CONFORMAL ACTIONS OF NILPOTENT GROUPS ON PSEUDO-RIEMANNIAN MANIFOLDS

CHARLES FRANCES AND KARIN MELNICK

Abstract. We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p, q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p + 1, assuming p ≤ q; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p, q), compact, conformally flat space, up to finite or cyclic covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure.

1. Introduction

Let (M, σ) be a compact pseudo-Riemannian manifold—that is, the tangent bundle of M is endowed with a type-(p, q) inner product, where p + q = n = dim M. We will always assume p ≤ q. The conformal class of σ is

[σ] = {e^{h}σ : h : M → R smooth}

Denote by Conf M the group of conformal automorphisms of M—the group of diffeomorphisms f of M such that f^{∗}σ ∈ [σ]. If n ≥ 3, then Conf M endowed with the compact-open topology is a Lie group (see [Ko, IV.6.1] for the Riemannian case; the proof is similar for p > 0). A basic question, first addressed by A. Lichnerowicz, is to characterize the pseudo-Riemannian manifolds (M, σ) for which Conf M does not preserve any metric in [σ]; in this case, Conf M is essential. The pseudo-Riemannian Lichnerowicz conjecture says that if M is compact and Conf M is essential, then M is conformally flat—that is, locally conformally equivalent to R^{p,q} with the translation-invariant, type-(p, q) metric. A stronger result in the Riemannian case was proved by Lelong-Ferrand [LF1]. See for example [Fr2] for more background on this conjecture.

Date: September 8, 2009.

One difficulty for general type (p, q) is that no characterization of essential conformal groups exists. In the Riemannian case, on the other hand, for M compact, Conf M is essential if and only if it is noncompact. For p ≥ 1, noncompactness is only a necessary condition to be essential. Now, a first approach to the conjecture is to exhibit sufficient conditions on a group of conformal transformations which ensure it is essential, and to test the conjecture on groups satisfying the given condition. From the conformal point of view, the Einstein universe is the most symmetric structure of type (p, q). These spaces, denoted Ein^{p,q}, are defined in section 2.1 below. The group Conf Ein^{p,q} is isomorphic to PO(p + 1, q + 1), and it is essential. The Einstein spaces are conformally flat. Thanks to [Zi1], we know that a simple noncompact real Lie group acting isometrically on a compact pseudo-Riemannian manifold (M, σ) of type (p, q) satisfies rk H ≤ p, where rk H denotes the real rank. For H < Conf M noncompact and simple, the rank

rk H ≤ p + 1 = rk PO(p + 1, q + 1)

This was first proved in [Zi1], also in [BN], and for H not necessarily simple in [BFM, 1.3 (1)]. Thus, conformal actions of simple groups H, with rk H = p + 1, on type-(p, q) compact pseudo-Riemannian manifolds cannot preserve any metric in the conformal class. The results of [BN], together with [FZ], give that when H < Conf M attains this maximal rank, then M is globally conformally equivalent to Ein^{p,q}, up to finite covers when p ≥ 2; for p = 1, M is conformally equivalent to the universal cover Ẽin^{1,n−1}, up to cyclic and finite covers. In particular, M is conformally flat, so this result supports the pseudo-Riemannian Lichnerowicz conjecture. The interested reader can find a wide generalization of this theorem in [BFM, 1.5]. Actions of semisimple Lie groups often exhibit rigid behavior partly because the algebraic structure of such groups is itself rigid. The structure of nilpotent Lie groups, on the other hand, is not that well understood; in fact, a classification of nilpotent Lie algebras is available only for small dimensions. From this point of view, it seems challenging to obtain global results similar to those above for actions of nilpotent Lie groups. Observe also that a pseudo-Riemannian conformal structure does not naturally define a volume form, so that the nice tools coming from ergodic theory are not available here.

For a Lie algebra 𝔥, we adopt the notation 𝔥_{1} = [𝔥, 𝔥], and 𝔥_{k} is defined inductively as [𝔥, 𝔥_{k−1}]. The degree of nilpotence d(𝔥) is the minimal k such that 𝔥_{k} = 0. For a connected, nilpotent Lie group H, define the nilpotence degree d(H) to be d(𝔥). If a connected Lie group H is nilpotent and acts isometrically on a type-(p, q) compact pseudo-Riemannian manifold M, where p ≥ 1, then the nilpotence degree d(H) ≤ 2p (when p = 0, then d(H) = 1). This was proved in the Lorentzian case in [Zi2], and in broad generality in [BFM, 1.3 (2)]. Theorem 1.3 (2) of [BFM] also implies d(H) ≤ 2p + 2 for H < Conf M. This bound is actually not tight, and the first result of the paper is to provide the tight bound, which turns out to be 2p + 1, the maximal nilpotence degree of a connected nilpotent subgroup in PO(p + 1, q + 1).

Theorem 1.1. Let H be a connected nilpotent Lie group acting conformally on a compact pseudo-Riemannian manifold M of type (p, q), where p ≥ 1, p + q ≥ 3. Then d(H) ≤ 2p + 1.

By theorem 1.3 (2) of [BFM], a connected nilpotent group H such that d(H) = 2p + 1 cannot act isometrically on a compact pseudo-Riemannian manifold of type (p, q). The following theorem says that if this maximal nilpotence degree is attained in Conf M, then M is a complete conformally flat manifold, providing further support for the pseudo-Riemannian Lichnerowicz conjecture.

Theorem 1.2. Let H be a connected nilpotent Lie group acting conformally on a compact pseudo-Riemannian manifold M of type (p, q), with p ≥ 1, p + q ≥ 3. If d(H) = 2p + 1, then M is conformally equivalent to Ẽin^{p,q}/Φ, where Φ < Õ(p + 1, q + 1) is trivial or isomorphic to Z_{2} when p ≥ 2, or isomorphic to Z when p = 1.

Here, Ẽin^{p,q} denotes the universal cover of Ein^{p,q} and Õ(p + 1, q + 1) = Conf Ẽin^{p,q}. When p ≥ 2, the center of Õ(p + 1, q + 1) has order two, while for Õ(2, q + 1), q ≥ 2, the center is infinite cyclic. From this theorem follows a complete description of the H-action: let Φ′ be the intersection of Φ with the center of Õ(p + 1, q + 1). Then the conformal group of the quotient Ẽin^{p,q}/Φ is N/Φ′, where N is the normalizer of Φ in Õ(p + 1, q + 1). The conformal diffeomorphism given by the theorem conjugates the H-action on M to one on Ẽin^{p,q}/Φ, via a representation H → N/Φ′, which is faithful if the original H-action on M is.

We do not treat the Riemannian case p = 0 in this paper, since the results in this case are a trivial consequence of Ferrand’s theorem. Section 2 below provides brief background on the geometry of the Einstein universe, as well as an algebraic study of nilpotent subalgebras of 𝔬(p + 1, q + 1). In section 2.3, we introduce the notion of Cartan geometry, which is central in all the proofs, and recall the interpretation of type-(p, q) conformal structures, where p + q ≥ 3, as Cartan geometries infinitesimally modeled on Ein^{p,q}. Section 3 uses results of [BFM] to prove theorem 1.1. Actually, we prove here a stronger statement, theorem 3.2, which gives also the starting point to prove theorem 1.2: whenever a connected nilpotent group of maximal nilpotence degree acts conformally on M, then some point has nontrivial stabilizers, containing special elements called lightlike translations. We explain the role of these elements and outline the proof of theorem 1.2 at the end of section 3.

2. Ein^{p,q} as a homogeneous space for PO(p + 1, q + 1)

In this section, we introduce some basic notation used throughout the paper, and provide background on the geometry of the Einstein universe, as well as an algebraic study of nilpotent subalgebras of 𝔬(p + 1, q + 1).

2.1. Geometry of Ein^{p,q}. Let R^{p+1,q+1} be the space R^{p+q+2} endowed with the quadratic form

Q^{p+1,q+1}(x_{0}, …, x_{n+1}) = 2(x_{0}x_{p+q+1} + ⋯ + x_{p}x_{q+1}) + Σ_{p+1}^{q}x_{i}^{2}

We consider the null cone

N^{p+1,q+1} = {x ∈ R^{p+1,q+1} | Q^{p+1,q+1}(x) = 0}

and denote by N̂^{p+1,q+1} the cone N^{p+1,q+1} with the origin removed. The projectivization P(N̂^{p+1,q+1}) is a smooth submanifold of RP^{p+q+1}, and inherits from the pseudo-Riemannian structure of R^{p+1,q+1} a type-(p, q) conformal class (more details can be found in [Fr1], [BCDGM]). We call the Einstein universe of type (p, q), denoted Ein^{p,q}, this compact manifold P(N̂^{p+1,q+1}) with this conformal structure. Note that Ein^{0,q} is conformally equivalent to the round sphere (S^{q}, g_{S^{q}}). When p ≥ 1, the product (S^{p} × S^{q}, −g_{S^{p}} ⊕ g_{S^{q}}) is a conformal double cover of Ein^{p,q}. The projective orthogonal group of Q^{p+1,q+1}, isomorphic to PO(p+1, q +1), acts projectively on Ein^{p,q} and is the full conformal group of Ein^{p,q}.

2.1.1. Lightcones, stereographic projection, and Minkowski charts. A lightlike, timelike, or spacelike curve of a pseudo-Riemannian manifold (M, σ) is a C^{1} γ : I → M such that σ_{γ(t)}(γ′(t), γ′(t)) is 0, negative, or positive, respectively, for all t ∈ I. It is clear that the notion of lightlike, timelike and spacelike curves is a conformal one. Lightlike curves are sometimes also called null. It is a remarkable fact that all metrics in [σ] have the same null geodesics, as unparametrized curves (see for example [Fr6] for a proof). Thus it makes sense to speak of null—or lightlike—geodesics for pseudo-Riemannian conformal structures. Given a point x ∈ M, the lightcone of x, denoted C(x), is the set of all lightlike geodesics passing through x. The lightlike geodesics of Ein^{p,q} are the projections on Ein^{p,q} of totally isotropic 2-planes in R^{p+1,q+1}. Hence every null geodesic is closed. If x ∈ Ein^{p,q} is the projection of y ∈ N^{p+1,q+1}, the lightcone C(x) is just P(y^{⊥} ∩N^{p+1,q+1}). Such a lightcone is not smooth, but C(x)\{x} is smooth and diffeomorphic to R × S^{p−1} × S^{q−1} (see figure 1).

Figure 1. the lightcone of a point in Ein^{1,2}

For any (p, q), there is a generalized notion of stereographic projection. Consider φ : R^{p,q} → Ein^{p,q} given in projective coordinates of RP^{n+2} by

φ : x ↦ [−\frac{1}{2}Q^{p,q}(x, x) : x_{1} : ⋯: x_{n} : 1]

Then φ is a conformal embedding of R^{p,q} into Ein^{p,q}, called the inverse stereographic projection with respect to [e_{0}]. The image φ(R^{p,q}) is a dense open set of Ein^{p,q} with boundary the lightcone C([e_{0}]). Since the action of PO(p + 1, q + 1) is transitive on Ein^{p,q}, it is clear that the complement of any lightcone C(x) in Ein^{p,q} is conformally equivalent to R^{p,q}. Such an open subset of Ein^{p,q} will be called a Minkowski component, and denoted M(x). Its identification with R^{p,q} via stereographic projection with respect to x is a Minkowski chart.

Here we explain how images of lightlike lines of R^{p,q} under φ reach the boundary (see also [Fr1, ch 4]). Lightlike lines of R^{p,q} are identified via φ with traces on M([e_{0}]) of lightlike geodesics in Ein^{p,q}. If γ : R → R^{p,q} is a lightlike line, then lim_{t→∞} φ(γ(t)) = lim_{t→−∞} φ(γ(t)) = x_{γ}, where x_{γ} ∈ C([e_{0}])\{[e_{0}]}. For lightlike lines γ(t) = c + tu and β(t) = b + tv, the limits x_{γ} = x_{β} if and only if u = v and ⟨b − c, u⟩ = 0. In other words, M([e_{0}]) ∩ C(x) is a degenerate affine hyperplane for x ∈ C([e_{0}]) \ {[e_{0}]}.

2.1.2. A brief description of 𝔬(p + 1, q + 1). The Lie algebra 𝔬(p + 1, q + 1) consists of all (n + 2) × (n + 2) matrices X, where n = p + q, such that

X^{t}J_{p+1,q+1} + J_{p+1,q+1}X = 0

where J_{p+1,q+1} is the matrix of the quadratic form Q^{p+1,q+1}. It can be written as a sum 𝔲^{−} ⊕ 𝔯 ⊕ 𝔲^{+} (see [Ko, IV.4.2] for p = 0; the case p > 0 is a straightforward generalization), where

𝔯 = \{( \begin{matrix}a & & 0 \\ & M \\ & & −a \end{matrix} ) : \begin{matrix}a ∈ R \\ M ∈ 𝔬(p, q) \end{matrix} \}

𝔲^{+} = \{( \begin{matrix}0 & −x^{t}.J_{p,q} & 0 \\ & 0 & x \\ & & 0 \end{matrix} ) : x ∈ R^{p,q} \}

and

𝔲^{−} = \{( \begin{matrix}0 \\ x & 0 \\ 0 & −x^{t}.J_{p,q} & 0 \end{matrix} ) : x ∈ R^{p,q} \}

Thus 𝔯 ≅ 𝔠𝔬(p, q), and there are two obvious isomorphisms i^{+} and i^{−} from 𝔲^{+} and 𝔲^{−}, respectively, to R^{p,q}, given by the matrix expressions above. The standard basis of R^{p,q} corresponds under i^{−} to the basis of 𝔲^{−}

U_{i} = \{ \begin{matrix}E_{i}^{0} − E_{n+1}^{n+1−i} & i ∈ \{1, …, p\} ∪ \{q + 1, …, n\} \\ E_{i}^{0} − E_{n+1}^{i} & i ∈ \{p + 1, …, q\} \end{matrix}

where E_{i}^{j} is the (n + 2)-dimensional square matrix with all entries 0 except for a 1 in the (i, j) place.

The parabolic Lie algebra 𝔭 ≅ 𝔯 ⋉ 𝔲^{+} is the Lie algebra of the stabilizer P of [e_{0}] in PO(p + 1, q + 1), and similarly for 𝔭^{−} ≅ 𝔯 ⋉ 𝔲^{−}, the Lie algebra of the stabilizer of [e_{n+1}]. The groups P and P^{−} are isomorphic to the semidirect product CO(p, q) ⋉ R^{p,q}, and i^{+} (respectively i^{−}) intertwines the adjoint action of P on 𝔲^{+} (respectively 𝔲^{−}) with the conformal action of CO(p, q) on R^{p,q}.

2.1.3. Translations in PO(p + 1, q + 1). Let U^{+} be the closed subgroup of PO(p + 1, q + 1) with Lie algebra 𝔲^{+}.

Definition 2.1. A translation of PO(p + 1, q + 1) is an element which is conjugate in PO(p + 1, q + 1) to an element of U^{+}. A translation of 𝔬(p + 1, q + 1) is an element generating a 1-parameter group of translations of PO(p + 1, q + 1).

This terminology is justified because a translation is a conformal transformation of Ein^{p,q} fixing a point, say x, and acting as a translation on M(x). Notice that there are three conjugacy classes of translations in O(p+1, q+1): lightlike (we will also say null), spacelike, and timelike. An example of a null translation is the element T = (i^{+})^{−1}(1, 0, …, 0) of 𝔲^{+}. Since any null translation of 𝔭 is conjugate under P to T, the reader will easily check the following fact, that will be used several times below.

Fact 2.2. Let T ∈ 𝔭 be a nontrivial null translation and 𝔠(T) the centralizer of T in 𝔬(p + 1, q + 1). Then 𝔠(T) ∩ 𝔭 is of codimension one in 𝔠(T).

2.2. Bounds in PO(p + 1, q + 1). The first step for proving theorem 1.1 is to show that any nilpotent subalgebra of 𝔬(p + 1, q + 1) has degree ≤ 2p + 1. We will actually prove more:

Proposition 2.3. For a nilpotent subalgebra 𝔥 ⊂ 𝔬(p + 1, q + 1), the degree d(𝔥) ≤ 2p + 1. Assuming p ≥ 1, if d(𝔥) = d ≥ 2p, then 𝔥 contains a translation in its center; in fact, 𝔥_{d−1} consists of null translations.

2.2.1. Preliminary results. The following definitions will be relevant below. Let 𝔩 ⊂ 𝔤𝔩(n) be a subalgebra. The set of all compositions Π_{1}^{k}X_{i}, where X_{1}, …, X_{k} ∈ 𝔩, will be denoted 𝔩^{k}. We say that 𝔩 is a subalgebra of nilpotents if there exists k ≥ 1 such that 𝔩^{k} = 0. The minimal such k will be called the order of nilpotence of 𝔩, denoted o(𝔩). By Lie’s theorem, subalgebras of nilpotents coincide with those subalgebras of 𝔤𝔩(n), the elements of which are nilpotent matrices. If 𝔥 is a nilpotent Lie algebra, then ad 𝔥 ⊂ 𝔤𝔩(𝔥) is a subalgebra of nilpotents and d(𝔥) = o(ad 𝔥). For V a vector space with form B, a Lie subalgebra 𝔥 ⊂ 𝔠𝔬(V) is infinitesimally conformal if for all u, v ∈ V and X ∈ 𝔥,

B(Xu, v) + B(u, Xv) = λ(X)B(u, v)

for some infinitesimal character λ : 𝔥 → R. Of course, the Lie algebra of a subgroup of CO(V) ⊂ GL(V) acts by infinitesimally conformal endomorphisms of V.

Lemma 2.4. Let 𝔩 be a Lie algebra and V a finite dimensional 𝔩-module. Let Y ∈ 𝔥_{k−1} and v ∈ V. Then Y (v) ∈ 𝔩^{k}(V).

This lemma is easily proved by induction, using the Jacobi identity.

Lemma 2.5. Let V be a vector space with a symmetric bilinear form B of type (p, q). If \overline{𝔲} ⊂ 𝔠𝔬(V) is a subalgebra of nilpotents, then o(\overline{𝔲}) ≤ 2p + 1.

Proof: Note that \overline{𝔲} is infinitesimally isometric because there are no nontrivial infinitesimal characters \overline{𝔲} → R. If p = 0, then \overline{𝔲} ⊂ 𝔬(n), in which case \overline{𝔲} must be trivial. Now assume p ≥ 1. Let U be the connected group of unipotent matrices in CO(p, q) with Lie algebra \overline{𝔲}. Because U consists of unipotent matrices, it lies in a minimal parabolic subgroup of CO(p, q), hence leaves invariant some isotropic p-plane N ⊂ V. The order of \overline{𝔲} on both N and V/N^{⊥} is at most p, because each is dimension p. Because N^{⊥}/N inherits a positive-definite inner product that is infinitesimally conformally invariant by \overline{𝔲}, the order of \overline{𝔲} on it is 1. Then o(\overline{𝔲}) ≤ 2p + 1, as desired. ♢

2.2.2. Proof of proposition 2.3. Let H be the connected subgroup of G = PO(p + 1, q + 1) with Lie algebra 𝔥. We recall some facts from the theory of algebraic groups. First, the nilpotence degrees of a connected group and its Zariski closure are the same, so that there is no loss of generality assuming H Zariski closed. Then there is an algebraic Levi decomposition of the Lie algebra 𝔥 ≅ 𝔯 × 𝔲, where 𝔯 is abelian and comprises the semisimple elements of 𝔥, and 𝔲 consists of nilpotents (see for example [Bo, thm 10.6]). If 𝔲 is trivial, 𝔥 is abelian and proposition 2.3 is proved. If not, d(𝔥) = d(𝔲), so we will assume that 𝔥 = 𝔲. Moreover, 𝔲 is contained in a minimal parabolic subalgebra of 𝔬(p+1, q+1), and so conjugating 𝔲 if necessary, we have 𝔲 ⊂ 𝔭. Thanks to i^{+} (see 2.1.2), we identify 𝔲 with a subalgebra of 𝔠𝔬(p, q) ⋉ R^{p,q}. Denote by \overline{𝔲} the projection to 𝔠𝔬(p, q), which is actually in 𝔬(p, q) since \overline{𝔲} is a subalgebra of nilpotents. For any natural number k,

(1)  𝔲_{k} ⊆ \overline{𝔲}_{k} + \overline{𝔲}^{k}(R^{p,q})

The proof by induction of this relation is straightforward using lemma 2.4 for 𝔩 = 𝔲 and V = R^{p,q}, and is left to the reader. When p = 0, then any nilpotent subalgebra \overline{𝔲} ⊂ 𝔬(1, q + 1) is abelian, by the remarks above combined with lemma 2.5. We thus have d(𝔲) ≤ 2p + 1 when p = 0. Now proceed inductively on p, using lemma 2.5 and relation (1) to obtain d(𝔲) ≤ o(\overline{u}) ≤ 2p + 1 whenever 𝔲 ⊂ 𝔬(p + 1, q + 1) is nilpotent, for all p ∈ N. Next suppose that d(𝔲) = d ≥ 2p ≥ 2. Since \overline{𝔲} is a nilpotent subalgebra of 𝔬(p, q), its nilpotence degree is at most 2p − 1 by the first part of the proof. Since d ≥ 2p, \overline{𝔲}_{d−1} = 0 and

0 ≠ 𝔲_{d−1} ⊆ \overline{𝔲}^{d−1}(R^{p,q})

so an element of 𝔲_{d−1} can be written

w = Y_{1} ⋯ Y_{d−1}(v) for Y_{1}, …, Y_{d−1} ∈ \overline{𝔲}, v ∈ R^{p,q}

Further, any Y ∈ \overline{𝔲} annihilates w. Because Y_{1} ∈ \overline{𝔲} is infinitesimally conformal and nilpotent, it is infinitesimally isometric. Then

Q^{p,q}(w, w) = ⟨Y_{1} ⋯ Y_{d−1}(v), Y_{1} ⋯ Y_{d−1}(v)⟩ = −⟨Y_{2} ⋯ Y_{d−1}(v), Y_{1}(w)⟩ = 0

and so w is a null translation. ♢

2.3. Conformal structures as Cartan geometries. In the sequel, it will be fruitful to study pseudo-Riemannian structures in the setting of Cartan geometries. A Cartan geometry modeled on some homogeneous space X = G/P is a curved analogue of X.

Definition 2.6. Let G be a Lie group with Lie algebra 𝔤 and P a closed subgroup of G such that Ad P is faithful on 𝔤. A Cartan geometry (M, B, ω) modeled on (𝔤, P) is

(1) a principal P -bundle π : B → M

(2) a 𝔤-valued 1-form ω on B satisfying • for all b ∈ B, the restriction ω_{b} : T_{b}B → 𝔤 is an isomorphism

• for all b ∈ B and Y ∈ 𝔥, the evaluation ω_{b}( \frac{d}{dt} |_{0} be^{tY}) = Y
• for all b ∈ B and h ∈ P, the pullback R_{h}^{∗}ω = Ad h^{−1} ∘ ω

For the model X, the canonical Cartan geometry is the triple (X, G, ω_{G}), where ω_{G} denotes the left-invariant 𝔤-valued 1-form on G, called the Maurer-Cartan form. A conformal structure (M, [σ]) of type (p, q) with p + q ≥ 3, defines, up to isomorphism, a canonical Cartan geometry (M, B, ω) modeled on Ein^{p,q}— that is, on (𝔬(p + 1, q + 1), P). The interested reader will find the details of this solution, originally due to E. Cartan, of the so-called equivalence problem in [Sh, ch 7]. The group Aut M comprises the bundle automorphisms of B preserving ω. Any conformal diffeomorphism lifts to an element of Aut M, maybe not unique, but the fibers of the projection from Aut M to the conformal group of M are discrete. The Lie algebras of Aut M and Conf M are thus isomorphic. We will not distinguish in notation between an element f ∈ Conf M and the corresponding lift to Aut M.

3. General degree bound: proof of theorem 1.1

In this section, we use the interpretation of conformal structures as Cartan geometries to prove theorem 1.1. Let (M, B, ω) be a Cartan geometry modeled on (𝔤, P), and H < Aut M a connected Lie group. Since H acts on B, each vector X ∈ 𝔥 defines a Killing field on B, and for every b ∈ B, we will call X(b) the value of this Killing field at b. Thus, each point b ∈ B determines a linear embedding

s_{b} : 𝔥 → 𝔤
X ↦ ω_{b}(X(b))

The injectivity of s_{b} comes from the fact that H preserves a framing on B, hence acts freely (see [Ko, I.3.2]). The image s_{b}(𝔥) will be denoted 𝔥^{b}, and, for X ∈ 𝔥, the image s_{b}(X) will be denoted X^{b}. In general, s_{b} is not a Lie algebra homomorphism, except with respect to stabilizers (see [Sh, 5.3.10]): for any X, Y ∈ 𝔥 and b ∈ B such that Y^{b} ∈ 𝔭,

[X, Y]^{b} = [X^{b}, Y^{b}]

Observing that Y belongs to the stabilizer 𝔥(π(b)) if and only if Y^{b} ∈ 𝔭, we deduce the following fact.

Fact 3.1. If 𝔥^{b}∩𝔭 is codimension at most 1 in 𝔥^{b}, then 𝔥^{b} is a Lie subalgebra of 𝔤, isomorphic to 𝔥.

The following result implies theorem 1.1. It is more precise and will be useful for the proof of theorem 1.2:

Theorem 3.2. Let (M, [σ]) be a compact manifold with a type-(p, q) conformal structure, and let (M, B, ω) be the associated Cartan geometry. Let H < Aut M be a connected nilpotent Lie group. Then d(H) ≤ 2p + 1. If d(H) = 2p + 1, then every H-invariant closed subset F ⊂ M contains a point x such that

(1) The dimension of the orbit H.x is at most 1.

(2) For every b ∈ π^{−1}(x), 𝔥^{b} is a subalgebra of 𝔬(p + 1, q + 1).

(3) There exists X ∈ 𝔥 such that X^{b} is a lightlike translation in 𝔭 for every b ∈ π^{−1}(x), and X^{b} is in the center of 𝔥^{b}.

A consequence of this theorem is that when d(H) = 2p + 1, there are points with nontrivial stabilizers, because X as in (3) generates a 1-parameter subgroup of the stabilizer H(x). We will study the dynamics near x of this flow in the proof of theorem 1.2.

Proof: (of theorem 3.2) Let F ⊂ M be closed and H-invariant. The group H is amenable, so it preserves a finite Borel measure on F. Then the embedding theorem of [BFM, thm 4.1] with S = H gives x ∈ F and an algebraic subgroup Š < Ad_{𝔤}P such that, for all b ∈ π^{−1}(x),

(1) 𝔥^{b} is Š-invariant

(2) s_{b} intertwines the Zariski closure of Ad H in Aut 𝔥 with Š|_{𝔥^{b}}

Since the adjoint representation of PO(p + 1, q + 1) is algebraic and faithful, Š is the image of an algebraic subgroup of P, which we will also denote Š. We denote the corresponding Lie algebra by 𝔰̌ ⊂ 𝔭. The embedding theorem says that for any X ∈ 𝔥, there exists X̌ ∈ 𝔰̌ such that for all Y ∈ 𝔥,

[X, Y]^{b} = [X̌, Y^{b}]

Suppose that d = d(𝔥) ≥ 2p + 1. Because 𝔰̌ is algebraic, there is a decomposition 𝔰̌ ≅ 𝔯 ⋉ 𝔲 with 𝔯 reductive and 𝔲 consisting of nilpotent elements (see [WM, 4.4.7]). Because ad 𝔥 consists of nilpotents, the subalgebra 𝔯 is in the kernel of restriction to 𝔥^{b}, and 𝔲 maps onto ad 𝔥. Therefore, for l = d(𝔲),

2p ≤ d − 1 = d(ad 𝔥) ≤ l ≤ 2p + 1

where the upper bound comes from proposition 2.3. Also by this proposition, 𝔲_{l−1} consists of null translations. Whether l = d − 1 or d, we will show that 𝔥^{b} centralizes a null translation in 𝔭, from which fact we will obtain the bound and points (1) and (3). First suppose l = d − 1. Then 𝔲_{d−2} consists of null translations and acts on 𝔥^{b} as ad 𝔥_{d−2}, which means it centralizes (𝔥_{1})^{b}. Then by facts 2.2 and 3.1, (𝔥_{1})^{b} embeds homomorphically in 𝔬(p + 1, q + 1). The order of 𝔲 on (𝔥_{1})^{b} is d − 1; further, 𝔲 and (𝔥_{1})^{b} generate a nilpotent subalgebra 𝔫 of order d − 1, in which (𝔥_{1})^{b} is an ideal. Since d − 1 ≥ 2p, proposition 2.3 implies that the commutators 𝔫_{d−2} are all null translations. But 𝔫_{d−2} contains

𝔲^{d−2}(𝔥_{1})^{b} = (𝔥_{d−1})^{b}

Because 𝔲 preserves (𝔥_{1})^{b} ∩ 𝔭 and acts by nilpotent transformations on (𝔥_{1})^{b}/((𝔥_{1})^{b} ∩ 𝔭), which is 1-dimensional,

𝔲^{1}(𝔥_{1})^{b} = (𝔥_{2})^{b} ⊂ 𝔭

Thus (𝔥_{k})^{b} ⊂ 𝔭 as soon as k ≥ 2, so for any X ∈ 𝔥, Y ∈ 𝔥_{k}, we have [X, Y]^{b} = [X^{b}, Y^{b}]. In particular, (𝔥_{d−1})^{b}, the image under s_{b} of the center of 𝔥, commutes with 𝔥^{b}, so that 𝔥^{b} is in the centralizer of a nonzero null translation. Next suppose l = d. Then 𝔲_{d−1} centralizes 𝔥^{b} because it acts as ad 𝔥_{d−1}. By proposition 2.3, 𝔲_{d−1} consists of null translations, so 𝔥^{b} commutes with a nonzero null translation in 𝔭. Given that 𝔥^{b} centralizes a nonzero null translation in 𝔭, 2.2 implies point (1); moreover, (𝔥^{b})_{1} ⊂ 𝔭. By fact 3.1, s_{b} : 𝔥 → 𝔬(p + 1, q + 1) is a homomorphic embedding. The assumption d ≥ 2p+1 and proposition 2.3 forces d = 2p+1, proving the bound. Also by proposition 2.3, (𝔥^{b})_{2p}, which is central in 𝔥^{b}, consists of null translations. Then (𝔥^{b})_{2p} ⊂ (𝔥^{b})_{1} ⊂ 𝔭, and (3) is proved. Finally, item (2) of the theorem is a consequence of item (1) and fact 3.1. ♢ 3.1. Outline of the proof of theorem 1.2. For x ∈ M, denote by H(x) the stabilizer of x in H. For each b ∈ π^{−1}(x), the action of H by automorphisms of the principal bundle B gives rise to an injective homomorphism ρ_{b} : H(x) → P. Theorem 3.2 says that if d(H) = 2p + 1, then each Hinvariant closed set F contains a point x_{0}, such that for some 1-parameter group h^{s} in H(x_{0}) and b_{0} ∈ π^{−1}(x_{0}), the image ρ_{b_{0}}(h^{s}) is a 1-parameter group τ^{s} of null translations in P. The dynamics of null translations are studied in section 4.1 and summarized in fact 4.1. In section 4.2, we make the crucial link between the dynamics of τ^{s} on Ein^{p,q} and those of h^{s} on M, via the respective actions on special curves, called geodesics, in the two Cartan bundles. The actions on these curves are conjugate locally by the exponential maps of the two Cartan geometries. In section 4.3, we deduce from this relationship the dynamics of the h^{s}-action and develop a method to precisely compute the differential of h^{s} near x_{0}; see proposition 4.5. In section 5 we use our description of the h^{s}-action to show that M is conformally flat, namely locally modeled on Ein^{p,q} (see proposition 5.1). First we reduce the claim to showing flatness on a neighborhood of x_{0}. Proposition 5.2 of section 5.1 establishes flatness on a nonempty open subset. In section 5.2, we make a technical modification of proposition 5.2 in order to show that this flat set includes a neighborhood of x_{0} (see proposition 5.7). The purpose of section 6 is to understand the global structure of M, using classical techniques for (G, X)-structures. We again use the dynamics of h^{s} to show that the subset of M tending under the forward or reverse flow to a fixed curve develops to the complement of the fixed set of τ^{s} in Ein^{p,q}; the Lorentzian case is treated first in section 6.4, while the more complicated case p ≥ 2 is in section 6.5. In both cases, we apply the the theorem [Fr5, thm 1.8] on boundaries of embeddings of flat Cartan geometries to conclude. This final section completes the proof of theorem 1.2.

4. Conformal dynamics

This section establishes properties of the flow h^{s} that will later be used to show M is conformally flat. Along the way, we develop some general tools to relate the behavior of an automorphism of a Cartan geometry with the behavior of the corresponding holonomy on the model space.

4.1. Dynamics of null translations on Ein^{p,q}. The first task is to describe the dynamics of 1-parameter groups of null translations in the model space Ein^{p,q}. Let τ^{s} be the flow generated by the null translation T = (i^{+})^{−1}(1, 0, …, 0) of section 2.1.3. The action of τ^{s} on Ein^{p,q} is given in projective coordinates by

τ^{s} : [y_{0} : ⋯: y_{n+1}] ↦ [y_{0} + sy_{n} : y_{1} − sy_{n+1} : y_{2} : ⋯: y_{n+1}]

The fixed set is

F = P(e_{0}^{⊥} ∩ e_{1}^{⊥} ∩ N^{p+1,q+1})

When p = 2, it has codimension 2, and contains a singular circle

Λ = P(span{e_{0}, e_{1}}) ⊂ F

When p = 1, then F = Λ; since p + q ≥ 3, the codimension of F is also at least 2 in this case. If y ∉ F, then

τ^{s}.y → [y_{n} : −y_{n+1} : 0 : ⋯: 0] ∈ Λ as s → ∞

Every point x ∈ Ein^{p,q} lies in some C(y) for y ∈ Λ, and y is unique when x ∉ F. We summarize the dynamics of τ^{s} near Λ; see also figure 2:

Fact 4.1. The complement of the closed, codimension-2 fixed set F of τ^{s} in Ein^{p,q} is foliated by subsets of lightcones Č(y) = C(y)\(C(y)∩F), for y ∈ Λ. Points x ∈ Č(y) tend under τ^{s} to y along the lightlike geodesic containing x and y; in particular, τ^{s} preserves setwise all null geodesics emanating from points of Λ.

Figure 2. local picture of flow by null translation τ^{s} 4.2. Geodesics and holonomy. In this section (M, B, ω) will be a Cartan geometry modeled on G/P. The form ω on B determines special curves, the geodesics. Here they will be defined as projections of curves with constant velocity according to ω—that is, γ : (−ϵ, ϵ) → M is a geodesic if γ(t) = π(γ̂(t)) where

ω(γ̂′(t)) = ω(γ̂′(0)) for all t ∈ (−ϵ, ϵ)

Geodesics on the flat model space (G/P, G, ω_{G}) are orbits of 1-parameter subgroups. Note that this class of curves is larger than the usual set of geodesics in case the Cartan geometry corresponds to a pseudo-Riemannian metric or a conformal pseudo-Riemannian structure (see [Fi], [Fri], [Fri-S] for a definition of conformal geodesics). The exponential map is defined on B × 𝔤 in a neighborhood of B × {0} by

exp(b, X) = exp_{b}(X) = γ̂_{X,b}(1)

where γ̂_{X,b}(0) = b and ω(γ̂′_{X,b}(t)) = X for all t. Let h ∈ Aut M. Then h carries geodesics in M to geodesics:

ĥ ∘ γ̂_{X,b} = γ̂_{X,ĥ(b)}

Suppose that h^{s} is a 1-parameter group of automorphisms Then for any b_{0} ∈ B, the curve parametrized by the flow γ̂(s) = h^{s}.b_{0} projects to a geodesic γ(s) in M: it is easy to see that γ̂(s) has ω-constant velocity.

Definition 4.2. If h ∈ Aut M fixes x, and b ∈ π^{−1}(x), then the element g ∈ P such that h.b = bg is the holonomy of h with respect to b. More generally, given a local section σ : U → B, the holonomy of h ∈ Aut M with respect to σ at some point x ∈ U ∩ h^{−1}.U is g such that h.σ(x) = σ(h.x)g

If H(x) is the stabilizer of x in H < Aut M and b ∈ π^{−1}(x), then the holonomy with respect to b gives a monomorphism ρ_{b} : H(x) → P. Replacing b with bp has the effect of post-composing with conjugation by p^{−1}. For automorphisms fixing a point x_{0}, the holonomy with respect to some b_{0} ∈ π^{−1}(x_{0}) tells a lot about the action in a neighborhood of x_{0} via the exponential map. If, moreover, an automorphism h fixes x_{0} and preserves the image of a geodesic γ emanating from x_{0}, then the holonomy at x_{0} determines the holonomy along γ, as follows. For X ∈ 𝔤, denote by e^{X} the exponential of X in G.

Proposition 4.3. Suppose that h ∈ Aut M fixes a point x_{0} and has holonomy g_{0} with respect to b_{0} ∈ π^{−1}(x_{0}). Let γ(t) = π(exp(b_{0}, tX)) for X ∈ 𝔤, defined on an interval (α, β) containing 0. Suppose there exist

• a path g : (α, β) → P with g(0) = g_{0}
• a diffeomorphism c : (α, β) → (α′, β′)

such that, for all t ∈ (α, β),

g_{0}e^{tX} = e^{c(t)X}g(t)

Then

(1) The curves exp(b_{0}, c(t)X) and γ(c(t)) are defined for all t ∈ (α, β), and h.γ(t) = γ(c(t)).

(2) Viewing exp(b_{0}, tX) as a section of B over γ(t), the holonomy of h at γ(t) with respect to this section is g(t).

Proof: In G, reading the derivative of g_{0}e^{tX} with ω_{G} gives (see [Sh, 3.4.12])

X = (Ad g(t)^{−1})(c′(t)X) + ω_{G}(g′(t))

Because h preserves ω, the derivative of

h. exp(b_{0}, tX) = exp(b_{0}g_{0}, tX)

according to ω is X for all t ∈ (α, β). On the other hand, it is also true in B that whenever t ∈ (α, β) and exp(b_{0}, c(t)X) is defined,

ω((exp(b_{0}, c(t)X)g(t))′) = (Ad g(t)^{−1})(c′(t)X) + ω_{G}(g′(t))

This formula follows from the properties of ω in the definition 2.6 of a Cartan geometry; see [Sh, 5.4.12]. Therefore, because the two curves have the same initial value and the same derivatives, both are defined for t ∈ (α, β), and

h. exp(b_{0}, tX) = exp(b_{0}, c(t)X)g(t)

This proves (2); item (1) follows by projecting both curves to M. ♢ We record one more completeness result that will be useful below, for flows that preserve a geodesic, but do not necessarily fix a point on it.

Proposition 4.4. Let X ∈ 𝔤 and b_{0} ∈ B be such that exp(b_{0}, tX) is defined for all t ∈ (α, β), where α < 0 < β. Suppose that for some Y ∈ 𝔤, there exists g : (α, β) → P such that for all t ∈ (α, β),

e^{tX} = e^{c(t)Y} g(t)

in G, where c is a diffeomorphism (α, β) → R fixing 0. Then exp(b_{0}, tY) is defined for all t ∈ R.

Proof: In G, we have for all t ∈ (α, β)

Y = \frac{1}{c′(t)}[Ad g(t)][X − ω_{G}(g′(t))] (2)

Let c^{−1}(s) be the inverse diffeomorphism (−∞, ∞) → (α, β). Define, for s ∈ R,

γ̂(s) = exp(b_{0}, c^{−1}(s)X)g(c^{−1}(s))^{−1}

The derivative of the right-hand side is

\frac{1}{c′(c^{−1}(s))}[Ad g(c^{−1}(s))][X − ω_{G}(g′(c^{−1}(s))]

For t = c^{−1}(s), this reduces to the right-hand side of (2). Then exp(b_{0}, sY) equals γ̂(s) and is defined for all s ∈ R. ♢

4.3. Dynamics of h^{s} on M. We now return to the pseudo-Riemannian manifold (M, σ) with associated Cartan geometry (M, B, ω) modeled on the pair (𝔬(p + 1, q + 1), P) corresponding to the homogeneous space Ein^{p,q}. The subalgebra 𝔲^{−} complementary to 𝔭 and the basis U_{1}, …, U_{n} are as in section 2.1.2. Let N (𝔲^{−}) be the null cone with respect to Q^{−} := (i^{−})^{∗}(Q^{p,q}) in 𝔲^{−} (see 2.1.2 for the definition of i^{−}). The following proposition captures properties of the flow h^{s} given by theorem 3.2 that reflect properties of τ^{s} established above, and that will be used to prove flatness in the next section. Now we suppose the group H < Conf M is nilpotent of maximal degree 2p + 1. The point x_{0} is given by theorem 3.2, and h^{s}.x_{0} = x_{0}. We have b_{0} ∈ π^{−1}(x_{0}) for which the holonomy of h^{s} is τ^{s} as in section 4.1.

Proposition 4.5. Suppose H < Conf M is nilpotent of maximal degree 2p + 1. Let F ⊆ M be closed and H-invariant subset, and let x_{0} ∈ F and X ∈ 𝔥 be given by theorem 3.2. Let h^{s} be the flow generated by X. Then there is b_{0} ∈ π^{−1}(x_{0}) for which the following holds:

(1) Let Δ̂(v) = exp(b_{0}, vU_{1}) with domain I_{Δ} ⊆ R; let Δ = π ∘ Δ̂. Then the flow h^{s} fixes pointwise the geodesic Δ, and, for v ∈ I_{Δ}, has holonomy at Δ(v) with respect to Δ̂ equal τ^{s}.

(2) There is an open subset S ⊂ N (𝔲^{−}) such that S ∪ −S is dense in N (𝔲^{−}), and for all v ∈ I_{Δ} and U ∈ S, if the geodesic β(t) = π ∘ exp(Δ̂(v), tU) is defined on (−ϵ, ϵ), then the flow h^{s} preserves β and reparametrizes by

c(t) = \frac{t}{1 + st}

for t ∈ (−ϵ, ϵ). In particular, for t > 0 (t < 0),

h^{s}(β(t)) → Δ(v) as s → ∞ (s → −∞)

Moreover, β(t) is complete.

(3) There is a framing f_{1}(t), …, f_{n}(t) of M along β(t) for which the derivative

h_{∗}^{s}(f_{i}(t)) = (\frac{1}{1 + st})^{σ(i)} f_{i}(c(t))

where

σ(i) = \{ \begin{matrix}0 & i = 1 \\ 1 & i ∈ \{2, …, n − 1\} \\ 2 & i = n \end{matrix}

To prove this proposition, we will compute the relevant holonomies of τ^{s} in the model space and use proposition 4.3. We start with some algebraic facts pertaining to Ein^{p,q}. Let 𝔯 be as in section 2.1.2, a maximal reductive subalgebra of 𝔭.

Lemma 4.6. Let R ≅ CO(p, q) be the connected subgroup of P with Lie algebra 𝔯, and let S be the unipotent radical of the stabilizer in R of U_{1}.

(1) Fix(Ad τ^{s}) ∩ 𝔲^{−} = RU_{1}

(2) Let S = R_{>0}^{∗} · S.U_{n}. Then S ∪ −S is open and dense in N (𝔲^{−}).

(3) The subgroups S and τ^{s} commute.

Proof:

(1) Recall that T is the infinitesimal generator for τ^{s} defined in section 2.1.3. It suffices to show

RU_{1} = ker(ad T) ∩ 𝔲^{−}

We leave this basic linear algebra calculation to the reader.

(2) We will show that S consists of all U ∈ N (𝔲^{−}) with ⟨U, U_{1}⟩ > 0; these elements and their negatives form an open dense subset of N (𝔲^{−}).

First, if g ∈ S, then g.U_{n} ∈ N (𝔲^{−}) and

⟨g.U_{n}, U_{1}⟩ = ⟨g.U_{n}, gU_{1}⟩ = ⟨U_{n}, U_{1}⟩ = 1

Both N (𝔲^{−}) and the property ⟨U, U_{1}⟩ > 0 are invariant by the action of R_{>0}^{∗}, so S is contained in the claimed subset.

Next let U ∈ N (𝔲^{−}) be such that ⟨U, U_{1}⟩ > 0. Replace U with a positive scalar multiple so that ⟨U, U_{1}⟩ = 1. Define g ∈ S by

g : U_{1} ↦ U_{1}
U_{n} ↦ U
V ↦ V − ⟨V, U⟩ · U_{1} for V ∈ {U_{1}, U_{n}}^{⊥}

It is easy to see that g is unipotent and belongs to O(Q^{−}), and thus defines an element of R. Therefore U ∈ S.

(3) Both S and τ^{s} lie in the unipotent radical of P, which, in the chosen basis, is contained in the group of upper-triangular matrices. The commutator of any unipotent element with τ^{s} is I_{n+2} + cE_{0}^{n+1} for some c ∈ R. There is no such element of O(p + 1, q + 1) for any nonzero c, so the commutator is the identity.

♢

Proof: (of proposition 4.5) Because Ad τ^{s} fixes U_{1}, the corresponding 1-parameter subgroups commute in G:

τ^{s}e^{vU_{1}} = e^{vU_{1}}τ^{s}

Then by proposition 4.3, the flow h^{s} fixes Δ pointwise, and the holonomy of h^{s} with respect to Δ̂ at any Δ(v), v ∈ I_{Δ}, equals τ^{s}. This proves (1). To prove (2), first consider the null geodesic α(t) = π(e^{tU_{n}}) in G/P, and let α̂(t) = e^{tU_{n}}. Now it is possible to compute the holonomy of τ^{s} along α with respect to α̂:

τ^{s} · α̂(t) = τ^{s} · e^{tU_{n}}
= α̂(c(t)) · e^{−c(t)U_{n}} · τ^{s} · e^{tU_{n}}

Refer to the expression for τ^{s} in section 4.1, and compute directly in O(p + 1, q + 1)

e^{−c(t)U_{n}} · τ^{s} · e^{tU_{n}} = diag(1 + st, 1 + st, 1, …, 1, \frac{1}{1 + st}, \frac{1}{1 + st}) + sT

Denote this holonomy matrix by h(s, t). Now let S < G be as in lemma 4.6, and let U = (Ad g)(U_{n}) with g ∈ S. Let α̂(t) = e^{tU}. Because τ^{s} commutes with g by lemma 4.6 (3), we can compute the holonomy of τ^{s} with respect to α̂ along α:

τ^{s} · α̂(t) = τ^{s} · e^{tU} = τ^{s} · e^{(Ad g)(tU_{n})}
= τ^{s} · g · e^{tU_{n}} · g^{−1} = g · τ^{s} · e^{tU_{n}} · g^{−1}
= g · e^{c(t)U_{n}} · h(s, t) · g^{−1} = e^{(Ad g)(c(t)U_{n})} · g · h(s, t) · g^{−1}
= α̂(c(t)) · g · h(s, t) · g^{−1}

Let S be as in lemma 4.6 (2). Let U ∈ S. Let β̂(t) = exp(Δ̂(v), tU) and β = π ∘ β̂, and assume β̂ is defined on (−ϵ, ϵ). From (1), the holonomy of h^{s} at Δ(v) with respect to Δ̂ is τ^{s}. The above calculation, together with proposition 4.3 (1), implies

h^{s}.β(t) = β(c(t))

for all t ∈ (−ϵ, ϵ). Taking s = ±1/ϵ and again applying proposition 4.3 (1) proves completeness of β(t). Then point (2) is proved. By proposition 4.3 (2), the holonomy of h^{s} at β(t) with respect to β̂ is g · h(s, t) · g^{−1}. The adjoint of h(s, t) on 𝔤/𝔭 in the basis comprising the images of U_{1}, …, U_{n} is

diag(1, \frac{1}{1 + st}, …, \frac{1}{1 + st}, \frac{1}{(1 + st)^{2}} )

Since S is contained in P, for g ∈ S, the span of (Ad g)(U_{1}), …, (Ad g)(U_{n}) is transverse to 𝔭. The adjoint of g · h(s, t) · g^{−1} in the corresponding basis of 𝔤/𝔭 is of course the same diagonal matrix as for g = 1. For β̂ and β as above, define a framing f_{1}, …, f_{n} along β by

f_{i}(β(t)) = (π_{∗} ∘ ω_{β̂(t)}^{−1} ∘ Ad g)(U_{i})

Now we can compute the derivative of h^{s} along β in the framing (f_{1}, …, f_{n}). Recall the identity for a Cartan connection

ω_{p}^{−1} ∘ (Ad g) = R_{g^{−1}∗} ∘ ω_{pg}^{−1}

We will write f_{i}(t) in place of f_{i}(β(t)) below.

h_{∗}^{s}(f_{i}(t)) = (π_{∗} ∘ h_{∗}^{s} ∘ ω_{β̂(t)}^{−1} ∘ Ad g) (U_{i})
= (π_{∗} ∘ ω_{h^{s}·β̂(t)}^{−1} ∘ Ad g) (U_{i})
= (π_{∗} ∘ R_{g^{−1}∗} ∘ ω_{h^{s}·β̂(t)·g}^{−1}) (U_{i})
= (π_{∗} ∘ ω_{β̂(c(t))·g·h(s,t)}^{−1}) (U_{i})
= (π_{∗} ∘ R_{g·h(s,t)∗} ∘ ω_{β̂(c(t))}^{−1} ∘ Ad(g · h(s, t))) (U_{i})
= (π_{∗} ∘ ω_{β̂(c(t))}^{−1} ∘ Ad(g · h(s, t) · g^{−1}) ∘ Ad g) (U_{i})
= (\frac{1}{1 + st})^{σ(i)} f_{i}(c(t))

which proves (3). ♢

5. Maximal degree of nilpotence implies conformal flatness

This section is devoted to the proof of the following proposition, the next step towards theorem 1.2.

Proposition 5.1. If the group H and the pseudo-Riemannian manifold M satisfy the assumptions of theorem 1.2, then M is conformally flat.

Recall that a type (p, q) pseudo-Riemannian manifold is conformally flat whenever it is locally conformally equivalent to Ein^{p,q}. If dim M ≥ 4, conformal flatness is equivalent to the vanishing the Weyl curvature W, which is a conformally invariant (3, 1) tensor on M; if dim M = 3, then vanishing of the (3, 0) Cotton tensor characterizes flatness (see [AG, p 131]). For the canonical Cartan geometry (M, B, ω) associated to the conformal structure (M, [σ]), the Cartan curvature is defined as follows: the 2-form

dω + \frac{1}{2}[ω, ω]

on B vanishes on u ∧ v at b whenever u or v is tangent to the fiber of b. We define the Cartan curvature K to be the resulting function B → Λ^{2}(𝔤/𝔭)^{∗}⊗𝔤 (see [Sh, 5.3.22]). Vanishing of K on B is equivalent to M being conformally flat (see [Sh, ch 7] or [Ko, ch IV]). The function K is Aut M-invariant and P -equivariant; in particular, if K(b) = 0 for b ∈ π^{−1}(x), then K vanishes on the fiber of B over x. In this case we will also say that K vanishes at x.

To prove proposition 5.1, we suppose V ⊂ M is not flat. Then ∂V is a nonempty H-invariant closed subset. Under the assumptions of theorem 1.2, there exists a flow h^{s} of H, a point x_{0} ∈ ∂V, and b_{0} ∈ π^{−1}(x_{0}) such that the holonomy of h^{s} with respect to b_{0} is the null translation τ^{s} studied in 4.1. We use the differential of the flow h^{s} computed above plus an idea of [Fr3] to show vanishing of the Weyl and Cotton tensors along any null geodesic β emanating from a point on Δ. Next we examine geodesic triangles in this set of vanishing curvature to show in proposition 5.7 that in fact the Weyl and Cotton tensors vanish in a neighborhood of x_{0}—a contradiction. The proof of proposition 5.7 below will require several preliminary results exposed in subsections 5.1 and 5.2.

5.0.1. Notation. In the two following subsections, the points x_{0} and b_{0} and the 1-parameter groups h^{s} and τ^{s} are as in the paragraph above. The infinitesimal generator of τ^{s} is T given in subsection 4.1. Recall also 𝔲^{−} and the basis U_{1}, …, U_{n} first defined in section 2.1.2. See proposition 4.5 for the definitions of Δ, I_{Δ}, and S ⊂ N (𝔲^{−}). Recall that each curve β(t) = exp(Δ̂(v), tU) with U ∈ S and v ∈ I_{Δ}, is defined for all t ∈ R.

5.1. Vanishing on lightcones emanating from Δ. The aim of this subsection is the proof of:

Proposition 5.2. For every U ∈ S and v ∈ I_{Δ}, the Cartan curvature of (M, B, ω) vanishes on π^{−1}(β(t)) for all t ∈ R, where β(t) = π∘exp(Δ̂(v), tU). Consequently, the Cartan curvature vanishes on the lightcone of each point of Δ, in a sufficiently small neighborhood.

Proof: Choose v ∈ I_{Δ}. We will show that when p + q ≥ 4, the Weyl curvature vanishes on β, and the Cotton tensor vanishes when p + q = 3. These tensors are zero on a closed set, and S ∪ −S is dense in N (𝔲^{−}). The neighborhood V can be chosen to be π∘exp_{Δ̂(v)}, restricted to a neighborhood of the origin in 𝔲^{−}. Then vanishing on the entire lightcone C(Δ(v)) ∩ V will follow. From the discussion above, vanishing of the Weyl and Cotton tensors implies flatness, which implies vanishing of the Cartan curvature on the same subset. Let f_{i}(t) be the framing along β given by proposition 4.5 (3). We first assume n ≥ 4 and consider the Weyl tensor. The conformal action of the flow h^{s} obeys

W (h_{∗}^{s}f_{i}(t), h_{∗}^{s}f_{j}(t), h_{∗}^{s}f_{k}(t)) = h_{∗}^{s}W (f_{i}(t), f_{j}(t), f_{k}(t))

The left hand side is

(\frac{1}{1 + st})^{σ(i)+σ(j)+σ(k)} W (f_{i}(c_{s}(t)), f_{j}(c_{s}(t)), f_{k}(c_{s}(t)))

We assume t > 0, so that h^{s}.β(t) → β(0) = Δ(v) as s → ∞ by proposition 4.5 (2). (If t < 0, then make s → −∞.) Now

W (f_{i}(0), f_{j}(0), f_{k}(0)) = lim_{s→∞}(1 + st)^{σ(i)+σ(j)+σ(k)}h_{∗}^{s}W (f_{i}(t), f_{j}(t), f_{k}(t))

If i = j = k = 1, then the left side vanishes because W is skew-symmetric in the first two entries. Therefore, we may assume the σ(i) + σ(j) + σ(k) ≥ 1. Boundedness of the right hand side implies

h_{∗}^{s}W (f_{i}(t), f_{j}(t), f_{k}(t)) → 0 as s → ∞

Because h_{∗}^{s}(f_{1}(t)) = f_{1}(c(t)), the above limit means W (f_{i}(t), f_{j}(t), f_{k}(t)) cannot have a nontrivial component on f_{1}(t). Then

W (f_{i}(0), f_{j}(0), f_{k}(0)) ∈ span{f_{2}(0), …, f_{n}(0)}

Varying U over S, one sees that the Weyl curvature at Δ(v) has image in

⋂_{g∈S} π_{∗}ω_{Δ̂(v)}^{−1}(span{(Ad g)(U_{2}), …, (Ad g)(U_{n})})

= ⋂_{g∈S} π_{∗}ω_{Δ̂(v)}^{−1}(Ad g)(U_{n})^{⊥}

By (2) of lemma 4.6, the set of all (Ad g)(U_{n}) with g ∈ S is a dense set of directions in the null cone N (𝔲^{−}). Then the intersection above is 0, so W vanishes at Δ(v). Now

0 = lim_{s→∞}(1 + st)^{σ(i)+σ(j)+σ(k)}h_{∗}^{s}W (f_{i}(t), f_{j}(t), f_{k}(t))

If σ(i) + σ(j) + σ(k) ≥ 2, then

W (f_{i}(t), f_{j}(t), f_{k}(t)) = 0

because h_{∗}^{s} cannot contract any tangent vector at β(t) strictly faster than (1 + st)^{2}. If σ(i) + σ(j) + σ(k) = 1, then we may assume i = k = 1, and h_{∗}^{s} must contract the Weyl curvature strictly faster than (1 + st), which is possible only if

W (f_{1}(t), f_{i}(t), f_{1}(t)) ∈ Rf_{n}(t)

But, in this case, for any inner product ⟨, ⟩ in the conformal class,

⟨W (f_{1}(t), f_{i}(t), f_{1}(t)), f_{1}(t)⟩ = −⟨W (f_{1}(t), f_{i}(t), f_{1}(t)), f_{1}(t)⟩ = 0

which implies W (f_{1}(t), f_{i}(t), f_{1}(t)) = 0, and again W vanishes at β(t), as desired. When dim M = 3, the argument follows the same steps and is easier. We leave it to the reader. ♢

5.2. Vanishing on a neighborhood of x_{0}. The previous subsection established vanishing of the Cartan curvature K on the union of lightcones emanating from the null geodesic segment Δ containing x_{0}. This union does not, however, contain a neighborhood of x_{0} in general. In this subsection we will show that Δ, or a particular reparametrization of it, is complete, and that lightcones of points on Δ intersect a neighborhood of x_{0} in a dense subset. Then vanishing of K in a neighborhood of x_{0} will follow. We keep the notations of the previous section: there is a flow h^{s} of H fixing x_{0} with holonomy the lightlike translation τ^{s}. Recall that T denotes the infinitesimal generator of the one-parameter group τ^{s}.

Proposition 5.3. There exists g_{θ} in the centralizer of T such that (Ad g_{θ})(𝔲^{−}) is transverse to 𝔭 and such that the curve Δ̂(t) = exp(b_{0}, t(Ad g_{θ})(U_{1})) in B is defined for all time t.

Proof: Recall that x_{0} and τ^{s} were obtained by theorem 3.2, which ensured that 𝔥^{b_{0}} centralizes T (see the beginning of section 3 for the notation 𝔥^{b_{0}}). Recall the dynamics on Ein^{p,q} of τ^{s} (fact 4.1): for each y in the null geodesic Λ, an open dense subset of the cone C(y) tends under τ^{s} to y. Then any flow coming from the centralizer of T must leave Λ setwise invariant; in particular, 𝔥^{b_{0}} preserves Λ.

Lemma 5.4. Let 𝔫 be a nilpotent subalgebra of o(p + 1, q + 1) fixing two points on Λ. Then the nilpotence degree of 𝔫 is at most 2p.

Proof: The stabilizer in PO(p + 1, q + 1) of a lightlike geodesic Λ in Ein^{p,q} restricts to an action equivalent to PSL(2, R) on RP^{1}; in particular, the stabilizer acts transitively on pairs of distinct points on Λ. Thus we may assume 𝔫 fixes [e_{0}] and [e_{1}]. Fixing [e_{0}] means 𝔫 is a subalgebra of 𝔭 ≅ 𝔠𝔬(p, q) ⋉ R^{p,q}. Recall the inverse stereographic projection φ : R^{p,q} → Ein^{p,q} from section 2.1.1. Let u_{1}, …, u_{n} be the standard basis of R^{p,q}. Then lim_{t→∞} φ(tu_{1}) = [e_{1}]. As in section 2.1.1, the set of lines in R^{p,q} which tend to [e_{1}] all have the form {x + tu_{1}} for x ∈ u_{1}^{⊥}. This set of lines is invariant by the 𝔫-action on R^{p,q}, which means that the translational components of 𝔫 ⊂ 𝔠𝔬(p, q) ⋉ R^{p,q} are all in u_{1}^{⊥}, and the linear components preserve Ru_{1}, and therefore also u_{1}^{⊥}. Now by calculations similar to those in the proof of 2.3, we see that, if \overline{𝔫} is the projection of 𝔫 on 𝔠𝔬(p, q), then

𝔫_{k} ⊆ \overline{𝔫}_{k} + \overline{𝔫}^{k}(u_{1}^{⊥})

for each positive integer k. But the nilpotence degree of a nilpotent subalgebra \overline{𝔫} of 𝔠𝔬(p, q) is at most 2p − 1, while the order of \overline{𝔫} on u_{1}^{⊥} is easily seen to be at most 2p (compare with lemma 2.5). ♢ As in the proof above, the image of the restriction 𝔞 of 𝔥^{b_{0}} to Λ is isomorphic to a subalgebra of 𝔰𝔩(2, R). Because 𝔞 is nilpotent, dim 𝔞 ≤ 1. Let 𝔳 be the unipotent radical of 𝔥^{b_{0}}; it has nilpotence degree 2p + 1 since 𝔥^{b_{0}} does. The restriction of 𝔳 to Λ is generated by a parabolic element of 𝔰𝔩(2, R), and it must be nontrivial by lemma 5.4. Thus 𝔞 is of parabolic type. Let L ∈ 𝔥^{b_{0}} have nontrivial image in 𝔞; denote this image by L̄. Suppose L̄ is parabolic type and that it fixes [1 : 0] ∈ RP^{1}. Then L fixes [e_{0}] in Ein^{p,q}. The 1-parameter group e^{sL} preserves Λ(t) = π(e^{tU_{1}}) and reparametrizes it by t ↦ \frac{t}{1+st}. Suppose that Δ̂(t) = exp(b_{0}, tU_{1}) is defined on (−ϵ, ϵ). Take s_{∞} = −1/ϵ and s_{−∞} = 1/ϵ and apply proposition 4.3 (1) to see that exp(b_{0}, tU_{1}) is defined for all t ∈ R. Next suppose that L̄ fixes [0 : 1] ∈ RP^{1}. Then e^{tL̄}.[1 : 0] = [1 : t], and, in Ein^{p,q}, the orbit is e^{tL}.[e_{0}] = Λ(t). Then there exist g(t) ∈ P such that

e^{tL} = e^{tU_{1}} · g(t) (3)

Then there exist (α, β) ⊂ R, a diffeomorphism c : (α, β) → R, and a path g(t) ∈ P such that

e^{tL} = e^{c(t)U_{1}} · g(t)

for all t ∈ (α, β). The curve exp(b_{0}, tL) is the orbit of b_{0} under the lift of a conformal flow, so it is complete, and proposition 4.4 applies to give that exp(b_{0}, tU_{1}) is defined for all t ∈ R. Note that, because the two subgroups e^{tL} and e^{tU_{1}} have the same restriction to Λ, the path g(t) is in the subgroup P_{Λ} < P pointwise fixing Λ. Last, consider arbitrary L̄ of parabolic type. There exists ḡ_{θ} ∈ SL(2, R) a rotation such that (Ad ḡ_{θ})(L̄) fixes [0 : 1]. Let g_{θ} be the image of ḡ_{θ} under the standard embedding SL(2, R) → PO(p + 1, q + 1) given by

\{( \begin{matrix}A & 0 & 0 \\ 0 & I_{n−2} & 0 \\ 0 & 0 & A^{−1} \end{matrix} ) : A ∈ SL(2, R) \}

with respect to which the identification RP^{1} → Λ is equivariant. Then g_{θ} centralizes T. In PO(p + 1, q + 1),

g_{θ}e^{tL}g_{θ}^{−1} = e^{tU_{1}} · g(t)

where g(t) ∈ P_{Λ} is as in (3). So

e^{tL} = e^{(Ad g_{θ})(tU_{1})} · h(t)

where h(t) = g_{θ}g(t)g_{θ}^{−1}. The subgroup P_{Λ} is normalized by g_{θ}, so h(t) ∈ P_{Λ}. Proposition 4.4 applies to show exp(b_{0}, (Ad g_{θ})(tU_{1})) is complete, because exp(b_{0}, tL) is defined for all t. To prove the transversality claim, we show (Ad g_{θ})(𝔲^{−}) is still transverse to 𝔭, provided g_{θ} does not exchange [e_{0}] and [e_{1}] in Ein^{p,q}. Then we will take g_{θ} = 1 when L̄ fixes [1 : 0], and to be the above rotation when L̄ is parabolic but does not fix [1 : 0]. The subalgebra (Ad g_{θ})(𝔲^{−}) is transverse to 𝔭 if the orbit of [e_{0}] in Ein^{p,q} under it is n-dimensional. In the Minkowski chart M([e_{n+1}]), the point [e_{0}] is the origin, and Λ is a null line through the origin, meeting the lightcone at infinity in one point, [e_{1}]. The subalgebra 𝔲^{−} acts by translations. If g_{θ} does not exchange [e_{0}] and [e_{1}], then g_{θ}^{−1}[e_{0}] is a point on Λ still contained in M([e_{n+1}]). The orbit

(g_{θ}𝔲^{−}g_{θ}^{−1}).[e_{0}] = g_{θ}(M([e_{n+1}]))

which is n-dimensional. ♢

Proposition 5.5. Let g_{θ} ∈ PO(p+1, q +1) be given by proposition 5.3, and S as in proposition 4.5. Let Δ̂(v) = exp(b_{0}, v(Ad g_{θ})(U_{1})) and Δ = π ∘ Δ̂. Let S′ = (Ad g_{θ})(S). Then

(1) The flow h^{s} fixes Δ(v) pointwise and has holonomy at Δ(v) with respect to Δ̂ equal τ^{s}.

(2) For each U ∈ S′ and v ∈ R, the curve β̂(t) = exp(Δ̂(v), tU) is complete and projects to a null geodesic.

(3) For Δ as in (1) and β̂ as in (2), the Cartan curvature vanishes on the fiber of β̂(t) for all t ∈ R.

Proof:

(1) Since g_{θ} centralizes the null translation T, one can follow the same proof as for proposition 4.5 (1).

(2) Recall the embedding PSL(2, R) → PO(p + 1, q + 1) with image the stabilizer of Λ. The element g_{θ} corresponds to rotation by some angle θ on Λ = P(span{e_{0}, e_{1}}). Compute that for U_{n} = E_{n}^{0} − E_{n+1}^{1} as in lemma 4.6,

(Ad g_{θ})(U_{n}) = U_{n}

Next, note that the subgroup S of lemma 4.6 is contained in P_{Λ}, the pointwise stabilizer of Λ. Then

g_{θ}Sg_{θ}^{−1} < P_{Λ} < P

Now any element of S′ is of the form

λ(Ad g_{θ} ∘ Ad w)(U_{n})

for some λ ∈ R and w ∈ S, and can be written

λ(Ad(g_{θ}wg_{θ}^{−1}))(U_{n}) = U

Since g_{θ}wg_{θ}^{−1} ∈ P, the element U projects to a null vector in 𝔤/𝔭 ≅ R^{p,q}. Then π ∘ exp(Δ̂(v), tU) is a null geodesic.

Fix U = (Ad g_{θ}w)(U_{n}) ∈ S′; it suffices to prove (2) for such U, since the geodesic generated by λU is complete if and only if the geodesic generated by U is. Recall the matrices h(s, t), representing the holonomy of h^{s} along null geodesics based at π(exp(b_{0}, vU_{1})) with initial direction in S or −S. Straightforward computation shows that g_{θ} commutes with h(s, t). Recall also that e^{sT} commutes with S. Then we compute in PO(p + 1, q + 1),

e^{sT} e^{tU} = e^{c(t)U}(g_{θ}wg_{θ}^{−1})h(s, t)(g_{θ}wg_{θ}^{−1})^{−1}

where c(t) = \frac{t}{1+st}.

Now proposition 4.3 with part (1) implies that h^{s} reparametrizes β(t) by c(t) and has holonomy

g_{θ}wh(s, t)w^{−1}g_{θ}^{−1} = (g_{θ}wg_{θ}^{−1})h(s, t)(g_{θ}wg_{θ}^{−1})^{−1} ∈ P

along it with respect to β̂. Part (1) of proposition 4.3 gives the desired completeness.

(3) As above, we may assume U = Ad(g_{θ}w)(U_{n}). Define a framing along β(t) as in proposition 4.5 by

f_{i}(β(t)) = (π_{∗} ∘ ω_{β̂(t)}^{−1} ∘ (Ad g_{θ}w))(U_{i})

Recall that (Ad g_{θ})(𝔲^{−}) is transverse to 𝔭 by proposition 5.3, so Ad(g_{θ}w)(𝔲^{−}) is, as well. Now the derivative of h(s, t) along β(t) in this framing is computed as in the proof of 4.5 by the adjoint action of the holonomy g_{θ}wh(s, t)w^{−1}g_{θ}^{−1} on the (Ad g_{θ}w)(U_{i}), modulo 𝔭. The derivative has the same diagonal form as in proposition 4.5. In fact, all the conclusions of 4.5, and thus also the arguments of proposition 5.2, hold when Δ̂(v) = exp(b_{0}, (Ad g_{θ})(vU_{1})), and S is replaced by S′, so we conclude that the Cartan curvature vanishes along the desired geodesics.

♢

Definition 5.6. Let (M, B, ω) be a Cartan geometry modeled on G/P. Let γ be a piecewise smooth curve in B. The development Dγ is the piecewise smooth curve in G satisfying Dγ(0) = e and (Dγ)′(t) = ω(γ′(t)) for all but finitely many t.

Note that for any piecewise smooth curve γ in B, the development Dγ is defined on the whole domain of γ, because it is given by a linear first-order ODE on G with bounded coefficients.

Proposition 5.7. The Cartan curvature K vanishes on an open set of the form π^{−1}(V) for V a neighborhood of x_{0} in M.

Proof: Recall the basis U_{1}, …, U_{n} for 𝔲^{−} first introduced in section 2.1.2. Let Δ̂(v) = exp(b_{0}, (Ad g_{θ})(vU_{1})), where g_{θ} is given by proposition 5.3, so Δ̂ is complete. Recall that the curves exp(Δ̂(v), tU), where U ∈ (Ad g_{θ})(S) = S′ are complete, as well, from proposition 5.5. To show that the Cartan curvature vanishes on a neighborhood above x_{0}, it suffices to show that K = 0 on exp(b_{0}, V), for V a neighborhood of 0 in (Ad g_{θ})(𝔲^{−}), because π_{∗b_{0}} maps ω_{b_{0}}^{−1}(V) onto a neighborhood of 0 in T_{x_{0}}M by proposition 5.3. First suppose g_{θ} = 1. Recall from the proof of lemma 4.6 (2) that S consists of all U ∈ N (𝔲^{−}) with ⟨U, U_{1}⟩ > 0. Let

Y = aU_{1} + X + cU_{n} ∈ 𝔲^{−}

with X ∈ span{U_{2}, …, U_{n−1}}, and assume that c ≠ 0. Let b = ⟨X, X⟩. Define, for each 0 ≤ r ≤ 1, a piecewise smooth curve α_{r} in B by concatenating

Δ̂(t(b^{2}/c + 2a)), 0 ≤ t ≤ r/2

and

exp(Δ̂(rb^{2}/2c + ra), (2t − r)(cU_{n} + X − b^{2}/2c U_{1})), r/2 ≤ t ≤ r

Note that cU_{n} + X − b^{2}/2c U_{1} ∈ ±S because c ≠ 0. Define β(r) = α_{r}(r).

β  α_{r} α_{s}  Δ̂ b_{0} β̂^{r} β̌^{s}

Figure 3. components of the homotopy between β and α_{1}

Because 𝔲^{−} is an abelian subalgebra of 𝔤, the development

Dα_{r}(r) = e^{r( \frac{b^{2}}{2c} +a)U_{1}} · e^{r(cU_{n}+X− \frac{b^{2}}{2c} U_{1})} = e^{rY}

Denote by β̂^{s} the restriction of β to [0, s] and by β̌^{s} the restriction of β to [s, 1]. The curve β is homotopic to α_{1} through the family of concatenations α_{s} ∗ β̌^{s}; similarly, β̂^{r} is homotopic to α_{r} for all 0 ≤ r < 1. Because the curvature K vanishes on the images of these homotopies, the developments of the homotopic curves have the same endpoints (see [Sh, 3.7.7 and 3.7.8]):

Dβ̂^{r}(r) = Dα_{r}(r) = e^{rY} ∀ r ∈ [0, 1]

But

Dβ(r) = Dβ̂^{r}(r) = e^{rY}

which means that

β(r) = exp(b_{0}, rY)

Then K(exp(b_{0}, Y)) = 0. Varying Y over all sufficiently small aU_{1}+X +cU_{n} with c ≠ 0 and passing to the closure gives vanishing of K on exp(b_{0}, V), for V a neighborhood of 0 in 𝔲^{−}.

If g_{θ} ≠ 1, then consider

Y = (Ad g_{θ})(aU_{1} + X + cU_{n}) ∈ (Ad g_{θ})(𝔲^{−})

again with c ≠ 0. Define α_{r} by concatenating a portion of Δ̂ as above with

exp(Δ̂(rb^{2}/2c + ra), (2t − r)(Ad g_{θ})(cU_{n} + X − b^{2}/2c U_{1})), r/2 ≤ t ≤ r

note that (Ad g_{θ})(cU_{n} + X − b^{2}/2cU_{1}) ∈ ±S′. Therefore, by proposition 5.5 (2) and (3), the curves α_{r} are defined on [0, r] for all r and K vanishes on them. Then the same argument as above applies to give vanishing of curvature at exp(b_{0}, Y). The set of possible Y are dense in a neighborhood V of 0 in (Ad g_{θ})(𝔲^{−}), so we obtain the desired vanishing on exp(b_{0}, V). ♢

6. End of the proof of theorem 1.2: global structure of M

This section is again under the assumptions of theorem 1.2: H is a connected nilpotent Lie group acting conformally on a type (p, q) pseudo-Riemanniann manifold (M, σ), and d(H) = 2p + 1 with p + q ≥ 3. From proposition 5.1, the manifold (M, σ) is locally conformally equivalent to Ein^{p,q}, or in other words is endowed with a (G̃, Ẽin^{p,q})-structure, where G̃ = Conf Ẽin^{p,q}, a covering group of PO(p + 1, q + 1). The developing map of the structure is a conformal immersion of the universal cover of M

δ : M̃ → Ẽin^{p,q}

(see [Th], [Go] for an introduction to (G, X)-structures and the construction of the developing map). If (M̃, B̃, ω̃) is the canonical Cartan geometry associated to the lifted conformal structure (M̃, σ̃), then δ lifts to an immersion of bundles δ̂ : B̃ → G̃, where G̃ is seen as a principal bundle over Ẽin^{p,q}. This immersion satisfies δ̂^{∗}ωG̃ = ω̃, where ωG̃ denotes the Maurer-Cartan form on G̃. Let ρ be the holonomy morphism Conf M̃ → G̃, related to the developing map by the equivariance property

ρ(ϕ) ∘ δ = δ ∘ ϕ ∀ ϕ ∈ Conf M̃.

The H-action lifts to a faithful action of a connected covering group, which we will also denote H, on M̃. The group ρ(H) = Ȟ is a connected nilpotent subgroup of G̃, with Lie algebra 𝔥̌ isomorphic to 𝔥. In particular d(𝔥̌) = 2p + 1, and by proposition 2.3, we may assume that Ȟ contains the lifted 1-parameter group τ^{s} (see section 4.1).

The fundamental group π_{1}(M) is isomorphic to a discrete subgroup Γ < Conf M̃ acting freely and properly on M̃. Because Γ centralizes 𝔥, the image ρ(Γ) = Φ centralizes 𝔥̌. To complete the proof of theorem 1.2, we must prove that the developing map δ is a conformal diffeomorphism between M̃ and Ẽin^{p,q}. Subsections 6.1, 6.2, and 6.3 contain preliminary geometric and algebraic results to this end. The end of the proof of theorem 1.2 will be given in 6.4 for the Lorentzian case, and in 6.5 for the other types.

6.1. More on geometry and dynamics on Ẽin^{p,q}. When p ≥ 2, Ẽin^{p,q} is a double cover of Ein^{p,q}. It is N̂^{p+1,q+1}/R_{>0}^{∗}. The conformal group G̃ is O(p + 1, q + 1), and the stabilizer of [e_{0}] is an index-two subgroup of P. A lightcone C(x) in Ẽin^{p,q} has two singular points, and its complement has two connected components, each one conformally equivalent to R^{p,q}.

The Lorentz case p = 1 is more subtle since Ẽin^{1,n−1} is no longer compact. It is conformally equivalent to (R × S^{n−1}, −dt^{2} ⊕ g_{S^{n−1}}). Details about this space are in [Fr1, ch 4.2] and [BCDGM]. The group G̃ = Conf Ẽin^{1,n−1} is a twofold quotient of the universal covering group of O(2, n), with center Z ≅ Z. The space Ein^{1,n−1} is the quotient of Ẽin^{1,n−1} by the Z-action.

The lightlike geodesics and lightcones in Ẽin^{1,n−1} are no longer compact. Any lightlike geodesic can be parametrized γ(t) = (t, c(t)), where c(t) is a unit-speed geodesic of S^{n−1}. Lightlike geodesics are preserved by Z, which acts on them by translations; the quotient is a lightlike geodesic of Ein^{1,n−1}. Any lightcone C(x) ⊂ Ẽin^{1,n−1} has infinitely-many singular points, which coincide with the Z-orbit of x. The complement of C(x) in Ẽin^{1,n−1} has a countable infinity of connected components, each one conformally diffeomorphic to R^{1,n−1}. The center Z freely and transitively permutes these Minkowski components.

Recall that τ^{s} is the flow on Ẽin^{1,n−1} generated by the null translation T. The null geodesic Λ = P(span{e_{0}, e_{1}}) is the fixed set of τ^{s} on Ein^{1,n−1}. Let Λ̃ be the inverse image of Λ in Ẽin^{1,n−1}; it is noncompact and connected, and equals the fixed set of τ^{s} on Ẽin^{1,n−1}. Given x̃ ∈ Ẽin^{1,n−1} \ Λ̃, there are two distinct points x̃^{+} and x̃^{−} on Λ̃ such that

lim_{s→∞} τ^{s}.x̃ = x̃^{+}  and  lim_{s→−∞} τ^{s}.x̃ = x̃^{−}

Details about this material can be found in [Fr1, p 67].

6.2. About the centralizer of 𝔥̌. For arbitrary (p, q), let Λ̃ be the inverse image in Ẽin^{p,q} of Λ; it is connected and fixed by τ^{s}. Our first task is to find an algebraic restriction on Φ, using that it commutes with 𝔥̌.

Proposition 6.1. The centralizer C(𝔥̌) of 𝔥̌ in G̃ leaves Λ̃ invariant. The C(𝔥̌)-action on Λ̃ factors through a homomorphism to Z_{2} if p ≥ 2, and it factors through a homomorphism to Z when p = 1.

Proof: Denote 𝔠(𝔥̌) the Lie algebra of the centralizer of 𝔥̌. Observe first that both 𝔠(𝔥̌) and 𝔥̌ centralize τ^{s}. Recall that Λ̃ is a local attracting set for τ^{s} (see the proof of proposition 5.5), so 𝔠(τ^{s}) leaves Λ̃ invariant. The centralizer 𝔠(τ^{s}) in 𝔬(p + 1, q + 1) of {τ^{s}} thus consists of matrices of the form

( \begin{matrix}a & b & −x^{t}.J_{p−1,q−1} & s & 0 \\ c & −a & −y^{t}.J_{p−1,q−1} & 0 & −s \\ & & M & y & x \\ & & & a & −b \\ & & & −c & −a \end{matrix} ) \begin{matrix}a, b, c, s ∈ R \\ x, y ∈ R^{p−1,q−1} \\ M ∈ 𝔬(p − 1, q − 1) \end{matrix} (4)

As in the proof of proposition 5.5, the projection of 𝔥̌ on 𝔰𝔩(2, R) is a 1dimensional subalgebra of parabolic type. The projection of 𝔠(𝔥̌) lies in the same subalgebra; we wish to show it is zero. We may assume, by conjugating 𝔥̌ in the centralizer of τ^{s} if necessary, that 𝔠(𝔥̌) and 𝔥̌ are subalgebras of

𝔮 = {X ∈ 𝔠(τ^{s}) : a, c = 0}

This algebra is isomorphic to (R ⊕ 𝔬(p − 1, q − 1)) ⋉ 𝔥𝔢𝔦𝔰(2n − 3). Elements of 𝔮 are denoted u = (b, M, x, y, s), with b ∈ R, x, y ∈ R^{p−1,q−1}, and M ∈ 𝔬(p − 1, q − 1). Denote b = π_{1}(u), M = π_{2}(u), and (x, y) = π_{3}(u). Note that if π_{i}(u) = 0 for i = 1, 2, 3, then u is in the center of 𝔮. If u_{1} = (b_{1}, M_{1}, x_{1}, y_{1}, s_{1}) and u_{2} = (b_{2}, M_{2}, x_{2}, y_{2}, s_{2}) are in 𝔮, then an easy computation yields

• π_{1}([u_{1}, u_{2}]) = 0
• π_{2}([u_{1}, u_{2}]) = [M_{1}, M_{2}]
• π_{3}([u_{1}, u_{2}]) = (b_{1}y_{2} − b_{2}y_{1} − M_{1}.x_{2} + M_{2}.x_{1}, −M_{1}.y_{2} + M_{2}.y_{1})

Now, if u_{0} = (b_{0}, M_{0}, x_{0}, y_{0}, s_{0}) is in 𝔠(𝔥̌), each u = (b, M, x, y, s) ∈ 𝔥̌ must satisfy the relations:

• [M_{0}, M] = 0
• M_{0}.y = M.y_{0}
• b_{0}y − M_{0}.x = by_{0} − M.x_{0}

We claim that b_{0} = 0. If not, then from the last relation above, whenever u_{1} and u_{2} are in 𝔥̌, then

y_{1} = \frac{b_{1}}{b_{0}} y_{0} − \frac{1}{b_{0}} M_{1}.x_{0} + \frac{1}{b_{0}} M_{0}.x_{1}

and

y_{2} = \frac{b_{2}}{b_{0}} y_{0} − \frac{1}{b_{0}} M_{2}.x_{0} + \frac{1}{b_{0}} M_{0}.x_{2}

This implies

π_{3}([u_{1}, u_{2}]) = (\frac{b_{1}}{b_{0}} (−M_{2}.x_{0} + M_{0}.x_{2}) + \frac{b_{2}}{b_{0}} (−M_{0}.x_{1} + M_{1}.x_{0})
− M_{1}.x_{2} + M_{2}.x_{1}, −M_{1}.y_{2} + M_{2}.y_{1})

As in section 2.2.2, we may assume without affecting the nilpotence degree and without changing C(𝔥̌) that 𝔥̌ is Zariski closed. Write 𝔥̌ ≅ 𝔯̌ ⋉ 𝔲̌, where 𝔲̌ is an algebra of nilpotents, and d(𝔥̌) = d(𝔲̌). Let 𝔪̌ = π_{2}(𝔲̌). It is a nilpotent Lie subalgebra of 𝔬(p − 1, q − 1), and also an algebra of nilpotents, since 𝔲̌ is so. Using the equation above, we get by induction that π_{3}(𝔲̌_{k}) ⊂ 𝔪̌^{k}(R^{p−1,q−1}) × 𝔪̌^{k}(R^{p−1,q−1}). Moreover, π_{2}(𝔲̌_{k}) = 𝔪̌_{k}, and π_{1}(𝔲̌_{k}) = 0 as soon as k ≥ 1. By proposition 2.3, d(𝔪̌) ≤ 2p−3, and o(𝔪̌) ≤ 2p − 1 by lemma 2.5. As a consequence, π_{1}(𝔲̌_{2p−1}) = π_{2}(𝔲̌_{2p−1}) = π_{3}(𝔲̌_{2p−1}) = 0, which implies that 𝔲̌_{2p−1} is in the center of 𝔲̌, and finally d(𝔲̌) ≤ 2p. Since d(𝔥̌) = d(𝔲̌), we get a contradiction. Therefore, 𝔠(𝔥̌) is actually the subalgebra of 𝔠(τ^{s}) with b = 0 in (4). The projection of C(ȟ) to PO(p + 1, q + 1) commutes with the flow of parabolic type on Λ generated by 𝔥̌. The centralizer of such a 1-parameter subgroup in PSL(2, R) is itself; in particular, any finite subgroup of the centralizer is trivial. From the above calculation, on the other hand, the identity component of C(𝔥̌) acts trivially on Λ̃. It follows that the image of C(𝔥̌) in PO(p + 1, q + 1) acts trivially on Λ, so the restriction of C(𝔥̌) to Λ̃ factors through a homomorphism to the kernel of G̃ → PO(p + 1, q + 1). From the previous section, this kernel is isomorphic to Z_{2} when p ≥ 2 and equals Z when p = 1. This concludes the proof of the proposition. ♢ 6.3. Geometric properties of the developing map. Recall that {h^{s}} < H satisfies ρ(h^{s}) = τ^{s}. We now adopt the notations and results of subsection 4.3. The lightlike geodesic Δ(t) = π∘exp(b_{0}, tU_{1}) is pointwise fixed by h^{s} and is locally an attracting set for it (see proposition 4.5 (2)). Choose x̃_{0} ∈ M̃ over x_{0} and lift h^{s} to M̃. Let Λ̃ ⊂ Ẽin^{p,q} be as above: it is a closed subset, pointwise fixed by τ^{s}, and it is the attracting set for τ^{s}. Because of these dynamics, π_{M}^{−1}(Δ) ⊂ δ^{−1}(Λ̃), which is a closed, Γ-invariant, 1-dimensional, immersed submanifold. Let Δ̃ be the component of δ^{−1}(Λ̃) containing x̃_{0}. Denote Γ_{0} < Γ the subgroup leaving Δ̃ invariant.

Proposition 6.2. The image π_{M}(Δ̃) ⊂ M is closed. Therefore, Γ_{0} acts cocompactly on Δ̃.

Proof: We show π_{M}(Δ̃) is closed in π_{M}(δ^{−1}(Λ̃)), and therefore in M. Suppose π_{M}(x̃_{n}) → π_{M}(ỹ) with x̃_{n} ∈ Δ̃ and ỹ ∈ δ^{−1}(Λ̃). Let U be a neighborhood of ỹ that maps diffeomorphically to its images under π_{M} and under δ. There exist γ_{n} ∈ Γ such that γ_{n}.x̃_{n} → ỹ in U. Then δ(γ_{n}.x̃_{n}) → δ(ỹ) in Λ̃, and we may assume U is small enough that δ(U) ∩ Λ̃ is an open segment. Then γ_{n}.x̃_{n} and ỹ are in a common segment of δ^{−1}(Λ̃) ∩ U. Then for some γ_{n} = γ, the translate γ.ỹ ∈ Δ̃. ♢ As a consequence, we obtain the following result.

Proposition 6.3. The map δ is a covering map from Δ̃ onto Λ̃. When M is Lorentzian, δ is a diffeomorphism between Δ̃ and Λ̃.

Proof: First note that Δ̃ is open in δ^{−1}(Λ̃). For if δ^{−1}(Λ̃) were recurrent, then Λ̃ would be, as well; but Λ̃ is a closed, embedded submanifold of Ẽin^{p,q}. Therefore the image δ(Δ̃) is a connected open subset of Λ̃. By equivarience of δ and the previous proposition, ρ(Γ_{0}) preserves δ(Δ̃) and acts cocompactly on it. But ρ(Γ_{0}) centralizes 𝔥̌, so its action on Λ̃ factors through either a finite group, or the extension of a finite group by Z. In both cases, δ(Δ̃) must equal Λ̃. When M has Lorentz type, then δ : Δ̃ → Λ̃ must be a diffeomorphism, because all lightlike geodesics in Ẽin^{1,n−1} are embedded copies of R, as described in section 6.1; in particular, they have no self-intersection. Now assume p ≥ 2. On one hand, Γ_{0} acts cocompactly on Δ̃; on the other hand, the action of ρ(Γ_{0}) on Λ̃ factors through a finite group. Let Γ′_{0} ◁Γ_{0} be such that ρ(Γ′_{0}) is the kernel in ρ(Γ_{0}) of restriction to Λ̃. Then the restriction of δ factors

δ|Δ̃ : Δ̃ → Λ̃
↘ ↑
Δ̃/Γ′_{0}

Because Γ′_{0} acts freely and properly on Δ̃, the quotient map Δ̃ → Δ̃/Γ′_{0} is a covering, and because Γ′_{0} has finite index in Γ_{0}, its action on Δ̃ is cocompact. The map Δ̃/Γ′_{0} → Λ̃ is surjective because δ|Δ̃ is; it is a local diffeomorphism because δ|Δ̃ and Δ̃ → Δ̃/Γ′_{0} are. By compactness of Δ̃/Γ′_{0}, it follows that Δ̃/Γ′_{0} → Λ̃, hence Δ̃ → Λ̃, is a covering, as desired. ♢

6.4. Proof of theorem 1.2: the case of Lorentz manifolds. Suppose now that p = 1. Let

Ω = {z̃ ∈ M̃ \ Δ̃ | lim_{s→∞} h^{s}.z̃ exists and is in Δ̃}

Proposition 6.4. The set Ω is nonempty and open. It is mapped diffeomorphically by δ onto Ẽin^{1,n−1} \ Λ̃.

Proof: Let us first check that Ω is nonempty. Recall Δ̃ is pointwise fixed by h^{s}. Let z̃_{∞} ∈ Δ̃, and choose b̃_{∞} ∈ B̃ above z̃_{∞} such that the holonomy of h^{s} with respect to b̃_{∞} is τ^{s}. Let S be as in proposition 4.5 and U ∈ S. Consider the geodesic β(t) = π ∘ exp(b̃_{∞}, tU). It is complete by proposition 4.5 (2); further, for t > 0,

lim_{s→∞} h^{s}.β(t) = β(0) = z̃_{∞}

Then β(t) ∈ Ω for t > 0. To prove that Ω is open, choose z̃_{0} ∈ Ω. There exists z̃_{∞} ∈ Δ̃ such that lim_{s→∞} h^{s}.z̃_{0} = z̃_{∞}. Since the orbits of τ^{s} are lightlike geodesics in Ẽin^{1,n−1}, the same is true for the orbits of h^{s} on M̃. Then z̃_{0} lies on some lightlike geodesic emanating from z̃_{∞}. Any such geodesic not fixed by h^{s} has the form π ∘ exp(b̃_{∞}, tU) with U ∈ S. Then for s_{0} > 0 there exist ϵ > 0 and a diffeomorphism c : (s_{0}, ∞) → (0, ϵ) such that, for every s ∈ (s_{0}, ∞),

π ∘ exp(b̃_{∞}, c(s)U) = h^{s}.z̃_{0}.

There are a neighborhood I of z̃_{∞} in Δ̃, a segment Ĩ ⊂ B̃ lying over I, and an open neighborhood U of 0 in 𝔲^{−} such that the map

μ : I × (U ∩ S) → M̃
(z̃, u) ↦ π ∘ exp(b̃, u)

where b̃ ∈ Ĩ lies over z̃, is defined and is a submersion. Choosing s_{0} big enough, c(s_{0})U ∈ U ∩ S, so that V = μ(I × (U ∩ S)) is an open subset containing h^{s_{0}}.z̃_{0}. It follows immediately from proposition 4.5 that V ⊂ Ω. Since Ω is h^{s}-invariant, h^{−s_{0}}(V) ⊂ Ω. It is an open subset containing z̃_{0}, which shows that Ω is open. We now prove that δ is an injection in restriction to Ω. Assume that z̃ and z̃′ are two points of Ω satisfying δ(z̃) = δ(z̃′). Let z̃_{∞} = lim_{s→∞} h^{s}.z̃ and z̃′_{∞} = lim_{s→∞} h^{s}.z̃′. Then

δ(z̃_{∞}) = lim_{s→∞} τ^{s}.δ(z̃) = lim_{s→∞} τ^{s}.δ(z̃′) = δ(z̃′_{∞})

Because δ is injective on Δ̃ by proposition 6.3, z̃_{∞} = z̃′_{∞}. Choose U an open neighborhood of z̃_{∞} which is mapped diffeomorphically by δ on an open neighborhood V of z_{∞} = δ(z̃_{∞}). There exists s_{0} ≥ 0 such that for all t ≥ s_{0}, h^{t}.z̃ ∈ U and h^{t}.z̃′ ∈ U. Moreover, δ(h^{t}.z̃) = δ(h^{t}.z̃′) = τ^{t}.δ(z̃). Since δ is an injection in restriction to U, the images h^{t}.z̃ = h^{t}.z̃′, so z̃ = z̃′, as desired.

It remains to show that δ(Ω) = Ẽin^{1,n−1}\Λ̃. The inclusion δ(Ω) ⊂ Ẽin^{1,n−1}\ Λ̃ follows easily from the definition of Ω. Just note that any z̃ ∈ δ^{−1}(Λ̃) is fixed by h^{s}, so Ω cannot meet δ^{−1}(Λ̃). Now, pick z ∈ Ẽin^{1,n−1}. There exists z_{∞} ∈ Λ̃ such that lim_{s→∞} τ^{s}.z = z_{∞}. By proposition 6.3, there is a unique z̃_{∞} ∈ Δ̃ such that δ(z̃_{∞}) = z_{∞}. Also, there is a neighborhood U of z̃_{∞} mapped diffeomorphically by δ on some neighborhood V of z_{∞}. There exists s_{0} such that for s ≥ s_{0}, τ^{s}.z ∈ V. Let z̃ ∈ U be such that δ(z̃) = τ^{s_{0}}.z. Then for all s ≥ s_{0}, we have h^{s}.z̃ ∈ U and lim_{s→∞} h^{s}.z̃ = z̃_{∞}. Thus, z̃ ∈ Ω. Moreover, δ(h^{−s_{0}}.z̃) = z and since Ω is h^{s}-invariant, z ∈ δ(Ω), as desired. ♢

Remark 6.5. Notice that when we proved that Ω is nonempty, we showed that Δ̃ is in the closure of Ω.

The inverse of δ on Ω̌ = Ẽin^{1,n−1} \ Λ̃ is a conformal embedding λ : Ω̌ → M̃. Because n ≥ 3, ∂Ω̌ has codimension at least 2. Then theorem 1.8 of [Fr5] applies in our context. It yields an open subset Ω̌′ containing Ω̌ and a conformal diffeomorphism λ^{−1} : M̃ → Ω̌′, which coincides with δ on Ω. Two conformal maps which are the same on an open set of a connected pseudo-Riemannian manifold of dimension ≥ 3 must coincide, so λ^{−1} = δ. Now, Ω̌′ contains δ(Δ̃) = Λ̃ and Ẽin^{1,n−1} \ Λ̃, which yields Ω̌′ = Ẽin^{1,n−1}, and δ : M̃ → Ẽin^{1,n−1} is a conformal diffeomorphism.

Thus M is conformally diffeomorphic to the quotient Ẽin^{1,n−1}/Φ. Since Φ centralizes 𝔥̌, it leaves Λ̃ invariant by proposition 6.1, and the restriction of Φ to Λ̃ factors through a homomorphism to Z ≅ Z. Because Φ acts freely, this restriction homomorphism is injective. Since M is compact, Φ must be infinite, so theorem 1.2 is proved in the Lorentzian case.

6.5. Proof of theorem 1.2: the case p ≥ 2. The proof in the Lorentz case must be adapted for p ≥ 2 because in this case, δ is a priori just a covering map from Δ̃ to Λ̃ and no longer a diffeomorphism. Recall that τ^{s} fixes Λ̃ pointwise. Let p_{1} ∈ Λ̃. The lightcone C(p_{1}) has two singular points, p_{1}, and another point, p_{2} ∈ Λ̃, and its complement consists of two Minkowski components, M_{1} and M_{2}. Also, Λ̃ \ {p_{1}, p_{2}} has two connected components I_{1} and I_{2}, which can be defined by dynamical properties of τ^{s}:

∀z ∈ M_{1}, lim_{s→∞} τ^{s}.z ∈ I_{1} and lim_{s→−∞} τ^{s}.z ∈ I_{2}
∀z ∈ M_{2}, lim_{s→∞} τ^{s}.z ∈ I_{2} and lim_{s→−∞} τ^{s}.z ∈ I_{1}

If F is the set of fixed points of τ^{s}, then C(p_{1}) \ F splits into two connected components, C_{1} and C_{2}. Suppose p_{1} = [e_{0}]. Then C(p_{1}) is the quotient (e_{0}^{⊥} ∩ N̂^{p+1,q+1})/R_{>0}^{∗} and p_{2} = [−e_{0}]. Recall that

F = (e_{0}^{⊥} ∩ e_{1}^{⊥} ∩ N̂^{p+1,q+1})/R_{>0}^{∗}

The components of C(p_{1}) \ F correspond to {⟨x, e_{1}⟩ > 0} and {⟨x, e_{1}⟩ < 0}. As in section 4.1, τ^{s}.[x] → [e_{0}] as s → ∞ if ⟨x, e_{1}⟩ > 0 and τ^{s}.[x] → [−e_{0}] if ⟨x, e_{1}⟩ < 0; similarly, τ^{s}.[x] → [e_{0}] as s → −∞ if ⟨x, e_{1}⟩ < 0 and τ^{s}.[x] → [−e_{0}] as s → −∞ if ⟨x, e_{1}⟩ > 0. The dynamics are the same at any p_{1} ∈ Λ̃, because there is a conformal automorphism of Ẽin^{p,q} sending [e_{0}] to p_{1} and preserving F:

∀z ∈ C_{1}, lim_{s→∞} τ^{s}.z = p_{1} and lim_{s→−∞} τ^{s}.z = p_{2}
∀z ∈ C_{2}, lim_{s→∞} τ^{s}.z = p_{2} and lim_{s→−∞} τ^{s}.z = p_{1}

Let {p̃_{2i+1} : i ∈ J} = δ^{−1}(p_{1}) and {p̃_{2i} : i ∈ J} = δ^{−1}(p_{2}). Order the points p̃_{2i+1} and p̃_{2i} compatibly with an orientation of Δ̃, and in such a way that p̃_{2i} is between p̃_{2i−1} and p̃_{2i+1}. If the covering δ : Δ̃ → Λ̃ is finite, then J is finite; in this case, order each set of points cyclically. The segment of Δ̃ from p̃_{2i−1} to p̃_{2i} will be denoted I_{2i−1}, and the segment from p̃_{2i} to p̃_{2i+1} will be I_{2i}. Now the set Ω of the previous section will be replaced by the two sets

Ω_{1} = {z̃ ∈ M̃ \ Δ̃ | lim_{s→∞} h^{s}.z̃ exists and is in I_{1}}
Ω_{2} = {z̃ ∈ M̃ \ Δ̃ | lim_{s→∞} h^{s}.z̃ exists and is in I_{2}}

Using the dynamical characterization of I_{1} and I_{2} corresponding to that of I_{1} and I_{2} given above, one can reproduce the proof of proposition 6.4 to obtain

Proposition 6.6. The sets Ω_{1} and Ω_{2} are nonempty and open. Each Ω_{i} is mapped diffeomorphically by δ onto M_{i}, i = 1, 2.

Lemma 6.7. δ(∂Ω_{i}) ⊂ C(p_{i}), i = 1, 2.

Proof: Let z̃ ∈ ∂Ω_{1}. By proposition 6.6, δ(z̃) ∈ \overline{M}_{1}. If δ(z̃) ∈ M_{1}, the same proposition gives z̃′ ∈ Ω_{1} such that δ(z̃′) = δ(z̃). Then if U′ is a neighborhood of z̃′ in Ω_{1}, and if U is a neighborhood of z̃ in M̃, with U ∩ U′ = ∅,

δ(U ∩ Ω_{1}) ∩ δ(U′) ≠ ∅

contradicting the injectivity of δ on Ω_{1}. The same proof holds if z̃ ∈ ∂Ω_{2}. ♢ If U is an open set of a type-(p, q) pseudo-Riemannian manifold (N, σ), and if x ∈ N, denote by C_{U}(x) the set of points in U which can be joined to x by a lightlike geodesic contained in U.

Lemma 6.8. There exists U a neighborhood of p̃_{1} in M̃ such that

U \ C_{U}(p̃_{1}) = (U ∩ Ω_{2}) ⋃(U ∩ Ω_{1})

Proof: First choose U a neighborhood of p̃_{1} that is geodesically convex for some metric in the conformal class, so U \ C_{U}(p̃_{1}) is a union of exactly two connected components U_{1} and U_{2}. (Here we use the assumption p ≥ 2. In the Lorentz case, there would be three connected components.) We may choose U small enough that δ maps U diffeomorphically on its image V, and δ(C_{U}(p̃_{1})) = V ∩ C(p_{1}). First, U ∩ Ω_{1} and U ∩ Ω_{2} are both nonempty: remark 6.5 is easily adapted to the current context to show that I_{1} ⊂ \overline{Ω_{1}} and I_{2} ⊂ \overline{Ω_{2}}. Assume then that U_{1}∩Ω_{1} ≠ ∅. By lemma 6.7, if U_{1}∩∂Ω_{1} ≠ ∅, then δ(U_{1}∩∂Ω_{1}) ⊂ C(p_{1}). Since δ is injective on U, then U_{1} ∩ ∂Ω_{1} ⊂ C_{U}(p̃_{1}), a contradiction. Therefore, U_{1} ∩ ∂Ω_{1} = ∅, so U_{1} ⊂ Ω_{1}. Similarly, U_{2} ⊂ Ω_{2}. ♢ Let W_{U} = C_{U}(p̃_{1}) \ F, and define W = ⋃_{s∈R} h^{s}.W_{U}.

Lemma 6.9. The set Ω = Ω_{1} ∪ W ∪ Ω_{2} ⊂ M̃ is open, and is mapped diffeomorphically by δ to Ẽin^{p,q} \ F.

Proof: We first prove that Ω is open. By lemma 6.8, and the fact that Ω_{1} and Ω_{2} are open, the set Ω_{1} ∪ W_{U} ∪ Ω_{2} is open. Now, if z̃ ∈ W, there exists s_{0} ∈ R such that h^{s_{0}}.z̃ ∈ W_{U}. Then there is a neighborhood U′ of h^{s_{0}}.z̃ contained in Ω_{1} ∪ W_{U} ∪ Ω_{2} ⊂ Ω. Then h^{−s_{0}}.U′ is a neighborhood of z̃ contained in Ω. We now show that δ is injective on Ω. By lemma 6.6, δ is injective on Ω_{1} and Ω_{2}, and because δ(Ω_{1}) = M_{1} is disjoint from δ(Ω_{2}) = M_{2}, the map δ is actually injective on Ω_{1} ∪ Ω_{2}. Because δ(W) ⊂ C(p_{1}) is disjoint from δ(Ω_{1} ∪ Ω_{2}), it suffices to prove that δ is injective on W. Assume z̃, z̃′ ∈ W with δ(z̃) = δ(z̃′), and suppose this point is in C_{1}, so

lim_{s→∞} τ^{s}.δ(z̃) = lim_{s→∞} τ^{s}.δ(z̃′) = p_{1}

Since z̃ ∈ W, either lim_{s→∞} h^{s}.z̃ = p̃_{1} or lim_{s→−∞} h^{s}.z̃ = p̃_{1}. But if lim_{s→−∞} h^{s}.z̃ = p̃_{1}, then lim_{s→−∞} τ^{s}.δ(z̃) = p_{1}, contradicting δ(z̃) ∈ C_{1}. Therefore, lim_{s→∞} h^{s}.z̃ = p̃_{1}, and for the same reasons, lim_{s→∞} h^{s}.z̃′ = p̃_{1}. Then there exists s_{0} > 0 such that for all s ≥ s_{0}, both h^{s}.z̃ and h^{s}.z̃′ are in U. Since δ(h^{s}.z̃) = τ^{s}.δ(z̃) = τ^{s}.δ(z̃′) = δ(h^{s}.z̃′), and since δ is injective on U, we get h^{s}.z̃ = h^{s}.z̃′ and finally z̃ = z̃′. The proof is similar if δ(z̃) = δ(z̃′) is in C_{2}. It remains to understand the set δ(Ω). From proposition 6.6, M_{1} ∪ M_{2} ⊂ δ(Ω), and it is also clear that δ(Ω) ⊂ Ẽin^{p,q} \ F. If z ∈ C_{1}, then there exists s > 0 such that τ^{s}.z ∈ V. Hence, there is z̃ ∈ U such that δ(z̃) = τ^{s}.z, and finally δ(h^{−s}.z̃) = z. Since z̃ ∈ U, then h^{−s}.z̃ ∈ Ω, which proves z ∈ δ(Ω). In the same way, we show that if z ∈ C_{2}, then z ∈ δ(Ω). Finally δ(Ω) = Ẽin^{p,q} \ F = M_{1} ∪ M_{2} ∪ C_{1} ∪ C_{2}. ♢ The conclusion is essentially the same as in the Lorentzian case. Let Ω̌ be the complement of F in Ẽin^{p,q}. Then inverting δ on Ω̌ gives a conformal embedding λ : Ω̌ → M̃. Recall from section 4.1 that F = ∂Ω̌ has codimension 2. Then theorem 1.8 of [Fr5] gives an open subset Ω̌′ containing Ω̌ and a conformal diffeomorphism λ^{−1} : M̃ → Ω̌′, which coincides with δ on Ω.

As above, λ^{−1} = δ. Now, Λ̃ ⊂ Ω̌′, and since δ : M̃ → Ω̌′ is a diffeomorphism, the action of Φ on Ω̌′ is free and proper. In particular, the map associating to an element of Φ its restriction to Λ̃ is injective. By proposition 6.1, the group Φ is trivial or isomorphic to Z_{2}. Because M = M̃/Γ is compact, Φ acts cocompactly on Ω̌′, so Ω̌′ = Ẽin^{p,q}. Therefore M is conformally diffeomorphic to Ẽin^{p,q}/Φ, as was to be shown.

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Charles Frances Département de Mathématiques Université Paris-Sud XI Charles.Frances@math.u-psud.fr

Karin Melnick (partially supported by NSF fellowship DMS-855735) Department of Mathematics University of Maryland, College Park karin@math.umd.edu