on globally hyperbolic spacetimes and we recall some of their main properties. Section 3 deals with the model space of conformally flat Lorentzian structure, the so-called Einstein universe. After a quick description of its geometry, we characterize causally convex open subsets of its universal cover (see Sections 3.5 and 3.6). We devote Section 4 to the proof of Theorem 1: we construct an enveloping space (see Section 4.2) and we describe its causally convex globally hyperbolic open subsets (see Section 4.3). In Section 5, we propose a new proof of the existence and the uniqueness of the maximal extension of a globally hyperbolic conformally flat spacetime, using the notion of enveloping space. Section 6 is devoted to the proof of Theorem 2. Lastly, we establish a link between the notion of C 0 -maximality and the notion of eikonal functions in Section 7.

Throughout this paper, we denote by (p, q) the signature of a non-degenerate quadratic form Q where p and q are respectively the number of negative and positive coefficients in the polar decomposition of Q.

Spacetimes.

A Lorentzian metric on a manifold of dimension n is a non-degenerate symmetric 2-tensor g of signature (1, n -1). A manifold equipped with a Lorentzian is called Lorentzian.

In a Lorentzian manifold (M, g), we say that a non-zero tangent vector v is timelike, lightlike, spacelike if g(v, v) is respectively negative, zero, positive. The set of timelike vectors is the union of two convex open cones. When it is possible to make a continuous choice of a connected component in each tangent space, the manifold M is said timeorientable. The timelike vectors in the chosen component are said future-directed while those in the other component are said past-directed. A spacetime is an oriented and time-oriented Lorentzian manifold. Future, past. In a spacetime M , a differential curve is timelike, lightlike, spacelike if its tangent vectors are timelike, lightlike, spacelike. It is causal if its tangent vectors are either timelike or lightlike.

Given a point p in M , the future (resp. chronological future) of p, denoted J + (p) (resp. I + (p)), is the set of endpoints of future-directed causal (resp. timelike) curves starting from p. More generally, the future (resp. chronological future) of a subset A of M , denoted J + (A) (resp. I + (A)), is the union of J + (a) (resp. I + (a)) where a ∈ A.

An open subset U of M is a spacetime and the intrinsic causality relations of U imply the corresponding ones in M . We denote J + (A, U ) (resp. I + (A, U )) the future (resp. chronological future) in the manifold U of a set A ⊂ U . Then, I + (A, U ) ⊂ I + (A) ∩ U .

Dual to the preceding definitions are corresponding past versions. In general, past definitions and proofs follows from future versions (and vice versa) by reversing timeorientation.

Diamonds. We call diamond of M any intersection J -(p) ∩ J + (q), where p, q ∈ M such that p ∈ J + (q). We denote it J(p, q). Given two points p, q ∈ M such that p ∈ I + (q), the interior of the diamond J(p, q) is the intersection I -(p) ∩ I + (q) and is denoted I(p, q) (see [START_REF] Barrett | Semi-Riemannian geometry with applications to relativity[END_REF]Lemma 6,p.404]).

Achronal, acausal subsets.

A subset A of a spacetime M is called achronal (resp. acausal) if no timelike (resp. causal) curve intersects A more than once.

Causal convexity.

In Riemannian geometry, it is often useful to consider open neighborhoods which are geodesically convex. In Lorentzian geometry, there is, in addition, a causal convexity notion. A subset U of M is said causally convex if for every p, q ∈ U , any causal curve of M joining p to q is contained in U . Equivalently, if every diamond J(p, q) of M with p, q ∈ U is contained in U . It is easy to check that the intersection of two causally convex subsets is causally convex.

Cauchy developments.

Let A be an achronal subset of M . The future (resp. past) Cauchy development of A, denoted C + (A) (resp. C -(A)), is the set of points p of M such that every past-inextensible (resp. future-inextensible) causal curve through p meets A. The Cauchy development of A is the union of C + (A) and C -(A), denoted C(A).

Global hyperbolicity

all diamonds of M are compact.

It was proved by Sanchez in [START_REF] Antonio | Globally hyperbolic spacetimes can be defined as 'causal'instead of 'strongly causal[END_REF] that the first condition can be weaken to M is causal, that is M contains no causal loop.

A classical result of Geroch [START_REF] Paul | The domain of dependence[END_REF], later improved by Bernal and Sanchez [START_REF] Bernal | On smooth cauchy hypersurfaces and geroch's splitting theorem[END_REF], gives a characterization of global hyperbolicity involving the notion of Cauchy hypersurface. Definition 3. A topological (resp. smooth) Cauchy hypersurface is an achronal topological hypersurface (resp. an embedded Riemannian hypersurface) that is met by exactly once by every inextensible causal curve of M .

Theorem 3 ([6]). A spacetime M is globally hyperbolic if and only if it contains a topological Cauchy hypersurface.

Bernal and Sanchez [START_REF] Bernal | On smooth cauchy hypersurfaces and geroch's splitting theorem[END_REF] improved this result by proving the existence of a smooth Cauchy hypersurface.

Topological (resp. smooth) Cauchy hypersurfaces of a globally hyperbolic spacetime are homeomorphic (resp. diffeomorphic). Therefore, one can set the following definition.

Definition 4. A globally hyperbolic spacetime is said Cauchy-compact (or spatially compact) if it admits a compact Cauchy hypersurface.

A remarkable property of globally hyperbolic spacetimes is that causal convexity implies global hyperbolicity.

Proposition 1. Let M be a globally hyperbolic spacetime. Then, any causally convex open subset of M is globally hyperbolic.

Proof. Since M is globally hyperbolic, there is no causal loop in U . Since U is causally convex, the diamonds of U are exactly the diamonds of M contained in U . Thus, they are compact.

Shadows

In this section, we show that the causal structure of globally hyperbolic spacetimes is encoded by compact subsets of a Cauchy hypersurface, called shadows. 2Let M be a globally hyperbolic spacetime and let S ⊂ M be a Riemannian Cauchy hypersurface. Definition 5. Let p ∈ M . We call shadow of p on S, denoted by O(p, S), the set of points in S which are causally related to p. When there is no confusion about the Cauchy hypersurface S, we will simply write O(p) instead of O(p, S).

If p ∈ I ± (S) , then O(p, S) = J ∓ (p) ∩ S; if p ∈ S, O(p, S
) is reduced to {p}. Thus, by [START_REF] Barrett | Semi-Riemannian geometry with applications to relativity[END_REF]Lemma 40,p.423]), shadows are compact.

The main interest of the notion shadows is given by the following proposition proved by C. Rossi in her thesis (see [START_REF] Rossi | Espace-temps globalement hyperboliques conformément plats[END_REF]Prop. 2.6,chap. 4]).

Proposition 2. Suppose S is not compact. Then, two points p and q of M in the chronological future of S coincide if and only if their shadows on S are equal.

By Proposition 2, the shadows on S characterize completely the points of the globally hyperbolic spacetime M . This allows to reduce, in some situations, the study of the spacetime to the study of compact subsets of a Riemannian manifold.

Geometry of Einstein universe

In this section, we introduce the model space of conformally flat Lorentzian structures, the so-called Einstein universe, and we describe its causal structure.

The Klein model

Let R 2,n be the vector space R n+2 of dimension (n + 2) equipped with the nondegenerate quadratic form q 2,n of signature (2, n) given by

q 2,n (u, v, x 1 , . . . , x n ) = -u 2 -v 2 + x 2 1 + . . . + x 2
n in the coordinate system (u, v, x 1 , . . . , x n ) associated to the canonical basis of R n+2 . Definition 6. Einstein universe of dimension n, denoted by Ein 1,n-1 , is the space of isotropic lines of R 2,n with respect to the quadratic form q 2,n , namely

Ein 1,n-1 = {[x] ∈ P(R 2,n ) : q 2,n (x) = 0}.
In practice, it is more convenient to work with the double cover of the Einstein universe, denoted by Ein 1,n-1 :

Ein 1,n-1 = {[x] ∈ S(R 2,n ) : q 2,n (x) = 0}
where S(R 2,n ) is the sphere of rays, namely the quotient of R 2,n \{0} by positive homotheties.

Spatio-temporal decomposition of Einstein universe

The choice of a timelike plane of R 2,n , i.e. a plane on which the restriction of q 2,n is negative definite, defines a spatio-temporal decomposition of Einstein universe: Lemma 1. Any timelike plane P ⊂ R 2,n defines a diffeomorphism between S n-1 × S 1 and Ein 1,n-1 .

Proof. Consider the orthogonal splitting R 2,n = P ⊥ ⊕P and call q P ⊥ and q P the positive definite quadratic form induced by ±q 2,n on P ⊥ and P respectively. The restriction of the canonical projection R 2,n \{0} on S(R 2,n ) to the set of points (x, y) ∈ P ⊥ ⊕ P such that q P ⊥ (x) = q P (y) = 1 defines a map from S n-1 × S 1 to Ein 1,n-1 . It is easy to check that this map is a diffeomorphism.

For every timelike plane P ⊂ R 2,n , the quadratic form q 2,n induces a Lorentzian metric g P on S n-1 × S 1 given by

g P = dσ 2 (P ) -dθ 2 (P )
where dσ 2 (P ) is the round metric on S n-1 ⊂ (P ⊥ , q P ⊥ ) induced by q P ⊥ and dθ 2 (P ) is the round metric on S 1 ⊂ (P, q P ) induced by q P .

An easy computation shows that if P ⊂ R 2,n is another timelike plane, the Lorentzian metric g P is conformally equivalent to g P , i.e. q P and g P are proportionnal by a positive smooth function on S n-1 × S 1 . As a result, Einstein universe is naturally equipped with a conformal class of Lorentzian metrics. This Lorentzian conformal structure induces causality on Einstein universe. Indeed, changing the metric in the conformal class consists in multiplying by a positive function and so does not change the sign of the norm of a tangent vector. The causal structure of Einstein universe is trivial: any point is causally related to any other one (see e.g. [9, Cor. 2.10, Chap. 2]).

Let us point out that in general geodesics are not well-defined in a conformal spacetime. Indeed, a computation of the Levi Civita connexion shows that geodesics are not preserved by conformal changes of metrics. Nevertheless, lightlike geodesics are preserved as non-parametrized curves (see e.g. [4, Théorème 3]).

Causal structure of the universal cover

Let Ein 1,n-1 be the universal cover of Ein 1,n-1 . When n ≥ 3, every diffeomorphism between Ein 1,n-1 and S n-1 ×S 1 lifts to a diffeomorphism between Ein 1,n-1 and S n-1 ×R. The pull-back by the projection

S n-1 × R → S n-1 × S 1 of the conformal class [dσ 2 -dθ 2 ] on S n-1 × S 1 defined previously is the conformal class of the Lorentzian metric dσ 2 -dt 2
where dt 2 is the usual metric on R. This induces a natural conformally flat Lorentzian structure on Ein 1,n-1 . Definition 7. We call spatio-temporal decomposition of Ein 1,n-1 any conformal diffeomorphism between Ein 1,n-1 and S n-1 × R.

In what follows, we fix a spatio-temporal decomposition and we identify Ein 1,n-1 to

S n-1 × R.
The fundamental group of Ein 1,n-1 is isomorphic to Z, generated by the transformation δ : Ein 1,n-1 → Ein 1,n-1 defined by δ(x, t) = (x, t + 2π). That of Ein 1,n-1 is generated by the transformation σ : Ein 1,n-1 → Ein 1,n-1 such that σ 2 = δ, i.e. the map defined by σ(x, t) = (-x, t + π). Definition 8. Two points p and q of Ein 1,n-1 are said to be conjugate if one is the image under σ of the other.

While the causal structure of Ein 1,n-1 is trivial, the causal structure of Ein 1,n-1 is rich. We give a brief description below. We direct to [9, Chap. 2] for more details.

Lightlike geodesics of Ein 1,n-1 are the curves which can be written, up to reparametrization, as (x(t), t) where x : I → S n-1 is a geodesic of S n-1 defined on an interval I of R. The inextensible ones are those for which x is defined on R.

It turns out that the photons going through a point (x 0 , t 0 ) have common intersections at the points σ k (x 0 , t 0 ), for k ∈ Z; and are pairwise disjoint outside these points. The lightcone of a point (x 0 , t 0 ) is the set of points (x, t) such that d(x, x 0 ) = |t -t 0 | where d is the distance on the sphere S n-1 induced by the round metric. It disconnects Ein 1,n-1 in three connected components:

• The chronological future of (x 0 , t 0 ): this is the set of points (x, t) of S n-1 × R such that d(x, x 0 ) < t -t 0 .

• The chronological past of (x 0 , t 0 ): this is the set of points (x, t) of S n-1 × R such that d(x, x 0 ) < t 0 -t.

• The set of points non-causally related to (x 0 , t 0 ), i.e. the set of points (x, t) of We denote it Mink 0 (x 0 , t 0 ).

S n-1 × R such that d(x, x 0 ) > |t -t 0 |.
There are two other affine charts associated to the point (x 0 , t 0 ), namely:

• the set of points non-causally related to σ(x 0 , t 0 ), contained in the chronological future of (x 0 , t 0 ), denoted Mink + (x 0 , t 0 );

• the set of points non-causally related to σ -1 (x 0 , t 0 ), contained in the chronological past of (x 0 , t 0 ), denoted Mink -(x 0 , t 0 ).

The universal cover Ein 1,n-1 is globally hyperbolic: any sphere S n-1 × {t}, where t ∈ R, is a Cauchy hypersurface.

Conformal group

The subgroup O(2, n) ⊂ Gl n+2 (R) preserving q 2,n , acts conformally on Ein 1,n-1 . When n ≥ 3, the conformal group of Ein 1,n-1 is exactly O (2, n). This is a consequence of the following result, which is an extension to Einstein universe, of a classical theorem of Liouville in Euclidean conformal geometry (see e.g. [START_REF] Frances | Une preuve du théorème de Liouville en géométrie conforme dans le cas analytique[END_REF]): It is a classical fact that every conformal diffeomorphism of Ein 1,n-1 lifts to a conformal diffeomorphism of Ein 1,n-1 . Conversely, by Theorem 4, every conformal transformation of Ein 1,n-1 defines a unique conformal transformation of the quotient space Ein 1,n-1 = Ein 1,n-1 / < δ >.

Let Conf( Ein 1,n-1 ) denote the group of conformal transformations of Ein 1,n-1 . Let j : Conf( Ein 1,n-1 ) -→ O(2, n) be the natural projection. This is a surjective group morphism whose kernel is generated by δ.

Causally convex open subsets of Einstein universe

In this section, we characterize causally convex open subsets of Ein 1,n-1 in a spatiotemporal decomposition S n-1 × R. We denote by d the distance on S n-1 induced by the round metric.

Proposition 3.

Let Ω be a causally convex open subset of Ein 1,n-1 . Then, there exist two 1-Lipschitz functions f + and f -from an open subset U of S n-1 to R such that the following hold:

• f -< f + on U ;
• the extensions of f + and f -to ∂U coincide.

• Ω is the set of points (x, t) of Ein 1,n-1 such that f -(x) < t < f + (x).
The proof of Proposition 3 uses the following lemma. Lemma 2. For every points p and q in the closure of Ω such that p ∈ I + (q), the intersection

I -(p) ∩ I + (q) is contained in Ω.
Proof. Let p, q ∈ Ω such that p ∈ I + (q). There exist two sequences {p i } and {q i } of elements of Ω such that lim p i = p and lim q i = q. Let r ∈ I(p, q). Then, I + (r) is an open neighborhood of p and I -(r) is an open neighborhood of q. As a result, there exists an integer i 0 such that p i 0 ∈ I + (r) and

q i 0 ∈ I -(r). It follows that r ∈ I(p i 0 , q i 0 ) ⊂ Ω.
Proof of Proposition 3. Let U be the projection of Ω on the sphere S n-1 . Since Ω is causally convex, the intersection of Ω with any timelike line {x} × R, where

x ∈ U , is connected, i.e. it is a segment {x}×]f -(x), f + (x)[. This defines two functions f + and f -from U to R such that Ω is the set of points (x, t) such that f -(x) < t < f + (x). Fact 1. If there exists x ∈ U such that f + (x) = +∞ then f + ≡ +∞.
This is equivalent to prove that Ω is future-complete, i.e. I + (Ω) ⊂ Ω, as soon as it contains a timelike half-line α = {x} × [t, +∞[. Let p ∈ Ω and let q ∈ I + (p). Since α is future-inextensible, it intersects I + (q). Let q be a point in this intersection. Then, q ∈ J(q , p ) where p ∈ Ω ∩ I -(p). Thus, q ∈ Ω.

Fact 2. If f + is finite, it is 1-Lipschitz.
This is equivalent to prove that the graph of f + is achronal. Suppose there exist two distinct points p, q in the graph of f + such that p ∈ I + (q). Since p ∈ ∂Ω, we have I + (q) ∩ Ω = ∅. Then, q ∈ I(p , q ) where p ∈ I + (q) ∩ Ω and q ∈ Ω ∩ I -(q). Hence, q ∈ Ω. Contradiction. Fact 3. If f + and f -are finite and ∂U is non-empty, then the extensions of f + and f -to ∂U are equal.

Let f + (resp. f -) be the extension of f + (resp. f -) to U . Suppose there exists x ∈ U such that f -(x) < f + (x). Then, the timelike segment {x}×]f -(x), f + (x)[
is contained in the boundary of Ω. This contradicts Lemma 2.

Conversely:

Proposition 4. Let f + , f -: U ⊂ S n-1 → R be two 1-Lipschitz functions defined on an open subset U of S n-1 such that: • f -< f + on U ;
• the extensions of f + and f -to ∂U coincide.

Then, the set of points

(x, t) of Ein 1,n-1 such that f -(x) < t < f + (x), named Ω, is causally convex in Ein 1,n-1 .
The proof of Proposition 4 uses the following lemma.

Lemma 3. Let f + be the extension of f + to U . Then, for every point p in the graph of f + , the causal future of p is disjoint from Ω.

Proof. Let x ∈ U and set p = (x, f + (x)). Suppose there exists (y, s)

∈ Ω ∩ J + (p). Then, d(x, y) ≤ s -f + (x) < f + (y) -f + (x) ≤ d(x, y). Contradiction. Then, J + (p) ∩ Ω = ∅. Now, let x ∈ ∂U and let {x i } be a sequence of elements of U such that x = lim x i . Set p i = (x i , f + (x i )). Suppose there exists q ∈ J + (p) ∩ Ω.
Since Ω is open, there exists q ∈ I + (q) ∩ Ω. By transitivity, q ∈ I + (p). Then, I -(q ) is an open neighborhood of p. Since lim p i = p, we deduce that p i ∈ I -(q ) for i big enough. Equivalently, q ∈ I + (p i ). Thus,

I + (p i ) ∩ Ω = ∅. Contradiction.
There is an analogue statement for the extension of f -to U , denoted f -, with the reverse time-orientation.

Proof of Proposition 4. Since f

± is 1-Lipschitz, if f ± is infinite in a point of U , it is infinite on U . If f + ≡ +∞ and f -≡ -∞, we have U = S n-1 and Ω = Ein 1,n-1 .
Suppose f + < +∞ and f -≡ -∞. Let p, q ∈ Ω such that q ∈ J + (p). Let γ be a future causal curve of Ein 1,n-1 joining p to q. Suppose that γ Ω. Then, γ intersects the boundary of Ω, reduced in this case to the graph of f + , in a point r = (x, f + (x)) where x ∈ U . By Lemma 3, J + (r) is disjoint from Ω. Then, the segment of γ joining r to q is contained in J + (r). Thus, q ∈ Ω. Contradiction.

Suppose now that f + and f -are finite. If ∂U is empty, the proof is similar to the previous case. Otherwise, we call f the common extension of f + and f -to ∂U . In this case, the boundary of Ω is the union of the graphs of f + , f -and f . By Lemma 3, the points of Ω are not causally related to any point in the graph of f . Therefore, the previous arguments still hold.

Now, we describe Cauchy hypersurfaces of causally convex open subsets of Ein

1,n-1 . Let Ω := {(x, t) ∈ U × R : f -(x) < t < f + (x)} be a causally convex open subset of Ein 1,n-1
where U is an open subset of S n-1 and f + , f -are the functions from U to R given by Proposition 3.

Proposition 5.

Let h be a 1-Lipschitz real-valued function defined on U such that its extension to ∂U coincide with that of f + and f -and f -< h < f + on U . Then, the graph of h is a topological Cauchy hypersurface of Ω. Proposition 5 is a consequence of the following lemma. Lemma 4. Suppose f + and f -are finite. Then, every inextensible timelike curve of Ein 1,n-1 that intersects Ω meet each of the graphs of f + and f -.

Proof. Let γ be an inextensible timelike curve of Ein 1,n-1 that intersects Ω. Then, γ intersects the boundary of Ω. If ∂U is empty, ∂Ω is the union of the graphs of f + and f -. Otherwise, ∂Ω is the union of the graphs of f + , f + and f where f is the common extension of f + and f -to ∂U . Since the points of Ω are not causally related to any point of the graph of f (see Lemma 3), we deduce that in both cases, γ intersects the graph of f + or the graph of f -.

Suppose γ meets the graph of f + . Since Ω is not past-complete, γ leaves Ω and so intersects again the its boundary. Since the graph of f + is achronal, γ could not intersects the graph of f + a second time. Thus, γ intersects the graph of f -.

Proof of Proposition 5. Let γ be an inextensible timelike curve of Ω. Since Ω is causally convex, γ is the intersection of Ω with an inextensible timelike curve γ of Ein

Duality

In this section, we highlight a particular class of causally convex open subsets of Ein 1,n-1 involving a notion of duality in Einstein universe.

Duality in the Klein model. Recall that a subset of S(R 2,n ) is said to be convex if it is the projectivization of a convex subset of R 2,n . The convex hull of a subset A of

S(R 2,n ) is the smallest convex containing A. Let Λ ⊂ Ein 1,n-1 . Let us denote Conv(Λ) the convex hull of Λ in S(R 2,n ). The dual convex cone of Λ in S(R n+2 ) is Conv * (Λ) = {x ∈ S(R 2,n ) : < x, y > 2,n < 0 ∀y ∈ Conv(Λ)}.
Definition 9. We call dual of Λ the intersection of Ein 1,n-1 with the dual cone Conv * (Λ).

Notice that

Conv * (Λ) ∩ Ein 1,n-1 = {x ∈ Ein 1,n-1 : < x, y > 2,n < 0 ∀y ∈ Λ}.
Duality in the universal cover. Let π : Ein 1,n-1 → Ein 1,n-1 be the universal covering map. Lemma 5. Let Λ ⊂ Ein 1,n-1 . The restriction of the projection π to the set of points which are non-causally related to any point of Λ is injective. Furthermore, its image is contained in the dual of the projection of Λ in Ein 1,n-1 . If in addition, Λ is acausal, we have equality.

Proof. Set Ω := Ein 1,n-1 \(J + (Λ) ∪ J -(Λ)). By definition, Ω is the intersection of the affine charts Mink 0 (p) where p ∈ Λ. As a result, the restriction of π to Ω is injective and its image is contained in the dual of π(Λ) (see [START_REF] Smai | Anosov representations as holonomies of globally hyperbolic spatially compact conformally flat spacetimes[END_REF]Corollary 2.43]).

Suppose Λ is acausal. Then, π(Λ) is negative, i.e. for every x, y ∈ π(Λ), we have < x, y > 2,n < 0 for every representant x, y ∈ R 2,n of x, y (see [START_REF] Andersson | Cosmological time versus CMC time in spacetimes of constant curvature[END_REF]Lemma 10.13]).

Let x ∈ Ein 1,n-1 ∩ Conv * (π(Λ)). Set Λ 0 := {x} ∪ π(Λ). By definition, Λ 0 is a negative subset of Ein 1,n-1 . By [11, Proposition 2.47], there exists an acausal subset Λ 0 of Ein 1,n-1 which projects on Λ 0 . Furthermore, the proof of [START_REF] Smai | Anosov representations as holonomies of globally hyperbolic spatially compact conformally flat spacetimes[END_REF]Proposition 2.47] shows that we can choose such a Λ 0 such that it contains Λ. As a consequence, Λ 0 is the union of a lift p of x and Λ. Since Λ 0 is acausal, p is non-causally related to any point of Λ. The lemma follows.

Lemma 5 motivates the following definition. Definition 10. Let Λ be a subset of Ein 1,n-1 . We call dual of Λ, denoted by Λ • , the set of points which are non-causally related to any point of Λ. Lemma 6. Let Λ be a subset of Ein 1,n-1 such that its dual is non-empty. Then, the dual of Λ is causally convex. If in addition Λ is closed, its dual is open. Proof. Let p and q be two points in the dual Λ • , joined by a causal curve γ : I ⊂ R → M . Suppose there is t ∈ I such that γ(t) ∈ Λ • , in other words γ(t) is causally related to a point λ ∈ Λ. By transitivity, it follows that p or q is causally related to λ. Contradiction.

Suppose Λ is closed. If Λ is not compact, it would contain a causal curve inextensible in the future or in the past. Then, J + (Λ) ∪ J -(Λ) would be the whole space Ein 1,n-1 and Λ • would be empty. Therefore, Λ is compact. It follows that J ± (Λ) is closed. Hence, Λ • is open. Now, we characterize duals of achronal closed subsets of Ein 1,n-1 in a spatio-temporal decomposition S n-1 × R.

Let Λ be a closed achronal subset of Ein 1,n-1 . It is the graph of a 1-Lipschitz realvalued function f defined on a closed subset Λ 0 of the sphere S n-1 . Let f + , f -be the real-valued functions defined for every x ∈ S n-1 by:

f + (x) = inf x 0 ∈Λ 0 {f (x 0 ) + d(x, x 0 )} f -(x) = sup x 0 ∈Λ 0 {f (x 0 ) -d(x, x 0 )}. Notice that f -(x) < f + (x) for every x ∈ S n-1 \Λ 0 and that f + and f -are equal to f on Λ 0 . Proposition 6. The dual of Λ is the set of points (x, t) of Ein 1,n-1 such that f -(x) < t < f + (x).
Proof. Let (x, t) ∈ S n-1 × R be a point in the dual of Λ. By definition, (x, t) is noncausally related to any point (x 0 , f (x 0 )) where x 0 ∈ Λ 0 . In other words, for every

x 0 ∈ Λ 0 , we have d(x, x 0 ) > |t -f (x 0 )|, i.e. f (x 0 ) -t < d(x, x 0 ) < f (x 0 ) + t. Hence, sup x 0 ∈Λ 0 {f (x 0 ) -d(x, x 0 )} ≤ t ≤ inf x 0 ∈Λ 0 {f (x 0 ) + d(x, x 0 )}.
Since Λ 0 is compact, the supremum and infimum above are attained; the previous inequalities are then strict. Thus, we obtain f -(x) < t < f + (x). The converse inclusion is clear. From the causal point of view, it is more relevant to consider as model space the universal cover of Einstein universe with its group of conformal diffeomorphisms. As a result, a conformally flat Lorentzian structure on a manifold M of dimension n ≥ 3 is encoded by the data of a development pair (D, ρ) where D : M → Ein 1,n-1 is a developing map and ρ : π 1 (M ) → Conf( Ein 1,n-1 ) is the associated holonomy morphism3 . Proof. Let D : M → Ein 1,n-1 be a developing map. Let p ∈ M and let U be a neighborhood of p. Without loss of generality, we suppose that the restriction of D to U is a diffeomorphism on its image. Then, D(U ) is a neighborhood of D(p). Since Ein 1,n-1 is GH, it is in particular strongly causal. Thus, there exists a neighborhooh V of D(p) contained in D(U ) and causally convex in Ein 1,n-1 . Let V be the preimage of V under D |U . By definition, V is a neighborhood of p contained in U . Moreover, V is causall convex in M . Indeed, let γ be a causal curve of M joining two points q, q ∈ V . By Lemma 7, the image under D of γ is a causal curve of Ein 1,n-1 joining D(q) to D(q ). Since V is causally convex, D(γ) is contained in V . If γ is not contained in V , there exists r ∈ γ ∩ (U \V ). Hence, D(r) ∈ D(γ)\V . Contradiction.

Lemma 7. The restriction of the developing map

D to a causal curve of M is injective. Proof. Let γ : I ⊂ R → M be a causal curve. Since D is conformal, it follows that D • γ : I → Ein 1,n-1 is a causal curve.

Construction of an enveloping space

Let V be a simply-connected globally hyperbolic conformally flat spacetime of dimension n ≥ 3. In this section, we prove Theorem 1. Our proof still hold if we weaken the assumption simply-connected by developable.

Let (D, ϕ) be a pair where

• D : V → Ein 1,n-1 is a developing map; • ϕ : Ein 1,n-1 → S n-1 × R is a spatio-temporal decomposition of Ein 1,n-1 .
Throughout this section, we call π the projection of Ein 1,n-1 on S n-1 defined as π 0 • ϕ where π 0 : S n-1 × R → S n-1 is the projection on the first factor. Timelike foliation on V . Consider the vector field T on V defined as the pull-back by D of ∂ t . The flow of T defines a foliation of V by smooth timelike curves. Let B be the leaf space, namely the quotient space of V by the equivalence relation that identifies two points if they are on the same leaf. We denote by ψ : V → B the canonical projection. By definition, the map π • D is constant on each leaf, therefore it induces a map d from to B to S n-1 such that the following diagram commutes:

V D → Ein 1,n-1 ψ ↓ ↓ π B d → S n-1 that is d • ψ = π • D. Since π • D and ψ are submersions, d is a local homeomorphism.
Fiber bundle over the leaf space. Let E(V ) be the fiber bundle over B defined as the pullback by d : B → S n-1 of the trivial bundle π : Ein 1,n-1 → S n-1 , in other words:

E(V ) := {(p, b) ∈ Ein 1,n-1 × B : π(p) = d(b)}.
We denote by π : E(V ) → B the projection on the second factor.

Fact 2. The fiber bundle E(V ) is trivial.

Proof. Let f be the continuous map from B × R in E(V ) that sends (b, t) on (p, b) where p is the point of Ein 1,n-1 with coordinates (d(b), t) in the decomposition S n-1 × R. It is easy to see that f is bijective. Indeed, the inverse is the continuous map that sends

(p, b) ∈ E(V ) on (b, t) ∈ B × R where t is the projection of p ∈ Ein 1,n-1 S n-1 × R on R. Therefore, f is a homeomorphism. Clearly, the following diagram commutes R × B f → E(V ) π0 π B
i.e. π • f = π0 . In other words, f is an isomorphism of fiber bundles. The lemma follows.

The projection on the first factor D : E(V ) → Ein 1,n-1 is a local homeomorphism inducing a structure of conformally flat spacetime on E(V ). In particular, E(V ) is strongly causal (see Lemma 8). Conformal embedding of V in E(V ). Let i be the map from V to E(V ) defined by

i(p) := (D(p), ψ(p)) for every p ∈ V . Fact 4. The map i is a conformal embedding of V into E(V ).
Proof. Since the maps D and ψ are continuous, open and conformal, so does the map i. All we need to check is that i is injective. Let p, q ∈ V such that i(p) = i(q). Then, D(p) = D(q) and ψ(p) = ψ(q). This last inequality implies that p and q belongs to the same timelike leaf. Since the restriction of D to a leaf is injective (see Lemma 7), it follows from D(p) = D(q) that p = q. Remark 1. The restriction of D to i(M ) coincide with D, more precisely

D • i = D.
Now we prove that the image of i is causally convex in E(M ). The proof uses the following lemma.

Lemma 9. The image under i of a Cauchy hypersurface S of V is a spacelike hypersurface of E(V ) which deconnects E(V ).

Proof. Since i is a conformal embedding, i(S) is a spacelike embedded hypersurface of E(V ). Let ψ : V → B be the canonical projection of V on the leaf space B. Recall that the restriction of ψ to S is a homeomorphism on B. Clearly, the map i |S •ψ -1 |S : B → E(V ) is a section of π. Hence, i(S) is a global section of E(V ). The lemma follows.

Fact 5. The image i(V ) is causally convex in E(V ).

Proof. Let p 0 , p 1 ∈ V such that there is a causal curve γ of E(V ) joining p 0 to p 1 . Let γ be an inextensible causal curve of E(V ) containing γ. Each connected component of the intersection of γ with i(V ) is an inextensible causal curve of i(V ). We call γ 0 and γ 1 the connected components containing p 0 and p 1 respectively. To prove that γ is contained in i(V ) is equivalent to prove that γ 0 = γ 1 . Suppose that γ 0 and γ 1 are disjoint. Then, each one of them meets i(S) in a single point, x 0 and x 1 respectively, which are distinct. Therefore, the curve γ intersects i(S) in at least two distinct points. But, i(S) is acausal in E(M ) (see Lemma 9 and [8, chap. 14, Lemma 45 and Lemma 42]). Contradiction.

Remark 2.

All the results of this section stated until now are based on the existence of a developing map, and so are still valid if V is not-simply connected but developable.

Embedding of the conformally flat Cauchy extensions of V in E(V ).

Proposition 8. Let W be a conformally flat Cauchy-extension of V . Then, there is a conformal embedding i of W into E(V ) such that the following assertions hold:

• The image i (W ) is causally convex in E(V ) and contains i(V ); • Every Cauchy hypersurface of i(V ) is a Cauchy hypersurface of i (W ).
The proof of Proposition 8 uses the following lemma. S n-1 × R and let ψ : W → B be the canonical projection on the leaf space. Since f is a Cauchyembedding, f (V ) is causally convex in W (see [START_REF] Rossi | Maximal extension of conformally flat globally hyperbolic spacetimes[END_REF]Lemma 8]). Then, the image under f of every leaf of V is the intersection of a unique leaf of W with f (V ). It follows that the map ψ • f descends to the quotient in a diffeomorphism f : B → B . Let d : B → S n-1 be the developing map induced by D . It is clear that d := d • f is the developing map induced by D. Therefore, the map

F : E(V ) → E(W ) defined by F (p, b) = (p, f (p))
is a conformal diffeomorphism that sends every fiber of E(V ) on a fiber of E(W ). The lemma follows.

Proof of Proposition 8. Let j : W → E(W ) be the conformal embedding of W into E(W ) and let F : E(V ) → E(W ) be the isomorphism defined in the proof of Lemma 10. The map i := F -1 • j defines a conformal embedding of W into E(V ). By Lemma 5, i(V ) and i (W ) are causally convex in E(V ). Moreover, according to the proof of Lemma 10, the following diagram commutes:

V i → E(V ) f ↓ ↓ F W j → E(W ) that is F • i = j • f , i.e. i = F -1 • j • f = i • f . It follows that: • i(V ) = i (f (V )) ⊂ i (W ). • Every Cauchy hypersurface of i(V ) is a Cauchy hypersurface of i (W ). Indeed, since f is a Cauchy-embedding, if S is a Cauchy hypersurface of V , f (S) is a Cauchy hypersurface of W . Then, i(S) = i (f (S)) is a Cauchy hypersurface of i (W ).
The proposition follows.

Remark 3. If V is developable, it is easy to see that any conformally flat Cauchy extension of W is also developable. As a result, Proposition 8 is still true in this setting.

The C 0 -maximal extension of V . From now on, we identify V and the conformally flat Cauchy-extensions of V with their images in E(V ). Let S be a Cauchy hypersurface of V . Proof. Let V another C 0 -maximal extension of V . By Lemma 11, V , seen in E(V ), is contained in C(S). If this inclusion is strict, C(S) would be a Cauchy extension of V . This contradicts the C 0 -maximality of V .

Lemma
We proved again the existence and the uniqueness of the C 0 -maximal extension for simply-connected conformally flat globally hyperbolic flat spacetimes.

Corollary 3. If V is Cauchy-compact then the C 0 -maximal extension of V is conformally equivalent to Ein 1,n-1 .
Proof. By Lemma 1, the leaf space B is compact. Since d is a local homeomorphism, it follows that d is a covering. But, S n-1 is simply connected, so d is a homeomorphism. As a result, D : The construction of this fiber bundle depends on the choice of a pair (D, ϕ) where

E(V ) → Ein 1,n-1 is a conformal diffeomorphism. Indeed, let p ∈ Ein 1,
• D : V → Ein 1,n-1 is a developing map ; • ϕ : S n-1 × R → Ein 1,n-1 is a spatio-temporal decomposition of Ein 1,n-1 .
Given two such pairs (D, ϕ), (D , ϕ ), there exists a conformal transformation φ of Ein 1,n-1 such that D = φ • D. We say that (D, ϕ) and (D , ϕ ) are equivalent if ϕ = ϕ • φ -1 . Lemma 12. If (D, ϕ) and (D , ϕ ) are equivalent, the enveloping spaces E(V ) and E (V ) defined by (D, ϕ) and (D , ϕ ) are isomorphic, i.e. there exists a conformal diffeomorphism from E(V ) to E (V ) which sends fiber on fiber.

Example 1 (Enveloping space of Minkowski spacetime). Minkowski spacetime R 1,n-1 is conformally equivalent to the set of points of Ein 1,n-1 which are not causally related to a point p ∈ Ein 1,n-1 . Given a spatio-temporal decomposition ϕ : Ein 1,n-1 → S n-1 × R, the enveloping space of R 1,n-1 is the complement in Ein 1,n-1 of the fiber going through p of the trivial bundle π 0 • ϕ : Ein 1,n-1 → S n-1 . This is a trivial fiber bundle over

S n-1 \{π 0 • ϕ(p)} R n-1 .
Example 2 (Enveloping space of De-Sitter spacetime). Given a spatio-temporal decomposition ϕ : Ein 1,n-1 → S n-1 × R, De-Sitter spacetime is conformally equivalent to S n-1 ×]0, π[. Therefore, the enveloping space of De-Sitter spacetime relatively to this decomposition is π 0 • ϕ : Ein 1,n-1 → S n-1 where π 0 : S n-1 × R → S n-1 is the projection on the first factor.

Causally convex GH open subsets of the enveloping space

In this section, we describe causally convex open subsets of the enveloping space E(V )defined in the previous section -which are globally hyperbolic. Notice that since E(V ) is, a priori, not globally hyperbolic (see Example 1), causal convexity does not imply global hyperbolicity anymore.

By [START_REF] Rossi | Maximal extension of conformally flat globally hyperbolic spacetimes[END_REF]Theorem 10], causally convex GH open subsets of E(V ) which contain conjugate points, i.e. points whose image under D are conjugate in Ein 1,n-1 , are conformally equivalent to causally convex open subsets of Ein 1,n-1 . These are described in Section 3.5. This is why we only deal here with causally convex open subsets of E(V ) without conjugate points. We basically generalise the description of causally convex open subsets of Ein 1,n-1 .

The following definition introduces a class of sections of the enveloping space E(V ) → B, expressed in a global trivialization.

Definition 13. A real-valued function f defined on an open subset

U of B is said 1- Lipschitz if for every x ∈ U , there exists an open neighborhood U x of x contained in U such that the following hold: 1. the restriction of d to U x is injective; 2. the map f • d -1 |Ux : d(U x ) ⊂ S n-1 → R is 1-Lipschitz.
1. f -< f + on U ;
2. the extensions of f + and f -to ∂U coincide.

Proof. Let π the natural projection of E(V ) on B. We call U the projection of Ω on B.

Since Ω is causally convex, the intersection of Ω with any fiber π-1 (x), where x ∈ U , is connected, i.e. it is a segment {x}×]f -(x), f + (x)[. Notice that -∞ < f -(x) and f + (x) < +∞, otherwise Ω would contain conjugate points. Contradiction. This defines two real-valued functions f + , f -defined on U such that: Thus, f + is 1-Lipschitz. The proof is similar for f -with the reverse time-orientation. Lemma 2 is still valid for causally convex open subsets of E(V ). Hence, the same arguments used in the proof of Proposition 3 shows that the extensions of f + and f - to ∂U coincide.

Ω = {(x, t) ∈ U × R : f -(x) < t < f + (x)}. Let x ∈ U . Set p + = (x, f + (x)). Let p ∈ I -(p + ) ∩ Ω. Since Ω is GH,
Corollary 4. The graphs of f + and f -are achronal.

Proof. The proof is similar than that of Fact 2 in the proof of Proposition 3.

C 0 -maximal extensions of conformally flat globally hyperbolic spacetimes

In this section, we give a new proof of the existence and the uniqueness of the C 0 -maximal extension of a globally hyperbolic conformally flat spacetime V of dimension n ≥ 3, using the notion of enveloping space introduced in Section 4.

When V is simply-connected, the proof is given in Section 4.2 (see Proposition 9 and Corollary 2). We deal here with the case where V is not simply-connected. The proof consists to extend the action of π 1 V on Ṽ to a proper action on the C 0 -maximal extension of Ṽ . We prove then that the C 0 -maximal extension of V is the quotient of the C 0 -maximal extension of Ṽ by π 1 V .

Notations. Let D : Ṽ → Ein 1,n-1 be a developing map and let ρ : Γ → Conf( Ein 1,n-1 ) be the associated holonomy representation, where Γ := π 1 (V ).

Fix a decomposition ϕ : Ein 1,n-1 → S n-1 × R. Let E( Ṽ ) be the enveloping space related to the pair (D, ϕ) (see Section 4.2). We denote by π : E( Ṽ ) → B the projection on the second factor and by D : E( Ṽ ) → Ein 1,n-1 the projection on the first factor.

Fact 6. If Ṽ is Cauchy-compact, the C 0 -maximal extension of V is a finite quotient of Ein 1,n-1 .
Proof. This is an immediate consequence of Corollary 3.

Suppose now that Ṽ is not Cauchy-compact. Let S be a non-compact Cauchy hypersurface of Ṽ . The C 0 -maximal extension of Ṽ is the Cauchy development C( S) of S in E( Ṽ ) (see Proposition 9). In what follows, we extend the action of Γ to C( S).

Action of Γ on C( S). The points of C( S) are characterized by their shadows on S (see Proposition 2). As a result, we show that the action of Γ on S induces naturally an action of Γ on C( S):

Proposition 11. Let p ∈ C( S) and let γ ∈ Γ. There exists a unique point γ.p in C( S) such that its shadow on S is exactly the image under γ of the shadow of p on S. This defines an action of Γ on C( S) which satisfies the following properties:

• the restriction of the action of Γ on C( S) to Ṽ coincide with the usual action of Γ on Ṽ ;

• the action of Γ on C( S) preserves the causality relations, i.e. for every p ∈ C( S) and for every γ ∈ Γ, we have:

p ∈ J -(q) ⇔ γ.p ∈ J -(γ.q) p ∈ I -(q) ⇔ γ.p ∈ I -(γ.q);
• the restriction of D to C( S) is ρ-equivariant, i.e. for every p ∈ C( S) and for every γ ∈ Γ, we have:

D(γ.p) = ρ(γ) D(p).
We prove Proposition 11 by an analysis-synthesis reasoning. In the analysis, we suppose that γ.p exists and is unique, and we look for properties satified by γ.p that will characterize it. In the synthesis, we use the criteria found in the analysis to determine the point γ.p. In what follows, the shadow of a point p ∈ C( S) on S is denoted by O(p).

Analysis. Suppose that for every γ ∈ Γ and every p ∈ C( S), the point γ.p exists and is unique. For every γ ∈ Γ and every p ∈ Ṽ , we denote by γp (without the dot between γ and p) the usual action of the deck transformation γ on p. Let us start with this easy remark. Remark 5. Let p ∈ C( S) and let γ ∈ Γ. If p ∈ Ṽ , then γ.p = γp. Indeed, since the action of Γ on Ṽ respect the causality relations, we have J -(γp) = γJ -(p). Therefore,

O(γp) := J -(γp) ∩ S = γJ -(p) ∩ S = γ(J -(p) ∩ γ -1 S). Since S is Γ-invariant, we deduce that γ(J -(p)∩γ -1 S) = γO(p). Hence, O(γp) = γO(p), i.e. γ.p = γp.
Lemma 14. The map which associates to every (γ, p) ∈ Γ × C( S) the point γ.p ∈ C( S) is a group action. Moreover, this action respect causality relations.

Proof. The fact that the map (γ, p) ∈ Γ × C( S) → γ.p is a group action follows easily from the fact that the restriction to Γ × S is the usual group action of Γ on S.

Let γ ∈ Γ and let p, q ∈ C( S). Suppose that p, q ∈ J + (S). Then,

p ∈ J -(q) ⇔ O(p) ⊂ O(q) ⇔ O(γ.p) := γO(p) ⊂ γO(q) =: O(γ.q) ⇔ γ.p ∈ J -(γ.q).
By symmetry, the same arguments still hold if p, q ∈ J -( S). It remains the case where q ∈ I + ( S) and p ∈ I -( S). In this case, we have

p ∈ J -(q) ⇔ O(p) ∩ O(q) = ∅ ⇔ γO(p) ∩ γO(q) = ∅ ⇔ O(γ.p) ∩ O(γ.q) = ∅.
Since Γ preserves S, it preserves the chronological future/past of S in C( S). Thus, O(γ.p) ∩ O(γ.q) = ∅ ⇔ γ.p ∈ J -(γ.q). The lemma follows.

Lemma 15. The restriction of the developing map D : E( Ṽ ) → Ein 1,n-1 to C( S) is ρ-equivariant, i.e. for every p ∈ C( S) and for every γ ∈ Γ, we have D(γ.p) = ρ(γ) D(p).

Proof. Let γ ∈ Γ and let p ∈ C( S). Suppose that p ∈ I + ( S). Let q ∈ C( S) ∩ I + (p). Since C( S) is globally hyperbolic, the restriction of D to I -(q, C( S)) is injective and its image is causally convex in Ein

(I -(q) ∩ S) is an achronal hypersurface of D(I -(q, C( S))). Set Σ := D(I -(q) ∩ S). Since O(p) ⊂ I -(q, C( S)), the restriction of D to O(p) is injective and its image is exactly O( D(p), Σ). Thus D(γO(p)) = ρ(γ)D(O(p)) = ρ(γ)O( D(p), Σ) = O(ρ(γ) D(p), ρ(γ)Σ).
(

By definition, γO(p) = O(γ.p). From Lemma 14, we get O(γ.p) ⊂ I -(γ.q, C( S)). As above, we deduce that the restriction of D to O(γ.p) is injective and its image is exactly

O( D(γ.p), D(I -(γ.p)∩ S)). But, D(I -(γ.p)∩ S) = ρ(γ)Σ. Indeed, since O(γ.q) = γO(q) we have I -(γ.q) ∩ S = γ(I -(q) ∩ S). Thus, D(I -(γ.q) ∩ S) = ρ(γ)Σ. Hence, D(γO(p)) = D(O(γ.p)) = O( D(γ.p), ρ(γ)Σ). (2)
It follows from ( 1) and (2) that O(ρ(γ) D(p), ρ(γ)Σ) = O( D(γ.p), ρ(γ)Σ). By Proposition 2, we deduce that D(γ.p) = ρ(γ) D(p). If p ∈ I -( S), the same arguments hold with the reverse time-orientation. Lastly, if p ∈ S, the lemma follows from Remark 5. Remark 6. In the proof of Lemma 15, we can replace I -(q, C( S)) by any other causally convex open neighborhood U of p in C( S) such that:

• U intersects S;
• the restriction of D to U is injective;

• the image D(U ) is causally convex in Ein 1,n-1 .
Synthesis. Let p ∈ C( S) and let γ ∈ Γ. Without loss of generality, we suppose that p ∈ I + ( S). The idea is to reconstruct the proof of Lemma 15 to determine γ.p. More precisely, we choose a relevant causally convex open neighborhood U of p in C( S) satisfying the properties stated in Remark 6 and we construct the open subset that will turn out to be γ.U and will therefore contain γ.p (see Figure 1). Proof. Clearly, U ⊂ C( S) ∩ I -(q). Since C( S) is causally convex in E( Ṽ ), we have C( S) ∩ I -(q) = I -(q, C( S)). Since the restriction of D to I -(q, C( S)) is injective and its image is causally convex in Ein 1,n-1 (see [9, Prop. 2.7 and Cor. 2.8, p. 151]), the lemma follows.

Fix q ∈ C( S) ∩ I + (p). Let U be the Cauchy development of I -(q) ∩ S in E( Ṽ ). Lemma 16. The image under D of I -(q) ∩ S is achronal in Ein 1,n-1 . Proof. We have I -(q) ∩ S ⊂ I -(q, C( S)). Since C( S) is GH, the restriction of D to I -(q, C( S)) is injective (see [9, Prop. 2.
Let W be the Cauchy development of γ(I -(q) ∩ S) in E( Ṽ ).

Remark 7. The image under D of γ(I

-(q) ∩ S) is achronal in Ein 1,n-1 . Indeed, D(γ(I -(q) ∩ S)) = ρ(γ)D(I -(q) ∩ S).
The assertion follows then from Lemma 16.

Fact 8. The restriction of D to W is injective. Moreover, the image under D of W is equal to the Cauchy development of D(γ(I

-(q) ∩ S)) in Ein 1,n-1 .
Proof. Let r, r ∈ W such that D(r) = D(r ). Let E π(r) and E π(r ) be the fibers going through r and r . By definition, their images under D are the fibers of the trivial bundle Ein 1,n-1 → S n-1 going through D(r) and D(r ). Since D(r) = D(r ), these fibers coincide. We call this fiber ∆. SinceE π(r) and E π(r ) are inextensible timelike curves, they intersect γ(I -(q)∩ S in two points r 0 and r 0 . Hence, D(r 0 ), D(r 0 ) ∈ ∆∩D(γ(I -(q)∩ S)). But, D(γ(I -(q) ∩ S)) is achronal in Ein 1,n-1 (see Remark 7). Thus, the timelike line ∆ intersects D(γ(I -(q) ∩ S)) at most once. Hence, D(r 0 ) = D(r 0 ). The restriction of D to γ(I -(q) ∩ S) being injective, we deduce that r 0 = r 0 . Thus, E π(r) = E π(r ) . Since the restriction of D to any causal curve is injective (see Lemma 7), we get r = r .

We deduce easily from the injectivity of D on W that D(W ) is contained in the Cauchy development of D(γ(I -(q) ∩ S)) in Ein 1,n-1 . This inclusion is clearly a Cauchyembedding. Since W is C 0 -maximal (see Proposition 9), we deduce equality. Proof. We can suppose without loss of generality that p ∈ I + ( S). Let q ∈ I + (p) and set Σ := D(I -(q) ∩ S). We prove that D(I -(q, C( S))) is contained in the affine chart Mink -( D(q)) (see Section 3.3). Suppose there exists r ∈ I -(q, C( S)) such that D(r) ∈ Mink -( D(q)). Then, I( D(q), D(r)) contains conjugate points. Since the image under D of I -(q, C( S)) is causally convex in Ein 1,n-1 (see [9, Cor. 2.8, p. 151]), we deduce that I -(q, C( S) admits a photon whose image under D contains conjugate points. Therefore, by [ Since the action of Γ on S is free, we deduce that γ = id. This proves that the action of Γ on C( S) is free.

Suppose the action of Γ on C( S) is not properly discontinuous. Then, by [5, Proposition 1], there exist a sequence {p i } of points of C( S) converging to some point p ∞ ∈ C( S) and a divergent sequence {γ i } of elements of Γ such that {γ i .p i } converges to some point q ∞ ∈ C( S). Without loss of generality, we can suppose that p ∞ , q ∞ ∈ J + ( S).

Let p ∈ I + (p ∞ ). Since lim p i = p ∞ , all the p i belong to I -(p) except a finite number. Hence, O(p i ) ⊂ O(p) for every i ≥ i 0 where i 0 is a natural integer. Let x i ∈ O(p i ). Up to extracting, {x i } converges to some x ∞ ∈ O(p). Similarly, let q ∈ I + (q ∞ ); all the γ i p i belong to O(q) except a finite number. Hence, γ i O(p i ) = O(γ i p i ) ⊂ O(q). Therefore, up to extracting, {γ i x i } converges to some point y ∞ ∈ O(q). This contradicts the properness of the action of Γ on S (see [START_REF] Frances | Lorentzian kleinian groups[END_REF]Proposition 1]).

The C 0 -maximal extension of V . Proposition 13. The quotient space Γ\C( S) is a C 0 -maximal extension of V .

Proof. Let i : Ṽ toE( Ṽ ) the embedding of Ṽ in E( Ṽ ) defined in Section 4. The corestriction of i to C( S) is a Cauchy embedding, denoted by f . Let π : C( S) → Γ\C( S) be the canonical projection. Since the action of Γ on i( Ṽ ) coincide with the usual action of Γ on Ṽ , the map f descends to quotient in a Cauchy embedding f : V → Γ\C( S). It is easy to see that since C( S) is C 0 -maximal (see Proposition 9), the quotient Γ\C( S) is also C 0 -maximal. The proposition follows.

Corollary 5. The C 0 -maximal extension Γ\C( S) is unique up to conformal diffeomorphism.

Proof. Let f : V → W be a Cauchy embedding of V in a C 0 -maximal globally hyperbolic conformally flat spacetime W . It lifts to a Cauchy embedding f : Ṽ → W . By Proposition 8, W admits a conformal copy in E( Ṽ ) contained in C( S), where S is a Cauchy hypersurface of Ṽ (seen in E( Ṽ )). The inclusion of W in C( S) is a Cauchy embedding which descends to the quotient in a Cauchy embedding from W to Γ\C( S). By Proposition 13, this last one is surjective. The corollary follows.

The proof of Proposition 13 is based on the fact that if the universal covering of a globally hyperbolic spacetime is maximal then this spacetime is maximal. A priori, the converse assertion is not true in general. However, Proposition 13 allows to prove that it is true in the conformally flat setting. Corollary 6. Let V be a C 0 -maximal spacetime. Then, the universal covering of V is C 0 -maximal.

Proof. Let S be a Cauchy hypersurface of Ṽ . By Proposition 13, there is a Cauchy embedding f from V to Γ\C( S). This last one lifts to a Cauchy embedding f : Ṽ → C( S). Since V is maximal, f is surjective. Therefore, f is surjective. Since C( S) is C 0 -maximal (see Proposition 9), Ṽ is C 0 -maximal.

C 0 -maximal extensions respect inclusion

In this section, we show that the functor maximal extension respect inclusion in the setting of conformally flat spacetimes. More precisely, we establish the following result: Theorem 5. Let V be a conformally flat globally hyperbolic spacetime and let U be a causally convex open subset of V . Then, there is a conformal embedding from the C 0maximal extension Û of U into the C 0 -maximal extension V of V . Moreover, the image of this embedding is causally convex in V .

We first prove this result in the case where V is simply-connected in Section 6.1 before dealing with the general case in Section 6.2.

The simply-connected case

We prove Theorem 5 in the case where V is simply-connected: Proposition 14. Let V be a simply-connected conformally flat globally hyperbolic spacetime and let U be a causally convex open subset of V . Then, there is a conformal embedding from the C 0 -maximal extension Û of U into the C 0 -maximal extension V of V . Moreover, the image is causally convex in V .

The key idea is to realize the C 0 -maximal extensions of U and V in the enveloping space E(V ) so we can compare them. The proof of Proposition 14 uses the following lemma.

Let D : V → Ein 1,n-1 be a developing map.

Lemma 18. The inclusion map i : U → V induces a conformal embedding of E(U ) into E(V ) that sends every fiber of E(U ) on a fiber of E(V ) and such that the restriction to every fiber of E(U ) is surjective.

Proof. Consider the foliation of V by inextensible timelike curves induced by the pullback by the developing map D of the vector field ∂ t on Ein 1,n S n-1 × R. Since U is causally convex in V , the intersection of every leaf of V with U is an inextensible timelike curve of U . Therefore, the foliation of V induces a foliation of U by inextensible timelike curves. Notice that this foliation coincides with that induced by the pull-back of ∂ t by the restriction of D to U . Let ψ : V → B and ψ U : U → B U be the canonical projections on the leaf spaces. Then, the map ψ • i descends to the quotient in an embedding ī : B U → B. Let d : B → S n-1 (resp. d U : B U → S n-1 ) be the developing map induced by the developing map D (resp. the restriction of D to U ). It is clear that d • ī = d U . It follows that the map from E(U ) to E(V ) that sends (p, b) on (p, ī(b)) is a conformal embedding that sends every fiber of E(U ) on a fiber of E(V ). Moreover, the restriction to every fiber of E(U ) is clearly surjective.

In other words, Lemma 18 says that E(U ) can be seen as the union of the fibers of E(V ) over some open subset of B.

Proof of Proposition 14. We identify V (resp. U ) with its image in E(V ) (resp. E(U )), then we identify E(U ) with its image in E(V ). Let V (resp. Û ) be the C 0 -maximal extension of V (resp. U ). By Proposition 9, V (resp. Û ) is the Cauchy development C(S) of a Cauchy hypersurface S of V (resp. Σ of U ) in E(V ) (resp. E(U )). We prove that the Cauchy development C(Σ) of Σ in E(V ) is exactly Û , then we prove that C(Σ) ⊂ C(S):

Let x ∈ Û and let γ an inextensible causal curve of E(V ) going through x. The intersection of γ with E(U ) is a union of connected components. The component containing x is an inextensible causal curve of E(U ), then it intersects Σ in a single point. Hence, x ∈ C(Σ). This proves that Û ⊂ C(Σ). Actually, this inclusion is a Cauchy embedding. Since U is C 0 -maximal, we deduce that Û = C(Σ).

Let x ∈ C(Σ) and let γ an inextensible causal curve of E(V ) going through x. By definition, the curve γ meets Σ, hence V . Since V is causally convex in E(V ), the intersection of γ with V is an inextensible causal curve of V , thus it intersects S in a single point. It follows that x ∈ C(S). Hence, C(Σ) ⊂ C(S). The proposition follows.

The general case

In the previous section, we proved Theorem 5 in the case where V is simply-connected. In this section, we prove it for any conformally flat globally hyperbolic spacetime V . Without loss of generality, we suppose that V is C 0 -maximal. Let π : Ṽ → V be the universal cover of V . Set Γ := π 1 (V ). Let U be a connected component of π -1 (U ) and let Γ be the stabilizer of U in Γ.

Lemma 19. The open set U is causally convex in Ṽ .

Proof. Let p, q ∈ U and let γ be a causal curve in Ṽ joining p to q. The projection π(γ) of γ in V is a causal curve joining the points π(p) and π(q) of U . Since U is causally convex in V , the curve π(γ) is contained in U . Hence, γ is contained in U . The lemma follows.

Let U be the C 0 -maximal extension of U . By Proposition 14, U can be conformally identified to a causally convex open subset of Ṽ .

Lemma 20. The stabilizer of U in Γ is equal to Γ .

Proof. Since U ⊂ U , the stabilizer of U is contained in the stabilizer of U . Conversely, let γ be an element of Γ stabilizing U . Let p ∈ U . Consider a Cauchy hypersurface Σ of U going through p. Let ϕ be an inextensible causal curve of U going through p. Then, γϕ is an inextensible causal curve of U . Therefore, γϕ intersects Σ in a unique point q. We prove that q = γp.

Since U is causally convex in Ṽ , the projection π( U ) is causally convex in V and π(Σ) is a Cauchy hypersurface of π( U ). Moreover, π(ϕ) is an inextensible causal curve of π( U ), then it meets π(Σ) in a single point. Since π(p), π(q) ∈ π(ϕ) ∩ π(Σ), we deduce that π(p) = π(q). Then, q = γ p with γ ∈ Γ. Since γp, γ p ∈ γϕ, if γ = γ , the causal curve π(ϕ) would be closed. Contradiction. Hence, γ = γ , so q = γp. This shows that γp ∈ U . Thus, γ ∈ Γ .

The inclusion U ⊂ Ṽ descends to the quotient in a conformal embedding from Γ \ U to V . Since U is causally convex in Ṽ , the image of this embedding is causally convex in V . It remains to prove the following assertion to conclude.

Lemma 21. The quotient space Γ \ U is the C 0 -maximal extension of U .

Proof. The inclusion U ⊂ U is a Cauchy embedding which descends to the quotient in a Cauchy embedding from U to Γ \ U . We have to prove that Γ \ U is C 0 -maximal.

According to Theorem 5, the C 0 -maximal extension of Ω is conformally equivalent to a causally convex open subset Ω of Ein 1,n-1 such that Ω ⊂ Ω ⊂ Ω where each inclusion is a Cauchy embedding. Thus, there exist two 1-Lipschitz real-valued functions h -, h + defined on U , whose extensions to ∂U equal f , such that Ω is the set of points (x, t) such that h -(x) < t < h + (x). Hence, g -(x) ≤ h -(x) < h + (x) ≤ g + (x) for every x ∈ U (see 22). In other words, Ω ⊂ Ω . The proposition follows. The following proposition gives a geometrical characterisation of eikonal functions.

Proposition 16 (Geometrical criterion of eikonality). A real-valued function f + defined on an open subset U of the sphere S n-1 is future eikonal if and only if for every x ∈ U , there exists a past-directed lightlike geodesic starting from (x, f + (x)), entirely contained in the graph of f + and with no past endpoint in the graph of f + . Proof. Suppose f + is eikonal. We call f the extension of f + to ∂U . Let x ∈ U . There exists x 0 ∈ ∂U such that f + (x) = f (x 0 ) + d(x, x 0 ). Hence, f + (x) -f (x 0 ) = d(x, x 0 ), i.e. the points (x, f + (x)) and (x 0 , f (x 0 )) are joined by a past lightlike geodesic ϕ. This last one is contained in the graph of f + . Conversely, let us prove that f + equals the function g + defined by [START_REF] Antonio | Globally hyperbolic spacetimes can be defined as 'causal'instead of 'strongly causal[END_REF]. By Lemma 22, we have f + ≤ g + . Let x ∈ U . There exists a past-directed lightlike geodesic ϕ starting from (x, f + (x)), entirely contained in the graph of f + and with no past endpoint in the graph of f + . In Ein 1,n-1 , the geodesic ϕ admits a past endpoint (x 0 , f (x 0 )) with x 0 ∈ ∂U . Thus, d(x, x 0 ) = f + (x) -f (x 0 ). Hence, f + (x) = f (x 0 ) + d(x, x 0 ) ≥ g + (x). Then, f + = g + . Now, we show that eikonal is a local property. Definition 15. A real-valued function f + defined on an open subset U of the sphere S n-1 is locally future eikonal if every point x of U admits an arbitrarily small neighborhood V x such that the restriction of f + to V x is future eikonal. Remark 9. A locally future eikonal function is locally 1-Lipschitz, thus 1-Lipschitz.

Let f + be a 1-Lipschitz real-valued function defined on an open subset U of S n-1 . We call f its extension to ∂U . Lemma 23. If f + is future eikonal then f + is locally future eikonal.

Proof. Let V be an open subset of U . We call g the extension of f + |V to ∂V . Let g + (resp. g -) be the function defined by the expression (3) (resp. (4) after replacing U by V . By Lemma 22, we have f + |V ≤ g. Let us prove that f + |V ≥ g so we get the equality. Let Ω (resp. W ) be the C 0 -maximal causally convex open subset of Ein 1,n-1 bounded by the graphs of f + and f -(resp. g + and g -). The intersection W ∩ Ω is a causally convex open subset of Ω; its C 0 -maximal extension is W . By theorem 5, we get W ⊂ Ω. Hence, g + ≤ f + |V . Thus, g + = f + |V , then f + |V is future eikonal.

Lemma 24. If f + is locally future eikonal then f + is future eikonal.

Proof. We use the criterion given by Proposition 16 to prove that f + is eikonal. Let x ∈ U and let V ⊂ U be a neighborhood of x such that f + |V is future eikonal. Then, there is a past-directed lightlike geodesic ϕ starting from (x, f + (x)) with the properties of Proposition 16. In Ein 1,n-1 , the geodesic ϕ admits a past endpoint (x 0 , f + (x 0 )) where x 0 ∈ ∂V (we still denote by f + its extension to U ). If x 0 ∈ ∂U , the lemma is proved. Otherwise, we choose a neighborhood V 0 ⊂ U of x 0 such that f + |V 0 is future eikonal. Again, there is a past-directed lightlike geodesic ϕ 0 starting from (x 0 , f (x 0 )), entirely contained in the graph of f + |V 0 , with past endpoint (x 1 , f + (x 1 )) where x 1 ∈ ∂V 0 . The geodesic ϕ 0 extends ϕ; indeed, otherwise (x 1 , f + (x 1 )) would be in the chronological past of (x, f + (x)), i.e. we would have d(x, x 1 ) < f + (x) -f + (x 1 ). This contradicts the fact that f + is 1-Lipschitz. Consequently, we extend ϕ in a lightlike geodesic, entirely contained in the graph of f + , with no past endpoint in the graph of f + . The lemma follows.

We proved the following statement: Proposition 17. A real-valued function defined on an open subset of the sphere S n-1 is future eikonal if and only if it is locally future eikonal.

Eikonal functions on a conformally flat Riemannian manifold

Theorem 4 .

 4 Let n ≥ 3. Any conformal transformation between two open subsets of Ein 1,n-1 is the restriction of an element of O(2, n).

Fact 1 .

 1 The leaf space B is homeomorphic to a Cauchy hypersurface S of V .Proof. Every leaf is a timelike curve of V and so meets S in a unique point. Therefore, the restriction of ψ to S is a continuous bijection on B. The restriction ofψ to S is open. Indeed, any open subset U of S coincides with ψ -1 |S (ψ |S (U )), so ψ |S (U ) is open in B.Then, the restriction of ψ to S is a homeomorphism on B.

Fact 3 .

 3 The fibers of E(V ) are inextensible timelike curves. Proof. The fiber E b of E(V ) over a point b ∈ B is the set of points (p, b) such that π(p) = d(b). It is then easy to see that the restriction of D to E b is a homeomorphism on the fiber π -1 (d(b)) of π : Ein 1,n-1 → S n-1 . The lemma follows.

Lemma 10 .

 10 Let W be a conformally flat Cauchy-extension of V . Then, the enveloping spaces E(V ) and E(W ) are isomorphic. Proof. Let f : V → W be a Cauchy embedding and let D : W → Ein 1,n-1 be a developing map such that D •f = D. Consider the foliation of W by inextensible timelike curves induced by the pull-back by D of the vector field ∂ t on Ein 1,n-1

n- 1 .Definition 12 .

 112 There exists a unique b ∈ B such that d(b) = π(p). Thus, (p, b) is the unique point of E(V ) such that D(p, b) = p. The corollary follows. The trivial fiber bundle π : E(V ) → B is called an enveloping space of V .

Remark 4 .Proposition 10 .

 410 This definition generalizes the (local) notion of 1-Lipschitz functions defined on an open subset of the sphere S n-1 . The function f is 1-Lipschitz if and only if its graph is locally achronal. Lemma 13. Any 1-Lipschitz real-valued function f defined on an open subset U B admits a unique extension to U . Any causally convex GH open subset Ω of E(V ) is the domain bounded by the graphs of two 1-Lipschitz real-valued functions f + and f -defined on an open subset U of B such that:

  the restriction of D to I + (p, Ω) is injective and its image is causally convex in Ein 1,n-1 (see [9, Prop. 2.7 and Cor. 2.8, p.151]). It follows that: • D(I + (p, Ω)) is the domain bounded by the graphs of two 1-Lipschitz real-valued functions g -< g + defined on an open subset of S n-1 (see Proposition 3); • d is injective on U x := π(I + (p, Ω)) and g + = f + • d -1 |Ux .

Fact 7 .

 7 7, p. 151]). Then, D(I -(p) ∩ S) is achronal in D(I -(q, C( S))). This last one being causally convex in Ein 1,n-1 (see [9, Cor. 2.8, p. 151]), we deduce easily that D(I -(p) ∩ S) is achronal in Ein 1,n-1 . The restriction of D to U is injective. Moreover, the image under D of U is equal to the Cauchy development of D(I -(q) ∩ S) in Ein 1,n-1 .

Proof of Proposition 11 .

 11 Set Σ := D(I -(q) ∩ S). Since O(p) ⊂ U , it follows from Fact 7 that D(O(p)) is equal to the shadow O( D(p), Σ) of D(p) on Σ. HenceD(γO(p)) = ρ(γ)D(O(p)) = ρ(γ)O( D(p), Σ) = O(ρ(γ) D(p), ρ(γ)Σ). Hence, O(ρ(γ) D(p), ρ(γ)Σ) ⊂ D(W ). Since D(W ) is the Cauchy development of ρ(γ)Σ (see Fact 8), it follows that ρ(γ) D(p) ∈ D(W ).Thus, there exists a unique point p ∈ U such that D(p ) = ρ(γ) D(p). It is clear that O(p ) = γO(p). If there is another point p ∈ C( S) such that O(p ) = γO(p), by Proposition 2, we get p = p . Then, we set γ.p := p . Dynamical properties of the action of Γ on C( S). In this paragraph, we prove that the action of Γ on C( S) is free and properly discontinuous. Lemma 17. Let p be a point in the complement of S in C( S). Then, the shadow of p on S is a topological disk of dimension n -1.

10 ,

 10 Theorem 10], C( S) is conformally equivalent to Ein 1,n-1 . Contradiction. Then, D(p) and Σ are contained in Mink -( D(q)). As a result, the shadow O( D(p), Σ) is the intersection in Minkowski spacetime of the past causal cone of D(p) with Σ. Thus, the map which associates to every past causal direction at D(p), the intersection of the straight line tangent to this direction with Σ is a homeomorphism. Hence, O( D(p), Σ) is a topological (n -1)-disk. Since the restriction of D to O(p) is a diffeomorphism on O( D(p), Σ) (see [9, prop. 2.7, p. 151]), the lemma follows.

Figure 1 :Proposition 12 .

 112 Figure 1: Action of Γ on the Cauchy development of S in the enveloping space E( S).

Remark 8 .

 8 The domain Ω in the proof of Proposition 15 is a union of connected component of the dual of the graph of f (seeProposition 6). Therefore, Proposition 15 can be reformulated as: C 0 -maximal causally convex open subsets of Ein 1,n-1 are exactly unions of connected components of duals of closed achronal subsets of Ein 1,n-1 . Definition 14. The function g + (resp. g -) is called future eikonal (resp. past eikonal). This definition is motivated by the classical notion of eikonal function in analysis: a real-valued function f defined on an open subset U of R n is called eikonal if f is differentiable almost everywhere and satisfies the eikonal equation || f || = 1. It turns out that if U has a piecewise smooth boundary, then the function f (x) := d(x, ∂U ), where d is the usual distance on R n , is eikonal in this sense.

  Let B be a conformally flat Riemannian manifold of dimension (n -1) ≥ 2 and let d : B → S n-1 be a developing map. The notion of eikonal function on an open subset of the sphere S n-1 being local, it naturally generalizes to functions on an open subset of B: Definition 16. A real-valued function f + defined on an open subset U of B is said future eikonal if for every x ∈ U , there exists an open neighborhood U x of x contained in U such that the following hold: 1. the restriction of d to U x is injective; 2. the map f + • d -1 |Ux : d(U x ) ⊂ S n-1 → R is future eikonal.

  Definition 1. A spacetime M is said strongly causal if for every point p ∈ M and every neighborhood U of p, there exists a neighborhood V of p contained in U , which is causally convex in M .

	Definition 2. A spacetime M is said globally hyperbolic (abbrev. GH) if the two
	following conditions hold:
	1. M is strongly causal.

  1,n-1 . By Lemma 4, γ meets the graph of h. Moreover, since the graph of h is achronal, γ intersects it exactly once. The proposition follows from [8, Definition 28 and Lemma 29, p. 415].

4 Enveloping space of a simply connected GH conformally flat spacetime 4.1 Conformally flat spacetimes

  Ein 1,n-1 )-structure, is conformally flat. Conversely, by Theorem 4, any conformally flat spacetime of dimension n ≥ 3 admits a (O(2, n), Ein 1,n-1 )-structure. We deduce the following statement.

	(O(2, n),
	A spacetime is called conformally flat if it is locally conformal to Minkowski space-
	time. Einstein universe is conformally flat since any point of Einstein universe admits
	a neighborhood conformally equivalent to Minkowski spacetime (see Section 3.3). It
	follows that any spacetime locally modeled on Einstein universe, i.e. equipped with a

Proposition 7. A conformally flat Lorentzian structure on a manifold M of dimension n ≥ 3 is equivalent to a (O 0 (2, n), Ein 1,n-1 )-structure.

  Then, if there exist t 0 , t 1 ∈ I such that t 0 = t 1 and D(γ(t 0 )) = D(γ(t 1 )), the curve D • γ would be a causal loop. Contradiction. Definition 11. A conformally flat spacetime M is said developable if any developing map descends to the quotient, giving a local diffeomorphism from M to Ein 1,n-1 , called again developing map.

Lemma 8. Any developable conformally flat spacetime M is strongly causal.

  Let W be a Cauchy extension of V . Let x ∈ W and let γ be an inextensible causal curve of E(V ) going through x. Since W is causally convex in E(V ) (seeProposition 8), the intersection of γ with W is an inextensible causal curve γ of W . By Proposition 8, S is a Cauchy hypersurface of W , then γ intersects S in a single point. It follows that x belongs to the Cauchy development of S in E(V ).

	Proposition 9. The Cauchy development C(S) of S in E(V ) is a C 0 -maximal extension
	of V .

11. The Cauchy development of S in E(V ) contains all the Cauchy-extensions of V . In particular, it contains V . Proof. Proof. By [8, Theorem 38, p. 421], C(S) is a globally hyperbolic spacetime for which S is a Cauchy hypersurface. According to Lemma 11, it is a Cauchy extension of V . Let W be a conformally flat Cauchy-extension of C(S). In particular, W is a Cauchy-extension of V . Then, W embeds conformally in E(V ) and the image is a causally convex open subset of E(V ) containing C(S). By Lemma 11, C(S) is exactly W (seen in E(V )). Hence, C(S) is C 0 -maximal. Corollary 2. The C 0 -maximal extension C(S) is unique up to conformal diffeomorphism.

This terminology has been introduced by C. Rossi in her thesis [9, Chapitre 4].

We direct the reader not familiar with (G, X)-structures to[START_REF] Goldman | Geometric structures on manifolds[END_REF] Chapter 5].

Acknowlegment

I would like to thank my PhD advisor Thierry Barbot for all the discussions that make this work possible, for his remarks and his help. I am also grateful to Charles Frances for his interests in my work and his relevant comments to improve a first version of this paper. This work of the Interdisciplinary Thematic Institute IRMIA++, as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR-10-IDEX-0002), and by SFRI-STRAT'US project (ANR-20-SFRI-0012) under the framework of the French Investments for the Future Program.

7 Eikonal functions and C 0 -maximality

Eikonal functions on the sphere

In this section, we characterize causally convex open subsets of Ein 1,n-1 which are C 0maximal in a spatio-temporal decomposition S n-1 × R. We denote by d the distance on S n-1 induced by the round metric. By [START_REF] Rossi | Maximal extension of conformally flat globally hyperbolic spacetimes[END_REF]Theorem 10], causally convex open subsets of Ein 1,n-1 with conjugate points are conformally equivalent to Ein 1,n-1 . For this reason, we consider a causally convex open subset Ω without conjugate points.

Let f + and f -be the two 1-Lipschitz real-valued functions defined on the projection U of Ω in S n-1 such that:

We call f the common extension of f + and f -to ∂U . Let g + , g -be the real-valued functions defined for every x ∈ U by:

Proposition 15. The causally convex open subset Ω of Ein 1,n-1 is C 0 -maximal if and only if f + equals g + and f -equals g -.

The proof of Proposition 15 uses the following lemma.

Lemma 22. Any 1-Lipschitz real-valued function g defined on U , whose extension to ∂U equals f , is bounded by g -and g + .

Proof. Let x ∈ U and let x 0 ∈ ∂U . We denote by ḡ the extension of

. The lemma follows.

Proof of Proposition 15. Let us denote Ω the set of points (x, t) of Ein 1,n-1 such that g -(x) < t < g + (x). By Lemma 22 and Proposition 5, Ω is a Cauchy-extension of Ω.

Past eikonal functions on U are defined similarly with the reverse time orientation. The geometrical criterion of eikonality given in Proposition 16 is still valid in this setting.

Eikonality is closely related to C 0 -maximality. Indeed, let E be the pull-back of the trivial fiber bundle π : Ein 1,n-1 → S n-1 by d:

We denote by π : E → B the projection on the second factor. The construction of this fiber bundle is exactly the same than that of the enveloping space in Section 4.2. Then, E is a conformally flat spacetime of dimension n ≥ 3 sharing the same properties than that of the enveloping space. Let D : E → Ein 1,n-1 be the developing map defined as the projection on the first factor. A causally convex GH open subset Ω of E is given by

where f + and f -are 1-Lipschitz functions defined on an open subset U of B such that their extensions to ∂U coincide (see Section 4.3).

Proposition 18. The GH conformally flat spacetime

Since Ω is GH, the restriction of D to I + (p, Ω) is injective and its image is causally convex in Ein 1,n-1 (see [9, Prop. 2.7 and Cor. 2.8, p.151]). The C 0 -maximality of Ω implies that D(I + (p, Ω)) is also C 0 -maximal (see [START_REF] Rossi | Espace-temps globalement hyperboliques conformément plats[END_REF]Prop. 3.6,p.156]). It follows that:

• D(I + (p, Ω)) is the domain bounded by the graph of a future-eikonal function g + and the graph of a past-eikonal function g -defined on an open subset of S n-1 (see Proposition 15);

• d is injective on U x := π(I + (p, Ω)) and

Thus, f + is future-eikonal. The proof is similar for f -with the reverse time-orientation. Conversely, suppose that f + is future-eikonal and f -past-eikonal. Let S be a Cauchy hypersurface in Ω. By Proposition 9, the C 0 -maximal extension of Ω is conformally equivalent to a causally convex open subset Ω of E, containing Ω and for which S is a Cauchy hypersurface. Suppose that Ω is stricly contained in Ω. Then, there exists x ∈ U such that (x, f + (x)) ∈ Ω or (x, f -(x)) ∈ Ω. Suppose that (x, f + (x)) ∈ Ω. The proof is symmetric if (x, f -(x)) ∈ Ω. Since f + is future-eikonal, there exists a past-directed lightlike geodesic ϕ starting from (x, f + (x)), entirely contained in the graph of f + and with no endpoint in the graph of f + . Then, ϕ does not intersects S. Contradiction. Hence, Ω = Ω, in other words Ω is C 0 -maximal.