Compact Lorentz manifolds with local symmetry

Karin Melnick^{∗}

July 15, 2008

Abstract

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly fibers of a fiber bundle. A corollary is that if M has an open, dense, locally homogeneous subset, then M is locally homogeneous.

Acknowledgements: I thank my dissertation advisor, Benson Farb for his guidance and encouragement. I also thank Abdelghani Zeghib, with whom I had helpful conversations while working on this project.

In loving memory of Laura Sue-Jung Kang

1 Introduction

This work addresses the question, which compact Lorentz manifolds admit a positive-dimensional pseudogroup of local isometries? This question can be loosely rephrased as, which compact Lorentz manifolds have nontrivial local symmetry? For a real-analytic, complete Lorentz manifold, a positive-dimensional pseudogroup of local isometries is equivalent to a positive-dimensional isometry group on the universal cover.

∗Department of Mathematics, Yale University, New Haven, CT partially supported by NSF fellowship DMS-855735

Examples of compact Lorentz manifolds with local symmetry will be discussed below. Given such a Lorentz manifold, one may construct a new compact Lorentz manifold with at least as much local symmetry by forming a warped product.

Definition 1.1 For two pseudo-Riemannian manifolds (P, λ) and (Q, μ), a warped product P ×_{f} Q is given by a positive function f on Q : the metric at (p, q) is f(q)λ_{p} + μ_{q}. The factor P is called the normal factor.

If Isom(P) = G, then G also acts isometrically on the warped product P ×_{f} Q for any f. More generally, let f be any function Q → M, where M is the moduli space of G-invariant metrics (of a fixed signature) on P, with f(q) = λ(q). Then G acts isometrically on the twisted product P ×_{f} Q, where the metric at (p, q) is λ(q)_{p} + μ_{q}. Results of Farb and Weinberger stated below give conditions under which a compact Riemannian manifold is a twisted product P ×_{f} Q with P a locally symmetric space. Our main result (Theorem 1.3 below) gives conditions under which the universal cover of a compact Lorentz manifold has this form with P a Riemannian symmetric space or a complete Lorentz space of constant curvature. In both cases, the conditions are that the manifold have a large pseudogroup of local isometries. Pseudo-Riemannian metrics are examples of rigid geometric structures of algebraic type. For M a compact real-analytic manifold with such a structure, Gromov’s Stratification Theorem (stated as Theorem 5.1 below) describes the orbit structure of local symmetries of M. The celebrated Open-Dense Theorem, which is a corollary of this stratification, states that if a point of M has a dense orbit under local isometries, then an open dense subset of M is locally homogeneous. It would be interesting to find conditions on M under which existence of a dense orbit implies that M is locally homogeneous. Dumitrescu has proved in [D] that a compact, three-dimensional, real-analytic Lorentz manifold with an open, dense, locally homogeneous subset is locally homogeneous. More generally, one might seek a fibered version: when does existence of a local isometry orbit with positive-dimensional closure imply that M is roughly a fiber bundle with locally homogeneous fibers? Our main theorem (1.3) can be viewed as such a result, under some particular topological and geometric conditions on a compact real-analytic Lorentz manifold.

1.1 Riemannian case

For M a compact Riemannian manifold, Isom(M) is compact. For example, a compact locally symmetric space of noncompact type has finite isometry group (See [WM2] 5.43. In fact, any compact M with negative-definite Ricci curvature has finite isometry group—see [Ko] II.4.4). While such a group provides some information about M, the isometry group of the universal cover X of M tells much more. For example, if M is a locally symmetric space of noncompact type, then Isom(X) is a semisimple group with no compact factors. A homogeneous, contractible, Riemannian manifold with this isometry group must be a symmetric space. Recall that an aspherical manifold is one with contractible universal cover. Farb and Weinberger studied compact aspherical Riemannian manifolds M with universal cover X having Isom^{0}(X) ≠ 1. They proved several results characterizing warped products with locally symmetric factors, and locally symmetric spaces in particular. The following theorem is a weakened statement of their main theorem. Orbibundles will be defined later below in Definition 3.4.

Theorem 1.2 (Farb and Weinberger [FW]) Let M be a compact aspherical Riemannian manifold with universal cover X. Let G = Isom(X). If G^{0} ≠ 1, then M is a Riemannian orbibundle

Λ\G^{0}/K → M → Q

where Λ ⊂ G^{0} is a cocompact lattice, K is a maximal compact subgroup of G^{0}, and Q is aspherical. Further, if π_{1}(M) contains no normal free abelian subgroup, then Z(G^{0}) is finite, G^{0} is semisimple, and a finite cover of M is isometric to

Λ\G^{0}/K ×_{f} Q

for f : Q → M, the moduli space of locally symmetric metrics on Λ\G^{0}/K.

The aspherical assumption is required. A metric on the sphere S^{n} with a bump at one point, for example, has isometry group containing only rotations fixing that point. However, [FW] contains the statement of a similar theorem to the above, under a noncompactness assumption on the connected isometry group of the universal cover, for arbitrary closed Riemannian manifolds. Their proof relies on the theory of proper transformation groups, Lie theory, and remarkable cohomological dimension arguments.

1.2 Lorentz case

For Lorentz manifolds, a crucial difference from the Riemannian case is that the isometry group need not act properly; in particular, orbits may not be closed, and the group of deck transformations may not intersect G^{0} in a lattice. On the other hand, fantastic work has been done on nonproper Lorentz-isometric actions ([K1], [Ze1], [Ze2]), which implies a great deal of structure in that case. The Lorentz manifolds with the most symmetry are those of constant curvature, modeled on Minkowski space, de Sitter space, or anti-de Sitter space. Any irreducible Lorentzian locally symmetric space has constant curvature, as was proved in [CLPTV] and independently in [Ze3]. Each of the model spaces is a homogeneous space, G/H, where H is the stabilizer of a point. The isometry group, stabilizer, curvature, and diffeomorphism type for each are in the following table.

Min^{n}  dS^{n}  AdS^{n}
Isom O(1, n − 1) ⋉ R^{n}  O(1, n) O(2, n − 1)
Stab  O(1, n − 1)  O(1, n − 1) O(1, n − 1)
Curv  0  1  −1
Diff  R^{n}  S^{n−1} × R R^{n−1} × S^{1}

Note that AdS^{2} ≅ dS^{2} ≅ SO(1, 2)/A, where A ≅ R^{∗} is a maximal Rsplit torus. A result of Calabi and Markus states that no infinite subgroup of O(1, n) acts properly on dS^{n}, so there are no compact complete de Sitter manifolds ([CM]). Kulkarni noted that when n is odd, lattices in SU(1, (n − 1)/2) act freely, properly discontinously, and cocompactly on AdS^{n}. For n even, he proved that there is no cocompact, properly discontinuous, isometric action on AdS^{n} ([Ku]). Kowalsky, using powerful dynamical techniques, which are treated in detail in Section 4.1 below, proved that a simple group acting nonproperly on an arbitrary Lorentz manifold is locally isomorphic to O(1, n), n ≥ 2, or O(2, n), n ≥ 3 ([K1]). Adams has characterized groups that admit orbit nonproper isometric actions on arbitrary Lorentz manifolds in [A1] and [A2]; an action G × M → M is orbit nonproper if for some x ∈ M, the map g ↦ g.x from G to M is not proper. There are several recent results on the form of arbitrary Lorentz manifolds admitting isometric actions of certain semisimple groups. Witte Morris showed that a homogeneous Lorentz manifold with isometry group O(1, n) or O(2, n − 1) is dS^{n} or AdS^{n}, respectively ([WM1]). Arouche, Deffaf, and Zeghib, using totally geodesic, lightlike hypersurfaces, showed that if a semisimple group with no local SL_{2}(R)-factors has a Lorentz orbit with noncompact isotropy, then a neighborhood of this orbit is a warped product N ×_{f} L, where N is a complete, constant-curvature Lorentz space, and L is a Riemannian manifold ([ADZ]). Deffaf, Zeghib, and the author treat degenerate orbits with noncompact isotropy in [DMZ]. We conclude that any nonproper action of a semisimple group with finite center and no local SL_{2}(R)-factors has an open subset isometric to a warped product as in [ADZ], and we describe the global structure of such actions. The work here combines features and techniques of many of these papers, as well as those of [FW]. As in [FW], we consider universal covers of compact aspherical Lorentz manifolds and seek to describe those for which the identity component of the isometry group is nontrivial. Here is the main result.

Theorem 1.3 Let M be a compact, aspherical, real-analytic, complete Lorentz manifold with universal cover X. Let G = Isom(X), and assume G^{0} is semisimple.

(1) Orbibundle. Then M is an orbibundle

P → M → Q

where P is aspherical and locally homogeneous, and Q is a good aspherical orbifold.

(2) Splitting. Further, precisely one of the following holds: A. G^{0} acts properly on X: Then P = Λ\G^{0}/K where Λ is a lattice in G^{0} and K is a maximal compact subgroup of G^{0}. Further, if |Z(G^{0})| < ∞, then a finite cover of M is isometric to

P ×_{f} Q for f : Q → M, the moduli space of Riemannian locally symmetric metrics on P = Λ\G^{0}/K. The Lorentzian manifold Q has Isom^{0}(Q̃) = 1. B. G^{0} acts nonproperly on X: Then M is a Lorentzian orbibundle. The metric along G^{0}-orbits is Lorentzian, with

P = Λ\(ÃdS^{k} × G_{2}/K_{2})

where k ≥ 3, G_{2} ◁ G^{0} with maximal compact subgroup K_{2}, and

Λ ⊂ Õ^{0}(2, k − 1) × G_{2}

acts freely, properly discontinuously, and cocompactly on ÃdS^{k} × G_{2}/K_{2}. The good Riemannian orbifold Q has Isom^{0}(Q̃) = 1. There is a warped product

X ≅ ÃdS^{k} ×_{h} L

for some real-analytic function h : L → R^{+}. Further, if |Z(G_{2})| < ∞, then X is isometric to

(ÃdS^{k} × G_{2}/K_{2}) ×_{f} Q̃

where f : Q̃ → M, and M is the moduli space of G^{0}-invariant Lorentzian metrics on ÃdS^{k} × G_{2}/K_{2}.

Corollary 1.4 Let M and G^{0} be as above. If M has an open, dense, locally homogeneous subset, then M is locally homogeneous.

The appendix below contains an example illustrating the necessity of the hypothesis of finite center in (2) A in order to conclude that M splits locally along G^{0}-orbits as a metric product. In the example, Isom^{0}(X) is a noncompact, connected, semisimple group H^{0}; the center of H^{0} is infinite; H^{0} acts properly on X; and the metric on H^{0}-orbits varies among Riemannian, Lorentzian, and degenerate. Proof Outline for Theorem 1.3:

• The first step involves Gromov’s stratification for isometric actions on spaces with rigid geometric structure: there is a closed orbit in Y ⊆ X on which the group of deck transformations acts cocompactly (Propositions 5.4, 5.5).

The stabilizer of a point in this orbit then determines the dynamics of the isometry group on X.

• If the stabilizer is compact, then the group generated by G^{0} and the fundamental group acts properly. In this case, techniques of [FW] apply (Section 6.1).

• When the stabilizer is noncompact, then G^{0} acts nonproperly on X. In this case, we extend work of [Ze1] to show that totally geodesic lightlike foliations exist on X (Theorem 4.4). There are two subcases, depending on whether the dynamics of the group on the space of these foliations is strong or weak.

– In the case of strong dynamics on the space of foliations, results of [Ze2] are used to produce the warped product structure on X. From here, the argument resembles the case in which G^{0} is proper on X (Sections 6.2.1, 6.2.2).

– In the case of weak dynamics on the space of foliations, there is an invariant lightlike vector field tangent to the closed orbit Y. We argue by contradiction that this case cannot arise. Techniques of nonproper Lorentz dynamics, including ideas of Kowalsky [K1], are applied to give a fairly precise description of Y: it belongs to a family of spaces that do not admit cocompact, properly discontinuous, isometric actions, yielding the contradiction (Section 6.3).

2 Notation

Throughout, M is a compact, aspherical, real-analytic, complete Lorentz manifold. The universal cover of M is X, with Isom(X) = G. The group of deck transformations is Γ ≅ π_{1}(M). The identity component of G is a semisimple group G^{0}, and Γ_{0} = Γ ∩ G^{0}. Note G^{0} ◁ G and Γ_{0} ◁ Γ. The Lie algebra of G^{0} is 𝔤. Let 𝔤 = 𝔤_{1} ⊕ ⋯ ⊕ 𝔤_{l} be the decomposition of 𝔤 into simple factors. Let G_{i} be the corresponding subgroups of G^{0}. The projection 𝔤 → 𝔤_{i} will be denoted π_{i}, as will the projection G^{0} → G_{i}. For an arbitrary group H acting on a space Y, the stabilizer of y ∈ Y will be denoted H(y). In particular, G_{i}(y) = G^{0}(y) ∩ G_{i}, and 𝔤_{i}(y) = 𝔤(y) ∩ 𝔤_{i}.

3 Background and Terminology

3.1 Proper actions

The following facts follow from the existence of slices for smooth proper actions on manifolds. See [P] for definitions related to stratified spaces.

Proposition 3.1 Let H be a Lie group acting smoothly and properly on a connected manifold Y.

1. For any compact \overline{A} ⊂ H\Y, there is a compact A ⊂ Y projecting onto \overline{A}.

2. In general, H\Y is a Whitney stratified space with

dim(H\Y) = dim Y − dim H + dim H(y)

where dim H(y) is minimal over y ∈ Y.

3. If the stabilizers H(y) belong to the same conjugacy class for all y ∈ Y, then H\Y is a smooth manifold.

Proof:

1. Let π be the projection Y → H\Y. For any \overline{y} = π(y) in H\Y, the Slice Theorem (see [P] 4.2.6) gives a neighborhood \overline{U} of \overline{y} and a diffeomorphism φ_{y} : H ×_{H(y)} V_{y} → π^{−1}(\overline{U}), where V_{y} is an open ball in some R^{k}. A disk about 0 in V_{y} corresponds under φ_{y} to a compact D_{y} containing y, and projecting to a compact neighborhood of \overline{y} in H\Y. For a compact subset \overline{A}, there exist \overline{y}_{1}, …, \overline{y}_{n} such that int(π(D_{y_{1}})), …, int(π(D_{y_{n}})) cover \overline{A}. Then D_{y_{1}} ∪ ⋯ ∪ D_{y_{n}} is the desired compact A ⊂ Y.

2. The stratification is by orbit types: for each compact K ⊂ H, let

Y_{(K)} = {y ∈ Y : gH(y)g^{−1} = K for some g ∈ H}

and let Y_{K} be the fixed set of K. Then the pieces of the stratification of H\Y are the components of the quotients

H\Y_{(K)} = N_{H}(K)\Y_{K}

Each piece has the structure of a smooth manifold. See [P] 4.3.11 and 4.4.6. When K = H(y) is minimal, then Y_{(K)} is open, and the piece H\Y_{(K)} has maximal dimension dim Y − dim H + dim H(y).

3. If all stabilizers are conjugate to one compact subgroup K, then H\Y = H\Y_{(K)}, which consists of a single piece, because Y is connected.

q.e.d.

3.2 Orbifolds and orbibundles

Definition 3.2 An n-dimensional orbifold is a Hausdorff, paracompact space with an open cover {U_{i}}, closed under finite intersections, with homeomorphisms

φ_{i} : Ũ_{i}/Λ_{i} → U_{i}

where Ũ_{i} is an open subset of R^{n} and Λ_{i} is a finite group. The atlas (U_{i}, φ_{i}) must additionally satisfy the compatibility condition: whenever U_{j} ⊂ U_{i}, then there is a monomorphism Λ_{j} → Λ_{i} and an equivariant embedding Ũ_{j} → Ũ_{i} inducing a commutative diagram

Ũ_{j} → Ũ_{i}
↓  ↓
Ũ_{j}/Λ_{j} → Ũ_{i}/Λ_{i}
↓  ↓
U_{j} → U_{i}

A smooth orbifold is an orbifold for which the action of each Λ_{i} is smooth, and the embeddings Ũ_{j} → Ũ_{i} are smooth.

Definition 3.3 A good (pseudo-Riemannian) orbifold is the quotient of a (pseudo-Riemannian) manifold by a smooth, properly discontinuous, (isometric) action.

It is not hard to see using proper discontinuity that a good orbifold is a smooth orbifold.

Definition 3.4 A smooth orbibundle is a manifold M with a projection π to a good orbifold B, written

N → M →^{π} B

where N is a manifold, and the orbifold charts (U_{i}, φ_{i}) on B lift to

ψ_{i} : π^{−1}(U_{i}) → N ×_{Λ_{i}} Ũ_{i}

where Λ_{i} acts freely and smoothly on N × Ũ_{i}. A pseudo-Riemannian orbibundle is a pseudo-Riemannian manifold M with a projection π to a good pseudo-Riemannian orbifold B as above, such that the maps arising from ψ_{i}

Ũ_{i} → N ×_{Λ_{i}} Ũ_{i} → π^{−1}(U_{i})

are isometric immersions.

Note that, for a pseudo-Riemannian orbibundle M the type of the metric on M may be different from the type of the metric on the quotient orbifold B.

3.3 Rational cohomological dimension

Definition 3.5 The rational cohomological dimension of Λ is

cd_{Q}Λ = sup{n : H^{n}(Λ, A) ≠ 0, A a QΛ-module}

References for the following facts about rational cohomological dimension are [FW] and [Me].

Proposition 3.6 Let Λ be a discrete group.

1. Let Y be a contractible space on which Λ acts freely and properly with Λ\Y a finite CW-complex. Then

cd_{Q}Λ ≤ dim Y

If Λ\Y is a manifold, then there is equality.

2. If Λ is finite, then cd_{Q}Λ = 0.

3. Let Λ_{0} ◁ Λ. Then

cd_{Q}Λ ≤ cd_{Q}Λ_{0} + cd_{Q}(Λ/Λ_{0})

4. Let Λ act on a contractible CW complex Y properly and cellularly.

Then cd_{Q}Λ ≤ dim Y

3.4 Symmetric spaces

Recall that any connected Lie group G has a maximal compact subgroup K, unique up to conjugacy. This subgroup is always connected; further, the quotient G/K is contractible ([I] 6). We collect here some facts about symmetric spaces of noncompact type, which are homogeneous Riemannian manifolds of the form G/K, where G is semisimple with no compact local factors, connected, and has finite center. References for this proposition are [E], [Me], [WM2], or [Wo]. Recall that a lattice Γ in a Lie group G is a discrete subgroup such that G/Γ has finite volume with respect to Haar measure.

Proposition 3.7 Let G be a connected semisimple Lie group with finite center. Let K be a maximal compact subgroup of G.

1. There is an Ad(K)-invariant decomposition 𝔤 = 𝔭 ⊕ 𝔨. The Ad(K)- irreducible subspaces of 𝔭 correspond to the simple factors of 𝔤, and there is no one-dimensional Ad(K)-invariant subspace of 𝔭.

2. N_{G}(K) = K

3. For any torsion-free lattice Γ ⊂ G, the center Z(Γ) = 1, and |N_{G}Γ/Γ| < ∞, where N_{G}Γ is the normalizer in G of Γ.

4 Lorentz dynamics

4.1 Kowalsky’s argument

In [K1], Kowalsky relates the dynamics of Lorentz-isometric actions of a semisimple Lie group G with the adjoint representation on Sym^{2}(𝔤^{∗}). For each x ∈ X, there are linear maps

f_{x} : 𝔤 → T_{x}X
f_{x} : Y ↦ \frac{∂}{∂t}|_{0} e^{tY} x

Differentiating ge^{tY} x = ge^{tY} g^{−1}(gx) gives the relation

g_{∗x}f_{x}(Y) = f_{gx} ∘ Ad(g)(Y)

Let <, >_{x} denote the inner product on 𝔤 obtained by pulling back the Lorentz inner product on T_{x}X by f_{x}. Since the G^{0}-action is isometric,

< Y, Z >_{gx}=< Ad(g^{−1})Y, Ad(g^{−1})Z >_{x}

In Kowalsky’s argument, the dynamics of the nonproper group action imply that many root spaces of 𝔤 belong to the same maximal isotropic subspace for some <, >_{x}. We adapt this argument to obtain the following result. Recall that π_{i} is the projection of G or 𝔤 on the i^{th} (local) factor.

Proposition 4.1 Let G be a connected semisimple group acting isometrically on a Lorentz manifold. Suppose that for y ∈ X, there is a sequence g_{n} ∈ G(y) with Ad(g_{n}) → ∞. Then 𝔤 has a root system Δ and an R-split element A such that

⨁_{α∈Δ,α(A)>0} 𝔤_{α}

is an isotropic subspace for <, >_{y}. Suppose further that G^{0} preserves an isotropic vector field S^{∗} along the orbit G^{0}y, and let S ∈ 𝔤 be such that f_{y}(S) = S^{∗}(y). Then, with respect to <, >_{y},

( ⨁_{α(A)>0} 𝔤_{α}) ⊥ S

Proof: Let g_{n} = k̂_{n}â_{n}l̂_{n} be the KT K decomposition of g_{n}, where T is a maximal R-split torus in G, and K = Ad^{−1}(Ad(K)), for Ad(K) a maximal compact subgroup of Ad(G). Let Ad(g_{n}) = k_{n}a_{n}l_{n} be the corresponding decomposition in Ad(G). The condition Ad(g_{n}) → ∞ implies a_{n} → ∞. Let A_{n} = ln a_{n}. By passing to a subsequence, we may assume

• A_{n}/|A_{n}| → A for some R-split A ∈ 𝔤

• k_{n} → k

• l_{n} → l

Let Δ be a root system with respect to 𝔞 = ln T. Let α, β ∈ Δ be such that α(A), β(A) > 0. Let U ∈ 𝔤_{α} and V ∈ 𝔤_{β}. We have, for all n,

< U, V >_{â_{n}l̂_{n}y}=< U, V >_{k̂_{n}^{−1}y}  (1)

The left hand side is

< a_{n}^{−1}(U), a_{n}^{−1}(V) >_{l̂_{n}y} = e^{−α(A_{n})−β(A_{n})} < U, V >_{l̂_{n}y}
= e^{−α(A_{n})−β(A_{n})} < l_{n}^{−1}(U), l_{n}^{−1}(V) >_{y}

The inner products < l_{n}^{−1}(U), l_{n}^{−1}(V) >_{y} converge to < l^{−1}(U), l^{−1}(V) >_{y}; in particular, they are bounded. The factors e^{−α(A_{n})−β(A_{n})} converge to 0. Then the left side of (1) converges to 0. The right hand side of (1) converges to

< k(U), k(V) >_{y}

Therefore, the sum of root spaces

⨁_{α(A)>0} k(𝔤_{α})

is an isotropic subspace for <, >_{y}. Now replace Δ with Δ ∘ k^{−1} and A with k(A) to obtain the first assertion of the proposition. Now let S^{∗} be a G^{0}-invariant vector field along the orbit G^{0}y. If S is such that f_{y}(S) = S^{∗}(y), then f_{gy}(Ad(g)(S)) = S^{∗}(gy) for any g ∈ G^{0}. Let Ad(g_{n}) = k_{n}a_{n}l_{n} be the KT K decomposition as above, and let A = lim(A_{n}/|A_{n}|). Now suppose α is a root with α(A) > 0. For U ∈ 𝔤_{α}

< k_{n}U, k_{n}a_{n}l_{n}(S) >_{k̂_{n}â_{n}l̂_{n}y}=< k_{n}U, S >_{y}

The left hand side is

< a_{n}^{−1}(U), l_{n}(S) >_{l̂_{n}y}= e^{−α(A_{n})} < U, l_{n}(S) >_{l̂_{n}y}

This sequence converges to 0. The right hand side converges to

< k(U), S >_{y}

Then k(𝔤_{α}) ⊥ S with respect to <, >_{y}, yielding the desired result when A is replaced with k(A) and Δ with Δ ∘ k^{−1}. q.e.d.

Remark 4.2 Note that if, for the R-split element A given by Proposition 4.1, π_{i}(A) ≠ 0, then π_{i}(g_{n}) → ∞.

Remark 4.3 In the proof above, if we start with a KT K decomposition with 𝔞 = ln T, then the element A given by Proposition 4.1 belongs to Ad(K)(𝔞).

4.2 Totally geodesic codimension-one lightlike foliations

A lightlike submanifold of a Lorentz manifold is a submanifold on which the restriction of the metric is degenerate. A foliation is lightlike if each leaf is lightlike. In [Ze1], Zeghib shows that a compact Lorentz manifold M with a noncompact group G ⊂ Isom(M) has totally geodesic codimension-one lightlike (tgl) foliations. Fix a smooth Riemannian metric σ on M giving rise to a norm | · | and a distance d on M. Let x ∈ M and g_{n} be a sequence in G. The approximately stable set of g_{n} at x is

AS(x, g_{n}) = {v ∈ T_{x}M : v = lim v_{n} where v_{n} ∈ T M and |g_{n∗}v_{n}| is bounded}

Zeghib proves that any unbounded g_{n} has a subsequence for which the approximately stable set in T M forms an integrable codimension-one lightlike distribution with totally geodesic leaves. The resulting foliation F is Lipschitz, in the sense that there exists C > 0 such that

∠(T F_{x}, T F_{y}) ≤ C · d(x, y)

for all sufficiently close x, y ∈ M. Provided x and y are in a common normal neighborhood, we can define the angle above as

∠(T F_{x}, T F_{y}) = ∠_{σ}(P_{γ}T_{x}F_{x}, T_{y}F_{y})

where P_{γ} is parallel transport with respect to the Lorentzian connection along the geodesic γ from x to y. In fact, there exists C that serves as a uniform Lipschitz constant for all totally geodesic codimension-one foliations. We extend this work to obtain tgl foliations on X associated to a sequence g_{n} ∈ G unbounded modulo Γ. Let | · | be a smooth norm on X that is Γinvariant; such a norm can be obtained by lifting an arbitrary smooth norm from M. For x ∈ X and a sequence g_{n} ∈ G, define

AS(x, g_{n}) = {v ∈ T_{x}X : v = lim v_{n} where v_{n} ∈ T X and |g_{n∗}v_{n}| is bounded}

Note that AS(gx, g_{n}) = AS(x, γ_{n}g_{n}) for any sequence γ_{n} in Γ, so this set can be considered associated to a sequence in Γ\G. On the other hand, for g ∈ G,

AS(x, g_{n}g^{−1}) = g_{∗}(AS(x, g_{n}))

Theorem 4.4 Let g_{n} ∈ G be unbounded modulo Γ. Then there is a subsequence such that the set of AS(x, g_{n}), for x ∈ X, form an integrable distribution with totally geodesic codimension-one lightlike leaves. Moreover, the set T GL(X) of tgl foliations is uniformly Lipschitz: there exist C, δ > 0, such that, for any foliation F ∈ T GL(X), for any x, y ∈ X with d(x, y) < δ,

∠(F_{x}, F_{y}) ≤ C · d(x, y)

The proof is essentially the same as that in [Ze1]. We outline that proof and provide the observations relevant to our generalization in [Me]. For completeness, the uniformly Lipschitz property is proved in detail in the Appendix of [Me]. We begin with a definition.

Definition 4.5 Let X be a k-dimensional manifold endowed with a smooth, torsion-free connection ∇ and a smooth Riemannian metric σ. A radius-r codimension-one geodesic lamination on X consists of a subset X′ ⊂ X and a section f : X′ → Gr^{k−1}(T X)|_{X′}, satisfying

1. L_{x} = exp^{∇}(f(x) ∩ B_{σ}(0, r)) is ∇-geodesic for each x ∈ X′

2. L_{x} ∩ L_{y} is open in both L_{x} and L_{y} for all x, y ∈ X′

Proposition 4.6 Let X be the universal cover of a compact manifold M. Let ∇ be a smooth connection and σ a smooth Riemannian metric, both lifted from M. For any r > 0, there exist C, δ > 0 such that any radius-r, codimension-one geodesic lamination (X′, f) on X is (C, δ)-Lipschitz: any x, y ∈ X′ with d_{σ}(x, y) < δ are connected by a unique ∇-geodesic γ, and

∠_{σ}(P_{γ}f(x), f(y)) ≤ C · d_{σ}(x, y)

We record two consequences.

Corollary 4.7 For any radius-r codimension-one geodesic lamination (X′, f), the function f is uniformly continuous on X′.

Assuming C ≥ 1, two values f(x), f(y) will be ϵ-close in Gr^{k−1}(T X) provided d_{σ}(x, y) ≤ min{ϵ/2C, δ}.

Corollary 4.8 The space T GL(X) is compact.

The space T GL(X) can be identified with a closed subset of the space of sections f : X → Gr^{k−1}T X. Given any sequence f_{n} ∈ T GL(X), a diagonalization procedure gives a pointwise limit f_{∞} defined on a countable dense subset X′ ⊂ X. Since f_{∞} is uniformly continuous on X′, it extends uniquely to X. Because the f_{n} are equicontinuous, they converge uniformly on compact sets to f_{∞}.

5 A closed orbit

In this section, we consider a slightly more general setting. Let M be a compact, connected, real-analytic manifold with a real-analytic rigid geometric structure of algebraic type defining a connection (See [B], [Gr] or [DAG] for an introduction to Gromov’s theory of rigid geometric structures). Assume that this connection is complete—that is, that the exponential map is defined on all of T M. An example of a rigid geometric structure of algebraic type defining a connection is a pseudo-Riemannian metric. We will make use of Gromov’s stratification theorem and its consequences for realanalytic rigid geometric structures of algebraic type. Let G be the group of automorphisms of the lifted structure on the universal cover X; it is a finite-dimensional Lie group ([Gr] 1.6.H). As usual, let G^{0} be the identity component of G; let Γ ⊂ G be the group of deck transformations of X; and let Γ_{0} = Γ ∩ G^{0}. Let J be the pseudogroup of germs of local automorphisms of M. For x ∈ M, let J_{x} be the pseudogroup of germs at x of local isometries. Call the J-orbit of x ∈ M the equivalence class of x under the relation x ∼ y when jx = y for some j ∈ J_{x}. Gromov’s stratification theorem says the following:

Theorem 5.1 ([Gr] 3.4) There is a J-invariant stratification

∅ = M_{−1} ⊂ M_{0} ⊂ ⋯ ⊂ M_{k} = M

such that, for each i, 0 ≤ i ≤ k, the complement M_{i}\M_{i−1} is an analytic subset of M_{i}. Further, each M_{i}\M_{i−1} is foliated by J-orbits, and the J-orbits are properly embedded in M_{i}\M_{i−1}.

Corollary 5.2 ([Gr] 3.4.B, c.f. [DAG] 3.2.A (iii)) There exists a closed J-orbit in M.

The stratification above is obtained from similar stratifications invariant by infinitesimal isometries of order k, for arbitrary sufficiently large k. It is shown in [Gr] 1.7.B that orbits of infinitesimal isometries of increasing order eventually stabilize to J-orbits. For any x ∈ M, the infinitesimal isometries of order k fixing x form an algebraic subgroup of GL(T_{x}M), because the given H-structure is of algebraic type. Then stabilization of infinitesimal isometries to local isometries implies that the group J(x) of germs in J_{x} fixing x has algebraic isotropy representation on T_{x}M (see [DAG] 3.5, [Gr] 3.4.A); in particular, J(x) has finitely-many components.

The aim of this section is to establish that the properties of J-orbits discussed above apply also to images in M of G^{0}-orbits on X. The main reason for this correspondence is the fact, proved by Nomizu [N], Amores [Am], and, in full generality, Gromov [Gr], that local Killing fields on X can be uniquely extended to global Killing fields. Because the connection on X is complete, any global Killing field integrates to a one-parameter subgroup of G (see [KN] VI.2.4). Thus there is a correspondence between local Killing fields near any point of M and elements of 𝔤. A group H ⊆ GL(V) will be called locally algebraic if 𝔥 is the Lie algebra of an algebraic subgroup of GL(V).

Proposition 5.3 For any y ∈ X, the image of the isotropy representation of G(y) is a finite-index subgroup of an algebraic subgroup of GL(T_{y}X); the same is true for G^{0}(y). In particular, G^{0}(y) is locally algebraic.

Proof: Denote by π the covering map from X to M. There is an obvious homomorphism φ : G(y) → J(z), where z = π(y). A tangent vector at the identity to J(z) corresponds to the germ of a local Killing field at z. Local Killing fields near z can be lifted to X, extended, and integrated, giving a linear homomorphism T_{e}(J(z)) → 𝔤 inverse to D_{e}φ. Then φ is a local diffeomorphism near the identity, and so it is a local isomorphism G(y) → J(z). By rigidity, any g ∈ G(y) with trivial germ at y is trivial, so φ is an isomorphism onto its image. The image is a union of components of J(z); because the latter group is algebraic, the proposition follows for G(y). The restriction of φ to G^{0}(y) is also an isomorphism onto its image. q.e.d.

Proposition 5.4 There is an orbit G^{0}y in X with closed image in M.

Proof: Let z ∈ M have closed J-orbit, and choose any y ∈ X with π(y) = z. The image π(G^{0}y) is a connected submanifold of Jz, though it is not a priori closed. Denote by J^{0}z the component of z in Jz. This is the orbit of z under local Killing fields on M—that is, all points of M that can be reached from z by flowing along a finite sequence of local Killing fields. Because each local Killing field on M corresponds to a 1-parameter subgroup of G^{0}, this component J^{0}z is contained in π(G^{0}y). They are therefore equal, and closed in M, because Jz has finitely-many components and is closed in M. q.e.d.

Proposition 5.5 Let y be as in the previous proposition, so G^{0}y has closed image in M. The subgroup Γ_{0} = G^{0} ∩ Γ ⊂ G^{0} acts freely, properly discontinuously, and cocompactly on G^{0}/G^{0}(y).

Proof: Let G_{y} be the subgroup of G leaving invariant the orbit G^{0}y, and Γ_{y} = G_{y} ∩Γ; note that Γ_{y} acts cocompactly on G^{0}y. Because G^{0}y is a closed submanifold of X, the orbit map G^{0}/G^{0}(y) → G^{0}y is a homeomorphism onto its image (See [Gl]). It therefore suffices to show that Γ_{0} has finite index in Γ_{y}. Now G^{0}y = G_{y}y is also the homeomorphic image of G_{y}/G(y), which is then connected. As in Proposition 5.3, G(y) has finitely-many components; then so does G_{y}. Thus G^{0} is a finite-index subgroup of G_{y}, so Γ_{0} is a finite-index subgroup of Γ_{y}, as desired. q.e.d.

Corollary 5.6 If G^{0} has no compact orbits on X, then Γ_{0} is an infinite normal subgroup of Γ.

6 Proof of main theorem

Let Y = G^{0}y be the orbit given by Proposition 5.4 with closed projection to M.

6.1 Proper case

If G^{0}(y) is compact, then G^{0} acts properly; in fact, so does the group G′ generated by G^{0} and Γ.

Proposition 6.1 Let G′ be the closed subgroup of G generated by G^{0} and Γ. If G^{0}(y) is compact, then G′ acts properly on X.

Proof: If G^{0}(y) is compact, then by Proposition 5.5, Γ_{0} is a cocompact lattice in G^{0}. Let F be a compact fundamental domain for Γ_{0} containing the identity in G^{0}; note F is also a compact fundamental domain for Γ in G′. Let A be a compact subset of X, and G′_{A} the set of all g in G′ with gA ∩ A ≠ ∅. Any g ∈ G′_{A} is a product γf where f ∈ F and γ ∈ Γ_{F A}. Since F A is compact, Γ_{F A} is a finite set {γ_{1}, …, γ_{l}}. Then G′_{A} is a closed subset of the compact set γ_{1}F ∪ ⋯ ∪ γ_{l}F, so it is compact. q.e.d. The first statement in the proper case of Theorem 1.3 is that M is an orbibundle

Λ\G^{0}/K_{0} → M → Q

We prove this statement, with Λ = Γ_{0}, in three steps.

Step 1: Γ/Γ_{0} proper on G^{0}\X. Let \overline{A} be a compact subset of G^{0}\X, and let

(Γ/Γ_{0})\overline{A} = \{[γ] ∈ Γ/Γ_{0} : [γ]\overline{A} ∩ \overline{A} ≠ ∅\}

The aim is to show this set is finite. There is a compact subset A of X projecting onto \overline{A} by Proposition 3.1 (1) because G^{0} acts properly on X. Let

Γ_{A,G^{0}A} = {γ ∈ Γ : γA ∩ G^{0}A ≠ ∅}

Note that Γ_{A,G^{0}A} is invariant under right multiplication by Γ_{0}, and

(Γ/Γ_{0})\overline{A} = Γ_{A,G^{0}A}/Γ_{0}

Let F be a compact fundamental domain for Γ_{0} in G^{0}. Since F A is compact and Γ acts properly, the set Γ_{A,F A} is finite. Then (Γ_{A,F A} · Γ_{0})/Γ_{0} = Γ_{A,G^{0}A}/Γ_{0} is finite, as well.

Let K_{0} be a maximal compact subgroup of G^{0}. Step 2: G^{0}(x) ≅ K_{0} for all x ∈ X. Any stabilizer G^{0}(x) is compact, so conjugate to a subgroup of K_{0}. Since K_{0} is connected, it suffices to show that dim K_{0} ≤ dim G^{0}(x). We follow the cohomological dimension arguments of Farb and Weinberger [FW]. By Proposition 3.6 (2) and 5.5,

cd_{Q}Γ = dim X  and  cd_{Q}Γ_{0} = dim(G^{0}/K_{0})

By the extension of [FW] (2.2) of the Conner conjecture ([O]), the quotient space G^{0}\X is contractible because G^{0} acts properly and X is contractible. For any x ∈ X,

dim X − dim(G^{0}/G^{0}(x)) ≥ dim(G^{0}\X)

by 3.1 (2). Because G′ acts properly and smoothly on X, the quotient G′\X is Whitney stratified, and so triangulable by [Go]. Then the action of Γ/Γ_{0} on G^{0}\X is proper and cellular (see [Me] 4.13). Then Proposition 3.6 (4) gives

cd_{Q}(Γ/Γ_{0}) ≤ dim(G^{0}\X)

Now the inequality 3.6 (3) gives for any x ∈ X,

dim X ≤ dim(G^{0}/K_{0}) + dim(G^{0}\X)
≤ −dim K_{0} + dim X + dim G^{0}(x)

so dim K_{0} ≤ dim G^{0}(x) for any x ∈ X, as desired.

Step 3: Orbibundle. Now from Step 2 and proposition 3.1 (3), G^{0}\X is a manifold on which Γ/Γ_{0} acts properly discontinuously. The foliation of X by G^{0}-orbits descends to M, and all leaves in M are closed. The leaf space is Q = (Γ/Γ_{0})\(G^{0}\X), a smooth orbifold. Given U open in Q, lift it to a connected Ũ in G^{0}\X. For U sufficiently small, the fibers of M over U are

Ũ ×_{ΛŨ} (Γ_{0}\G^{0}/K_{0})

where ΛŨ = {[γ] ∈ Γ/Γ_{0} : [γ]Ũ ∩ Ũ ≠ ∅} is a finite group. We have an orbibundle

Γ_{0}\G^{0}/K_{0} → M → Q

Now it remains to prove the second part of the theorem in the proper case, giving the metric on M, assuming Z(G^{0}) is finite. Step 4: Splitting of X. Let

ρ : X → G^{0}/K_{0}
ρ(x) = [g]  where  gK_{0}g^{−1} = G^{0}(x)

This map is well-defined and injective along each orbit because N(K_{0}) = K_{0} (3.7 (2)). Each fiber ρ^{−1}([g]) equals the fixed set Fix(gK_{0}g^{−1}). Each orbit is mapped surjectively onto G^{0}/K_{0}. Let L = ρ^{−1}([e]) = F ix(K_{0}), a totally geodesic submanifold of X. Under the quotient, L maps diffeomorphically to G^{0}\X, so L is connected. The map

G^{0}/K_{0} × L → X ([g], l) → gl

is a well-defined diffeomorphism. The restriction of the metric to each G^{0}-orbit must be Riemannian. Indeed, let x ∈ L and consider the isotropy representation of K_{0}. The map f_{x} : 𝔤 → T_{x}X gives a K_{0}-equivariant isomorphism 𝔤/𝔨 → T_{x}(G^{0}x), where K_{0} acts on 𝔤/𝔨 via the adjoint representation. If the inner-product on T_{x}(G^{0}x) is degenerate, then the kernel is 1-dimensional, and K_{0} is trivial on it. If T (G^{0}x) is Lorentzian, then K_{0} preserves a norm, so it fixes a minimal length timelike vector. Either way, the isotropy representation of K_{0} has a fixed vector. But Ad(K_{0}) has no one-dimensional invariant subspace in 𝔭 ≅ 𝔤/𝔨 (3.7 (1)), a contradiction.

For the same reason, L is orthogonal to each G^{0}-orbit. Indeed, let x ∈ L. The subspaces T_{x}(G^{0}x)^{⊥} and T_{x}L are both K_{0}-invariant complements to T_{x}(G^{0}x) in T_{x}X. If they are unequal, then there are nonzero vectors v ∈ T_{x}(G^{0}x) and w ∈ T_{x}L such that v − w ∈ T_{x}(G^{0}x)^{⊥}. Then

k(v − w) = kv − w ∈ T_{x}(G^{0}x)^{⊥}
⇒ kv − v ∈ T_{x}(G^{0}x)^{⊥}
⇒ kv = v

again contradicting that K_{0} has no one-dimensional invariant subspace in T_{x}(G^{0}x). Step 5: Splitting of Γ. The argument here is the same as in [FW]. The extension

Γ_{0} → Γ → Γ/Γ_{0}

is a subextension of

G^{0} → G′ → Γ/Γ_{0}

so the action Γ/Γ_{0} → Out(Γ_{0}) is the restriction of Γ/Γ_{0} → Out(G^{0}). Since G^{0} is semisimple, Out(G^{0}) is finite. Thus there is a finite-index subgroup Γ′ of Γ containing Γ_{0} such that conjugation by any γ ∈ Γ′ is an inner automorphism of Γ_{0}. The extension

Γ_{0} → Γ′ → Γ′/Γ_{0}

also determines a cocycle in H^{2}(Γ′/Γ_{0}, Z(Γ_{0})). But Z(Γ_{0}) is trivial (3.7 (3)). This extension is therefore a product

Γ′ ≅ Γ_{0} × Γ′/Γ_{0}

Since Γ′ has finite integral cohomological dimension, it is torsion-free, and thus so is Γ′/Γ_{0}. Then Γ′/Γ_{0} acts freely on G^{0}\X, and the quotient, which is a finite cover of Q, is a manifold Q′. The finite cover M′ = Γ′\X is diffeomorphic to Γ_{0}\G^{0}/K_{0} × Q′. The metric descends from X to M′ and has the form claimed in the theorem.

6.2 Nonproper case: if G^{0} has infinite orbit in T GL(X)

Now suppose that G^{0}(y) is noncompact, so G^{0} acts nonproperly; further, Γ\G is noncompact. By Theorem 4.4, there are tgl foliations on X. The set T GL(X) of all these foliations forms a G-space. Pick any F ∈ T GL(X) and let O be the G^{0}-orbit of F. Because G^{0} is connected, this orbit is connected, so it either equals {F} or is infinite. We first deduce the conclusion of the main theorem in case O is infinite.

6.2.1 Warped product

Consider the continuous map

φ : T GL(X) × X → P(T X)
φ : (F, x) ↦ (x, (T F_{x})^{⊥})

For each x ∈ X, the image φ(O ×{x}) is connected, so it is either infinite or just one point. The set D of all x for which |φ(O × {x})| = 1 is closed. The complement D^{c} ≠ ∅ because O is infinite. For x ∈ X, let C_{x} be the set of lightlike lines in T_{x}X normal to leaves through x of codimension-one, totally goedesic, lightlike hypersurfaces. For all x ∈ D^{c}, the set C_{x} is infinite. Now Theorem 1.1 of [Ze2] applies to give an open set U ⊆ D^{c} locally isometric to a warped product N ×_{h} L, where N is Lorentzian of constant curvature, and L is Riemannian. For each x ∈ U, the subspace generated by C_{x} equals T_{x}N_{x}, where N_{x} is the N-fiber through x (see the intermediate result [Ze2] 3.3). Since X is the universal cover of a compact, real-analytic manifold, Theorem 1.2 of [Ze2] implies that X is a global warped product N ×_{h} L, and both N and L are complete. Because G^{0} preserves the cone field x ↦ C_{x}, it also preserves the N-foliation. Then G_{1} = Isom^{0}(N) ◁ G^{0}, so it is semisimple. Since X is contractible, N and L are, as well. Then N must be isometric to ÃdS^{k} for some k, and G_{1} ≅ Õ^{0}(2, k − 1). The assumption that C_{x} ⊂ T_{x}N is infinite implies k ≥ 3.

6.2.2 Orbibundle

Now it remains to show that X → G^{0}\X is a fiber bundle, and that M is an orbibundle. Let G_{2} be the kernel of the homomorphism G^{0} → Isom^{0}(N); it is semisimple, and G^{0} ≅ G_{1} × G_{2} ⊆ Isom(N) × Isom(L). The G^{0}-orbit Y is isometric to N × L_{2} for a Riemannian submanifold L_{2} of L, and G_{2} is isomorphic to a connected subgroup of Isom(L_{2}). Clearly, G_{2}(x) is compact for all x ∈ X, so it is conjugate into K_{2}. We will show, using cohomological dimension, that G_{2}(x) ≅ K_{2} for all x ∈ X, where K_{2} is a maximal compact subgroup of G_{2}.

Since Γ_{0} acts properly discontinuously and cocompactly on Y ≅ N×G_{2}/G_{2}(y), it is also properly discontinuous and cocompact on N × G_{2}/K_{2}, where we assume G_{2}(y) ⊆ K_{2}. This latter space is contractible, so by Proposition 3.6 (2),

cd_{Q}Γ_{0} = k + dim(G_{2}/K_{2})

Next, the quotient G_{2}\L can be identified with G^{0}\X. Since L is contractible and G_{2} acts properly on it, either quotient is contractible by [FW] 2.2. We want to show that Γ/Γ_{0} acts properly discontinuously on this quotient. Suppose that a compact \overline{C} ⊂ G_{2}\L is given. The goal is to show that

(Γ/Γ_{0})\overline{C} = \{[γ] ∈ Γ/Γ_{0} : [γ]\overline{C} ∩ \overline{C} ≠ ∅\}

is finite. Denote by L_{y} the L-leaf containing y. There is a compact C ⊂ L_{y} ⊂ X projecting onto \overline{C} by 3.1 (1). Let LX be the bundle of Lorentz frames on X. We may assume C is small enough that LX|_{C} ≅ C × O(1, n − 1). Let A be the image of a continuous section of LX|_{C} split along the product X = N × L—that is, each frame in A has the first k vectors tangent to the N-foliation, and the succeeding vectors tangent to the L-foliation. Let B be the saturation A · (Z_{2} × Z_{2} × O(n − k)), where Z_{2} × Z_{2} ⊆ O(1, k − 1) acts transitively on orientation and time orientation of Lorentz bases, and O(n − k) ⊂ O(1, n − 1) is trivial on the first k basis vectors; now B is still compact. Since G acts properly on LX (see [Gr] 1.5.B or [Ko] 3.2), the set G_{A,B} is compact in G. Because N has constant curvature, G_{1} ≅ Isom^{0}(N) is transitive on Lorentz frames along N, up to orientation and time orientation. Then it is not hard to see

G_{C,G^{0}C} = G^{0} · G_{A,B} = G_{A,B} · G^{0}

Then G_{C,G^{0}C} consists of finitely many components of G. Now

(Γ/Γ_{0})\overline{C} = (Γ_{C,G^{0}C} · Γ_{0})/Γ_{0}

Distinct Γ_{0}-cosets in Γ occupy distinct components of G. Then Γ_{C,G^{0}C} consists of finitely many cosets of Γ_{0}, and (Γ/Γ_{0})\overline{C} is finite, as desired. Now, as in Step 2 of Section 6.1,

cd_{Q}(Γ/Γ_{0}) ≤ dim(G^{0}\X) = dim(G_{2}\L)

The inequality 3.6 (3) gives

k + dim L ≤ k + dim(G_{2}/K_{2}) + dim(G_{2}\L)

so dim G_{2}(x) = dim K_{2}, and G_{2}(x) is conjugate in G_{2} to K_{2} for all x. Then the quotient Q̃ = G^{0}\X is a contractible manifold by 3.1 (2). Since Γ/Γ_{0} acts properly discontinuously here, M is an orbibundle

Γ_{0}\G^{0}/H_{0} → M → Q

The homogeneous space G^{0}/H_{0} ≅ ÃdS^{k} × G_{2}/K_{2}.

6.2.3 Splitting

From section 6.2.1, we have

X ≅ ÃdS^{k} ×_{h} L

where the warping function h on L is G_{2}-invariant. The function h descends to a function h_{1} on Q̃. From the previous section, all G_{2}-orbits in L are equivariantly diffeomorphic to G_{2}/K_{2}. As in the proper case, if Z(G_{2}) is finite, we can define

ρ : X → G_{2}/K_{2}
ρ(x) = [g]  where  gK_{2}g^{−1} = G_{2}(x)

This map factors through the projection to L. As in the proper case, we can show that L ≅ G_{2}/K_{2} ×_{h_{2}} Q̃ for some h_{2} : Q̃ → M, the moduli space of G_{2}-invariant Riemannian metrics on G_{2}/K_{2}. Now h = (h_{1}, h_{2}) can be viewed as a function from Q̃ to the moduli space of G^{0}-invariant Lorentz metrics on ÃdS^{k} × G_{2}/K_{2}.

6.3 Nonproper case: fixed point in T GL(X)

Now suppose, as above, that G^{0}(y) is noncompact, so T GL(X) ≠ ∅, but every G^{0}-orbit in T GL(X) is a fixed point. Then G^{0} preserves a tgl foliation on X, so it preserves a lightlike line field on X. We will show that this is impossible. First, we may assume that this lightlike line field along Y is tangent to Y. Suppose that Y is either a fixed point or Riemannian. Then the kernel of the restriction of G^{0} to Y contains a noncompact semisimple local factor G_{1}. Recall that C_{y} is the set of lightlike lines in T_{y}X normal to codimension-one, totally geodesic, lightlike hypersurfaces through y. Now G_{1} acts on C_{y} via the isotropy representation, and by assumption, it preserves an isotropic line in C_{y}, but this is impossible if G_{1} is semisimple and noncompact. Therefore, the orbit Y is either Lorentzian or degenerate—T_{y}Y^{⊥}∩T_{y}Y ≠ 0 for all y ∈ Y. If Y is degenerate, then G^{0} preserves the lightlike line field T_{y}Y^{⊥} along Y. Suppose Y is Lorentzian. Now G^{0} preserves the projections of the isotropic line field y ↦ C_{y} onto T Y and (T Y)^{⊥}. If the second projection is nonzero, then the first is necessarily timelike. But if G^{0}(y) preserves a timelike vector in T_{y}Y, then G^{0}(y) must be compact, a contradiction. Therefore, we may assume G^{0} preserves a lightlike line field tangent to Y, and that Y is either a degenerate or Lorentz submanifold. We first collect some facts about the isotropy representation. Lemma 6.3 below says that the isotropy is either reductive or unimodular, in each case with a rather specific form. In the reductive case, Proposition 6.5 says that 𝔤(y) contains no nilpotents. On the other hand, Kowalsky’s argument almost always yields nilpotents in 𝔤(y). The only possibility then is that G has a direct factor locally isomorphic to SL(2, R). In this case, we show that the orbit Y is roughly dS^{2}. A generalization of the Calabi-Markus phenomenon implies Y admits no cocompact isometric actions, contradicting that Γ_{0}\Y is compact. In the unimodular case, Kowalsky’s argument yields root spaces in 𝔤(y), and some factor G_{1} of G acts nonproperly on Y. Kowalsky’s Theorem [K1] says that G_{1} is locally isomorphic to O(1, k) or O(2, k) for some k. We argue that the root lattice of O(2, k) is incompatible with properties of the adjoint representation of G_{1} established in Proposition 6.6. We conclude that Y is roughly equivalent to the light cone in Minkowski space, which also admits no cocompact isometric actions, a contradiction.

6.3.1 Properties of the isotropy respresentation

Fix an isometric isomorphism of T_{y}X with R^{1,n−1}, determining an isomorphism O(T_{y}X) ≅ O(1, n − 1). Let V be the image of T_{y}Y under this isomorphism, and let k = dim V. Let Φ : G^{0}(y) → O(1, n − 1) be the resulting isotropy representation. There is a filtration on V preserved by Φ. The notation U⊂_{i}V means U is a subspace of V with dim(V/U) = i. The invariant filtration is

0⊂_{1}V_{0}⊂_{k−1−i}V_{1}⊂_{i}V

where i = 0 or 1 depending on whether V is degenerate or Lorentz. The subspaces V_{0} and V_{1} are degenerate. Because Φ preserves the isotropic line V_{0} it descends to a quotient representation on V_{1}/V_{0}, which is orthogonal.

The image of Φ is conjugate in O(1, n − 1) to the minimal parabolic

P = (M × A) ⋉ U

where U ≅ R^{n−2} is unipotent, A ≅ R^{∗}, and M ≅ O(n − 2), with the conjugation action of M × A on U equivalent to the standard conformal representation of O(n − 2) × R^{∗} on R^{n−2}. Denote by 𝔭 the Lie algebra of P, and by 𝔪, 𝔞, 𝔲, the subalgebras corresponding to M, A, and U. Because G^{0} acts properly and freely on the bundle of Lorentz frames of X, the isotropy representation is an injective, proper map. By Corollary 5.3, the image Φ(G^{0}(y)) is locally algebraic. Let φ : 𝔤(y) → 𝔬(1, n − 1) be the Lie algebra representation tangent to Φ. Because im(φ) is algebraic, there is a Lie algebra decomposition

im(φ) ≅ 𝔯′ ⋉ 𝔲′

where 𝔯′ is reductive and 𝔲′ is unipotent ([WM3] 4.4.7). Any unipotent subalgebra of 𝔭 lies in 𝔲, so 𝔲′ ⊂ 𝔲. The reductive complement 𝔯′ is contained in a maximal reductive subalgebra, which is then conjugate into 𝔞 × 𝔪. Note that T_{y}Y can be identified with 𝔤/𝔤(y) by the map f_{y} as in Section 4.1, and there is the relation

g_{∗y} ∘ f_{y}(B) = f_{y} ∘ Ad(g)(B)

for B ∈ 𝔤 and g ∈ G^{0}(y). In other words, Φ restricted to V is equivalent to the representation \overline{Ad} of G^{0}(y) on 𝔤/𝔤(y) arising from the adjoint representation. Let \overline{ad} be the representation tangent to \overline{Ad}.

Proposition 6.2 There is a filtration of 𝔤 invariant by the adjoint of 𝔤(y):

0 ⊂ 𝔤(y) ⊂_{1} 𝔰(y) ⊂_{k−1−i} 𝔱(y) ⊂_{i} 𝔤

where i = 0 or 1 depending on whether Y is degenerate or Lorentz. The subspace 𝔰(y) is a subalgebra. The quotient representation for \overline{ad} on 𝔱(y)/𝔰(y) is skew-symmetric.

Proof: The φ-invariant filtration 0 ⊂ V_{0} ⊂ V_{1} ⊂ V of V corresponds to an \overline{ad}-invariant filtration of 𝔤/𝔤(y). Lifting to 𝔤 gives the desired ad(𝔤(y))- invariant filtration. That 𝔰(y) is a subalgebra follows from the facts that [𝔤(y), 𝔰(y)] ⊂ 𝔰(y) and dim(𝔰(y)/𝔤(y)) = 1. Orthogonality of Φ on V_{1}/V_{0} implies φ is skew-symmetric on V_{1}/V_{0}; skew-symmetry of \overline{ad} on 𝔱(y)/𝔰(y) follows. q.e.d. Now we show that the image of Φ is either contained in A × M or M ⋉ U.

Lemma 6.3 The image of φ is either reductive or consists of endomorphisms with no nonzero real eigenvalues.

Proof: Suppose there is B ∈ 𝔤(y) such that φ(B) has nonzero eigenvalue λ for some eigenvector v ∈ T_{y}X. The vector v is necessarily isotropic, and we may assume that v ∈ V_{0}. Otherwise, for any nonzero w ∈ V_{0}, the inner product < v, w >≠ 0, which implies that φ(B) has nonzero real eigenvalue on w, as well. Assume λ > 0; the case λ < 0 is similar. We may assume B ∈ 𝔯′. By considering φ(B) on the subquotients of the invariant filtration V_{0} ⊂ V_{1} ⊂ V, one sees that the trace of φ(B)|_{V} is nonnegative, and equals 0 if and only if V is Lorentz. Correspondingly, the trace of \overline{ad}(B) on 𝔤/𝔤(y) is nonnegative. If φ(B) ∈ 𝔭 has eigenvalue λ > 0, then the adjoint ad(φ(B)) has no negative eigenvalues on 𝔭. To simplify the argument, we will use that φ(B) = B_{1}+B_{2}, where 0 ≠ B_{1} ∈ 𝔞 and B_{2} ∈ 𝔪. It is easy to see that ad(B_{1}) has only real nonnegative eigenvalues on 𝔭. All eigenvalues of ad(B_{2}) are purely imaginary. Since ad(B_{1}) and ad(B_{2}) are simultaneously diagonalizable, their sum ad(φ(B)) cannot have a negative eigenvalue. Now suppose that im(φ) is not reductive, so 𝔲′ ≠ 0. Let m = dim(𝔲′). It is easy to compute that the trace of ad(φ(B)) on 𝔲′ is mλ. Since ad(φ(B)) has no negative eigenvalues, the trace of ad(φ(B)) on im(φ) ⊆ 𝔭 is positive. Then the trace of ad(B) on 𝔤(y) is positive. Finally, the trace of ad(B) on 𝔤 is positive, which is impossible because 𝔤 is semisimple, hence unimodular. q.e.d. Now we have that im(Φ) is either a reductive subgroup of A × M or has the form M′ ⋉ U′, where M′ ⊂ M and U′ ⊂ U.

6.3.2 Two examples with no compact quotient

Two-dimensional de Sitter space. The 2-dimensional de Sitter space dS^{2} has isometry group O(1, 2) and isotropy O(1, 1), which has an index-two subgroup isomorphic to R^{∗}. It is a well-known result of Calabi and Markus that no infinite subgroup of O(1, 2) acts properly on dS^{2}, so it has no compact quotient [CM]. More generally, if Y = dS^{2} × L for some Riemannian manifold L, then no subgroup of the product O(1, 2) × Isom(L) acts properly discontinuously and cocompactly on Y; this is proved in [Ze1] §15.1.

We will need an analogous result that also applies to the universal cover d̃S^{2}.

Proposition 6.4 Let S be a Lorentz manifold with universal cover d̃S^{2}. Let G ≅ Isom^{0}(S), and H be a connected Lie group. There is no subgroup Γ ⊂ G × H acting properly discontinuously and cocompactly on S × H.

Proof: It suffices to prove the proposition assuming S = d̃S^{2}. Let K = Ad^{−1}(SO(2)), where SO(2) is a maximal compact subgroup of Ad(G) ≅ O^{0}(1, 2). Let Z ≅ Z be the torsion-free factor of the center Z(G). Let \overline{K} be a compact fundamental domain in K for the Z-action with \overline{K} = \overline{K}^{−1}; for example, identifying SO(2) with S^{1} and K with R, we can take \overline{K} = [−1/2, 1/2]. Let A be a maximal R-split torus in G. We have G = KAK = Z\overline{G}, where \overline{G} = \overline{K}A\overline{K}. For any g ∈ \overline{G},

g\overline{K} ∩ \overline{K}A ≠ ∅

The translation number helps to sift the Z-action from the \overline{G}-action, to say that any Γ acting cocompactly has infinitely-many elements with uniformly bounded projection in H and G-projection intersecting \overline{G} in an infinite subset, thus contradicting properness. There is an isomorphism G ≅ S̃L_{2}(R), so G acts on the real line, with Z acting by integral translations. The translation number

τ : G → Z ≅ Z
τ : g ↦ lim_{n→∞} \frac{g^{n}(0)}{n}

is a quasi-morphism (see [Gh]): there exists D > 0 such that

|τ(gg′) − τ(g) − τ(g′)| < D for all g, g′ ∈ G

We can choose \overline{K} and A so that τ(\overline{K}) = [−1/2, 1/2], and τ(A) = 0. Therefore if g ∈ \overline{G}, then |τ(g)| ≤ 2D +1. Also note that for n ∈ Z ≅ Z and g ∈ G, then τ(ng) = n + τ(g).

Now suppose that \overline{C} ⊂ d̃S^{2} × H is a compact fundamental domain for Γ. Denote by ρ_{1} and ρ_{2} the projections onto d̃S^{2} and H, respectively. We may assume that the identity of H is in ρ_{2}(\overline{C}) = U. For n ∈ Z, let

S_{n} = \{(g, h) ∈ Γ : g ∈ n\overline{G}, hU ∩ U ≠ ∅\}

For a subset L ⊆ G, denote by [L] its image in d̃S_{2}. Note that, for any γ ∈ S_{n}, the intersection

γ([\overline{K}] × U) ∩ ([n\overline{K}] × U) ≠ ∅

Therefore, if Γ acts properly discontinuously, then |S_{n}| < ∞ for each n ∈ Z. On the other hand, we have [\overline{G}] × U ⊂ Γ · \overline{C}. Let C be a compact lift of ρ_{1}(\overline{C}) to G. Then we have

\overline{G} × U ⊂ Γ · (CA × U)

The restriction of |τ| to \overline{G}CA is bounded, so, for |n| sufficiently large,

n\overline{G}CA ∩ \overline{G} = ∅

It follows that \overline{G}×U is contained in the union of finitely many S_{n}·(CA×U), which is a union of finitely many translates γ ·(CA×U), which is impossible, because [\overline{G}] × U is not compact. q.e.d. The Minkowski light cone. A component of the light cone minus the origin in Minkowski space R^{1,k−1} is a degenerate orbit of O^{0}(1, k − 1), which we will momentarily denote by G^{0}. The stabilizer of an isotropic vector is isomorphic to M ⋉ U, where M, U ⊂ P are as above. We will show that no subgroup of G^{0} acts properly discontinuously and cocompactly on this orbit. Suppose that y is a point in the light cone and Γ ⊂ G^{0} is a discrete subgroup such that Γ\G^{0}/G^{0}(y) is a compact manifold. Then Γ\G^{0}/U is also compact; we may assume it is orientable. Because U is unimodular, the homogeneous space G^{0}/U has a G^{0}-invariant volume form (see [R] I.1.4). This form descends to Γ\G^{0}/U, where it has finite total volume. The subgroup A ≅ R^{∗} of P normalizes U, with generator a acting by Ad(a)(Y) = e^{2}Y for all Y ∈ 𝔲. Then a acts on Γ\G^{0}/U and scales the volume form by 1/e^{2(k−2)} at every point, which is impossible for a diffeomorphism of a compact manifold. In the next section, we will show that, if im(Φ) is reductive, then Ỹ is related, by proper G̃^{0}-equivariant maps, to d̃S^{2} × H, where H is a connected Lie group. In case im(Φ) is unimodular, we will show that there is a proper G^{0}-equivariant map (O(1, k − 1)/U) × G_{2} → Y, where U is the unipotent radical of the minimal parabolic of O(1, k − 1), and G_{2} is a local factor of G^{0}. In both the reductive and unimodular cases, no subgroup of G^{0} can act properly discontinuously and cocompactly on Y. Both cases involve studying the representation Φ and applying dynamical results from Section 4.1. An element B of 𝔤 is called nilpotent if ad(B) is nilpotent. An element B is semisimple if ad(B) is diagonalizable over C, and B is R-split if ad(B) is diagonalizable over R.

6.3.3 Reductive case

In this case, im(Φ) ⊂ A×M. Because G^{0}(y) is noncompact and Φ is proper, im(Φ) is not contained in M. The image is fully reducible on T_{y}X; it decomposes as a product A′×M′, where M′ is compact, A′ is one-dimensional, and A′ has nontrivial character on V_{0}. Note also that the exponential map is onto (A′)^{0} because it is onto both A^{0} and M^{0} (see [Kn] 1.104 and 4.48). Let Â × M̂ be the corresponding decomposition of G^{0}(y). Properness of Φ implies M̂ is compact. Continuity implies Ad(a^{n}) → ∞ for all nontrivial a ∈ Â^{0}: if Ad(a^{n}) were bounded, then \overline{Ad}(a^{n}) would be bounded, so Φ(a^{n}) would be bounded on V, a contradiction. Note the exponential map is onto Â^{0} because it is for (A′)^{0}.

Proposition 6.5

1. The restriction of the metric to Y is Lorentzian, so V_{1} ≠ V.

2. There is an \overline{ad}-invariant decomposition

\overline{𝔰}_{0}(y) ⊕ \overline{𝔰}_{1}(y) ⊕ \overline{𝔰}_{2}(y)

of 𝔤/𝔤(y) corresponding to the filtration in Proposition 6.2.

3. The stabilizer subalgebra 𝔤(y) contains no elements nilpotent in 𝔤; in particular, there are no root vectors of 𝔤 in 𝔤(y).

Proof:

1. Let B ∈ 𝔞̂, and let λ be the nonzero eigenvalue of φ(B) on V_{0}, which we assume is positive. If V is degenerate, then the trace of φ(B) on V_{1} = V is positive, so the trace of \overline{ad}(B) on 𝔤/𝔤(y) is positive. Now B ∈ 𝔷(𝔤(y)), so the trace of ad(B) on 𝔤(y) is 0. Then the trace of ad(B) on 𝔤 is positive, contradicting unimodularity of 𝔤.

2. Let 𝔰(y) ⊂ 𝔱(y) ⊂ 𝔤 be the 𝔤(y)-invariant subpaces in Proposition 6.2. Let \overline{𝔰}_{0}(y) be the projection of 𝔰(y) to \overline{𝔤} = 𝔤/𝔤(y). Let \overline{𝔱}(y) be the projection of 𝔱(y). Because \overline{ad}(𝔤(y)) is fully reducible, there is an invariant complement \overline{𝔰}_{1}(y) to \overline{𝔰}_{0}(y) in \overline{𝔱}(y). Let \overline{𝔰}_{2}(y) be an invariant complement to \overline{𝔱}(y) in \overline{𝔤}.

3. Suppose that X ∈ 𝔤(y) is nilpotent. Then \overline{ad}(X) is nilpotent, so φ(X) restricted to V is nilpotent. Because im(φ) contains no nilpotent elements, φ(X) is trivial on V. By (1), the inner product on V^{⊥} ⊂ R^{1,n−1} is positive definite, so φ(X) is skew-symmetric and generates a precompact subgroup of O(1, n − 1). Because Φ is proper, X should generate a precompact subgroup of G^{0}, a contradiction unless X = 0.

q.e.d. Now let b ∈ Â^{0}, so Ad(b^{n}) → ∞. By Proposition 4.1, there exists an R-split element B of 𝔤 and a root system such that

⨁_{α(B)>0} 𝔤_{α}

is isotropic for the pullback inner product <, >_{y} on 𝔤. By Proposition 6.5 (3), this sum of root spaces does not meet 𝔤(y). Therefore,

dim ( ⨁_{α(B)>0} 𝔤_{α}) = 1

Then there is exactly one root α with α(B) > 0. Let X_{α} and X_{−α} be root vectors spanning 𝔤_{α} and 𝔤_{−α}, respectively. Together with B, they generate a direct factor, say 𝔤_{1}, of 𝔤, isomorphic to 𝔰𝔩_{2}(R). Denote by L the null cone in 𝔤/𝔤(y) ≅ V. For b ∈ Â^{0}, the sequence \overline{Ad}(b^{n}) has unique distinct attracting and repelling fixed points, p^{+} and p^{−}, respectively, in the projectivization P(L); these correspond to the nontrivial eigenvectors of \overline{Ad}(b). For i = 1, …, l, denote by \overline{𝔤}_{i} the image of 𝔤_{i} modulo 𝔤(y); each such subspace is \overline{Ad}(G^{0}(y))-invariant. Similarly for X ∈ 𝔤, denote by \overline{X} its image in 𝔤/𝔤(y). Because X_{α} is isotropic for ⟨, ⟩_{y}, the projection \overline{X}_{α} ∈ L∩\overline{𝔤}_{1}. Then either the projectivization [\overline{X}_{α}] = p^{−}, or [\overline{Ad}(b^{n})(\overline{X}_{α})] → p^{+}. In either case, one of p^{−}, p^{+} is in [\overline{𝔤}_{1}], and \overline{Ad}(b) has an eigenvector with nontrivial real eigenvalue in \overline{𝔤}_{1}.

Denote by b_{1} the projection of b on G_{1}. It belongs to a 1-parameter subgroup e^{tZ} because the exponential map is onto Â^{0}, so Ad(b_{1}) fixes Z. Now Ad(b_{1}) has eigenvalues λ ≠ 1 and 1 on 𝔤_{1}, so λ^{−1} is also an eigenvalue, and b_{1} is R-split. The eigenvectors are nilpotent elements, so each has nontrivial projection modulo 𝔤_{1}(y). Then both p^{+} and p^{−} belong to [\overline{𝔤}_{1}], and they are the images of nilpotent elements of 𝔤_{1}. Now let Y be any nilpotent in 𝔤_{i} for i > 1. By 6.5 (3), \overline{Y} ≠ 0. If \overline{Y} ∉ \overline{s}_{1}(y), then [\overline{Ad}(b^{n})(\overline{Y})] converges in P(V) to p^{+}, or [\overline{Ad}(b^{−n})(\overline{Y})] converges to p^{−}; we assume the former. Then p^{+} would be the image of a nilpotent element from 𝔤_{1} and another from 𝔤_{i}. In the span of these two would be a nilpotent element of 𝔤(y), a contradiction. Therefore, \overline{𝔤}_{i} ⊆ \overline{𝔰}_{1}(y), and \overline{Ad}(b^{n}) is bounded on \overline{𝔤}_{i} for all i > 1. Since b_{1} ≠ 1 and Â^{0} is 1-dimensional, the intersection G_{i} ∩Â^{0} = 1 for all i > 1. It follows that 𝔤_{i}(y) ⊆ 𝔪′, so it is definite for the restriction of the Killing form κ_{i} of 𝔤_{i}, and Ad(G^{0}(y)) is bounded on 𝔤_{i}(y). The orthogonal of 𝔤_{i}(y) is an Ad(G^{0}(y))-invariant complement in 𝔤_{i}, which projects equivariantly and isomorphically to \overline{𝔤}_{i}. Now for all i > 1, the adjoint Ad(G^{0}(y)) is bounded on 𝔤_{i}, which implies that Ad(π_{i}(G^{0}(y))) is precompact. Because G^{0} preserves a Lorentz metric on Y ≅ G^{0}/G^{0}(y), any element of Z(G^{0}) ∩ G^{0}(y) would have trivial derivative along Y at y. Then Φ(Z(G^{0}) ∩ G^{0}(y)) is precompact. By properness of Φ, this group is finite. Therefore, π_{i}(G^{0}(y)) is precompact; let K_{i} denote the compact closure. Let K = K_{2} × ⋯ × K_{l} for i > 1. We have already established that the projection π_{1}(Â) contains a nontrivial R-split element. Because any other element of π_{1}(G^{0}(y)) must centralize this one, we conclude that π_{1}(G^{0}(y)) is isogenous to a maximal R-split subgroup A_{1} of G_{1}. Therefore, G^{0}(y) ⊆ A_{1} × K, and there is a G^{0}-equivariant proper map

Y ≅ G^{0}/G^{0}(y) → G^{0}/(A_{1} × K)

so Γ_{0} acts properly and cocompactly on both spaces. Then Γ_{0} acts properly and cocompactly on G_{1}/A_{1} × H, where H ≅ G_{2} × ⋯ × G_{l}. But now the universal cover of G_{1}/A is homothetic to d̃S^{2}, so Proposition 6.4 applies, giving a contradiction.

6.3.4 Unimodular case

Now assume im(Φ) = M′ ⋉ U′ with M′ compact and U′ unipotent. First we collect some algebraic facts for this case.

Proposition 6.6 Let B ∈ 𝔤(y).

1. B is not R-split.

2. If φ(B) is nilpotent, then B is nilpotent.

3. If B is nilpotent, then on the filtration in Proposition 6.2, \overline{ad}(B) carries each subspace to the next. In other words, \overline{ad}(B) is trivial on each factor of the associated graded space.

4. If φ(B) is nilpotent and 𝔤 ≠ 𝔱(y), then \overline{ad}(B) has nilpotence order 3.

Proof:

1. Suppose B is R-split. Let α be a root with α(B) ≠ 0 and X_{α}, X_{−α} nonzero elements of the corresponding root spaces generating a subalgebra of 𝔤 isomorphic to 𝔰𝔩_{2}(R). Because φ(B) can have no eigenvectors with nonzero real eigenvalue, X_{α} and X_{−α} are both contained in 𝔤(y). Then 𝔤(y) ⊂ 𝔭 contains a subalgebra isomorphic to 𝔰𝔩_{2}(R), a contradiction.

2. If φ(B) is nilpotent, then φ(B) ∈ 𝔲′. Then \overline{ad}(B) is nilpotent and ad(B) is nilpotent on 𝔤(y), which implies nilpotence of B.

3. If B is nilpotent, then ad(B) is trivial on both 𝔤/𝔱(y) and 𝔰(y)/𝔤(y), because they are both at most one-dimensional. Because ad(B) is skew-symmetric and nilpotent on 𝔱(y)/𝔰(y), this representation is also trivial: indeed, for any k ≥ 1,

(adB)^{2k} = (−1)^{k}(adB^{t} ∘ adB)^{k}

which is zero if and only if B = 0.

4. If φ(B) is nilpotent and 𝔤 ≠ 𝔱(y), then V is Lorentzian and the inner product on V^{⊥} is positive definite, so φ(B) is trivial on V^{⊥}. By injectivity of φ, the restriction of φ(B) to V is nontrivial, so \overline{ad}(B) is nontrivial. By item (3), \overline{ad}(B) has nilpotence order at most 3. Let W ∈ 𝔤\𝔱(y). We will show that ad^{2}(B)(W) ∉ 𝔤(y).

Denote by <, > the pullback of the inner product from T_{y}X to 𝔤. For any W, Z ∈ 𝔤

< ad(B)(W), Z > + < W, ad(B)(Z) >= 0

First we will show that ad(B)(W) ∉ 𝔰(y). Suppose it is. For any Z ∈ 𝔰(y)\𝔤(y), the inner product < W, Z >≠ 0. The identity

< ad(B)(W), W > + < W, ad(B)(W) >= 2 < ad(B)(W), W >= 0

implies ad(B)(W) ∈ 𝔤(y). Now ad(B)(𝔱(y)) ⊆ 𝔤(y) would imply \overline{ad}(B) is trivial, which cannot be. Then there must be some Z ∈ 𝔱(y) such that ad(B)(Z) ∈ 𝔰(y)\𝔤(y). Then

< ad(B)(W), Z >= − < W, ad(B)(Z) >≠ 0

But the left side above is zero if ad(B)(W) ∈ 𝔤(y), contradicting the original assumption that ad(B)(W) ∈ 𝔰(y).

Now ad(B)(W) must be in 𝔱(y)\𝔰(y), so

< ad(B)^{2}(W), W >= − < ad(B)(W), ad(B)(W) >≠ 0

which implies ad(B)^{2}(W) ∉ 𝔤(y), as desired.

q.e.d. Let M̂⋉Û be the decomposition of G^{0}(y) corresponding to im(Φ) = M′⋉U′. Again, because Φ is proper, M̂ is compact. Let 𝔪̂ and 𝔲̂ be the corresponding subalgebras of 𝔤. From item (2) above, û consists of nilpotent elements. Let J be the set of i such that π_{i}(𝔲̂) ≠ 0. We will show by induction that there exists X ∈ 𝔲̂ such that π_{i}(X) ≠ 0 if and only if i ∈ J. Let i_{1}, …, i_{k} be some order on the elements of J. Clearly, there is some X_{1} ∈ 𝔲̂ such that π_{i_{1}}(X_{1}) ≠ 0. Suppose X_{m} ∈ 𝔲̂ is such that π_{i_{j}}(X_{m}) ≠ 0 for all j ≤ m. There exists Y_{m} ∈ 𝔲̂ such that π_{i_{m+1}}(Y_{m}) ≠ 0. For some real number c, the element X_{m+1} = X_{m} + cY_{m} will have π_{i_{j}}(X_{m+1}) ≠ 0 for all j ≤ m + 1. Write this element X = ∑_{i∈J} X_{i} with X_{i} ∈ 𝔤_{i}. Note that nilpotence of X implies nilpotence of each X_{i}. The Jacobson-Morozov theorem (see [H] IX.7.4) yields, for each i ∈ J, an R-split element A_{i} ∈ 𝔤_{i} and a nilpotent element Y_{i} ∈ 𝔤_{i} such that

[A_{i}, X_{i}] = 2X_{i},  [A_{i}, Y_{i}] = −2Y_{i},  and [X_{i}, Y_{i}] = A_{i}

Then the elements A = ∑_{i} A_{i} and Y = ∑_{i} Y_{i} satisfy

[A, X] = 2X,  [A, Y] = −2Y,  and [X, Y] = A

The subalgebra generated by X, A, and Y is isomorphic to 𝔰𝔩_{2}. Let L be the corresponding subgroup of G^{0}. The adjoint of 𝔤(y) is trivial on 𝔰(y)/𝔤(y) by 6.6 (3), so G^{0} preserves a vector field tangent to the invariant isotropic line field along Y. Now Proposition 4.1 and Remark 4.3 for g_{n} = e^{nX} give some k ∈ L such that, for A′ = Ad(k)(A),

⨁_{α(A′)>0} 𝔤_{α} ⊂ 𝔰(y)

Let X′ = Ad(k)(X) ∈ 𝔰(y). We may assume k ∈ P SL_{2}(R). For

k = ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} )

and

X = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )

the bracket

[X, X′] = ( \begin{matrix}− sin^{2} θ & 2 cos θ sin θ \\ 0 & sin^{2} θ \end{matrix} )

This bracket belongs to 𝔤(y), which contains no R-split elements (6.6 (3) and (1)). Then sin θ must be 0, so Ad(k) is trivial, and

⨁_{α(A)>0} 𝔤_{α} ⊂ 𝔰(y)

Now we will show that this sum of root spaces is in fact contained in 𝔤(y). For Y the negative root vector as above, ad^{2}(X)(Y) ∈ 𝔤(y), so \overline{ad}(X) has order less than 3 on the corresponding element of 𝔤/𝔤(y). Then Y ∈ 𝔱(y) by Proposition 6.6 (4). Then [X, Y] = A ∈ 𝔰(y) by 6.6 (3), but A cannot be in 𝔤(y) by 6.6 (1), so

𝔰(y) = RA + 𝔤(y)

Now suppose α(A) > 0 and let X′ be an arbitrary element of 𝔤_{α} ⊆ 𝔰(y). Since [𝔰(y), 𝔰(y)] ⊆ 𝔤(y), the bracket [A, X′] = α(A)X′ ∈ 𝔤(y). Therefore, ⊕_{α(A)>0}𝔤_{α} ⊆ 𝔤(y), as desired; in particular, X_{i} ∈ 𝔤(y) for all i ∈ J.

Next we will show that |J| = 1. As above, Proposition 6.6 implies Y_{i} ∈ 𝔱(y) and A_{i} ∈ 𝔰(y) for all i ∈ J. If |J| > 1, then, for one i ∈ J and some nonzero c ∈ R, the difference cA−A_{i} is a nontrivial R-split element of 𝔤(y), contradicting 6.6 (1).

Now G^{0}(y) ≅ M̂ ⋉ Û with π_{i}(û) = 0 for all i except, say, 1. Then G^{0}(y) has precompact projection on all local factors except G_{1}. By Kowalsky’s Theorem ([K1]), 𝔤_{1} ≅ 𝔬(2, k), for some k ≥ 3, or 𝔬(1, k), for some k ≥ 2. We will deduce that 𝔤_{1} must be the latter, and that G_{1}(y) is as in the Minkowski light cone. The subspaces 𝔰_{1}(y) and 𝔱_{1}(y) will denote the intersections 𝔤_{1} ∩ 𝔰(y) and 𝔤_{1} ∩ 𝔱(y), respectively, below. Step 1: 𝔤_{1} ≅ 𝔬(1, 2) implies G_{1}y degenerate. If 𝔤_{1} ≅ 𝔬(1, 2), then it is generated by X, A, and Y from above. Recall X ∈ 𝔤_{1}(y); A ∈ 𝔰_{1}(y); and Y ∈ 𝔱_{1}(y). Then 𝔤_{1}/𝔤_{1}(y) is 2-dimensional and degenerate with respect to the inner product pulled back from T_{y}X; therefore, the orbit G_{1}y is also degenerate. Step 2: G_{1}y is degenerate in general. Assume that 𝔤_{1} is not isomorphic to 𝔬(1, 2), and suppose that the orbit G_{1}y ⊆ Y is of Lorentzian type. Then Theorem 1.5 of [ADZ] gives that G_{1}y is equivariantly homothetic, up to covers, to dS^{k} or AdS^{k} for some k ≥ 3; in either case, 𝔤_{1}(y) would be semisimple, a contradiction. Step 3: Case 𝔤_{1} ≅ 𝔬(2, k). Now suppose 𝔤_{1} ≅ 𝔬(2, k) for some k ≥ 3. Let Δ be a root system of 𝔤 as above. Let A ∈ 𝔰(y) be as above. Let α ∈ Δ be such that α(A) = 2. Let X ∈ 𝔤_{α} ∩ 𝔤(y). The root system of 𝔬(2, k) is generated by two simple roots, β and γ. The root spaces for β and γ are each (k − 2)-dimensional. The other positive roots are β − γ and β + γ, with one-dimensional root spaces. First suppose α = β, so X ∈ 𝔤_{β} ⊂ 𝔤(y). Let L be a generator of 𝔤_{−β−γ}. For any such X and L, the adjoint ad^{2}(X)(L) ≠ 0. Since the orbit G_{1}y is degenerate, L ∈ 𝔱(y), and ad(X)(L) ∈ 𝔰(y). Let W = ad(X)(L) ∈ 𝔤_{−γ}. Any nilpotent subalgebra of 𝔤_{1}(y) ⊆ 𝔭 is abelian, so W ∈ 𝔰(y)\𝔤(y). Then cW −A ∈ 𝔤_{1}(y) for some nonzero c ∈ R. But now L ∈ 𝔱(y)\𝔰(y) would be an eigenvector for this element with nonzero real eigenvalue. Then φ(cW − A) would have a nonzero real eigenvalue on V, contradicting Lemma 6.3. We conclude that X cannot be in 𝔤_{β}. The same argument shows X cannot be in 𝔤_{γ}; in fact, 𝔤_{1}(y) ∩ 𝔤_{ω} must be 0 for ω = ±β, ±γ. Now suppose that α = β ± γ, so either (β + γ)(A) or (β − γ)(A) equals 2. Then one of β(A) or γ(A) is nonzero, which again implies that one of 𝔤_{±β}, 𝔤_{±γ} is in 𝔤(y), a contradiction. G_{1}y is the Minkowski light cone Now we have that 𝔤_{1} ≅ 𝔬(1, k) for some k ≥ 2. Let α be the positive root of 𝔤_{1} with α(A) = 2. From above, 𝔤_{α} ⊂ 𝔲̂. Since this root space is a maximal abelian subalgebra of nilpotent elements in 𝔤_{1}, this containment is equality by 6.6 (2). There is a proper equivariant map

G^{0}/Û → G^{0}/(M̂ ⋉ Û) ≅ Y

so no subgroup of G^{0} acts properly discontinuously and cocompactly on Y, as in Section 6.3.2.

7 Appendix: Non-split example

As promised in the introduction, the following example illustrates the necessity of the hypothesis of finite center in (2) A in order to conclude that M splits locally along G^{0}-orbits as a metric product. In the example, Isom^{0}(X) ≅ S̃L(2, R), a noncompact semisimple group with infinite center; it acts properly on X; and the metric on S̃L(2, R)-orbits varies among Riemannian, Lorentzian, and degenerate. Consider the basis for 𝔰𝔩(2, R)

A = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )  K = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )  P = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

The brackets among these generators are

[A, P] = 2K  [A, K] = 2P  [K, P] = 2A

Denote by B_{λ} the inner product on 𝔰𝔩(2, R) in which A, P, and K are mutually orthogonal,

B_{λ}(A, A) = 1 = B_{λ}(P, P),  and  B_{λ}(K, K) = λ

When λ = −1, then B_{λ} is a constant multiple of the Killing form. Denote also by A, P, K the corresponding left-invariant vector fields on S̃L(2, R). Note that B_{λ} determines a left-invariant inner product on T S̃L(2, R).

Let X = S̃L(2, R) × R with the following Lorentz metric ν:

ν_{(x,t)}|_{span{A,P,K}} = B_{cos t}
ν_{(x,t)} (\frac{∂}{∂t}, \frac{∂}{∂t}) = − cos t
ν_{(x,t)} (\frac{∂}{∂t}, A) = 0 = ν_{(x,t)} (\frac{∂}{∂t}, P )
ν_{(x,t)} (\frac{∂}{∂t}, K) = sin t

The S̃L(2, R)-fibers are Riemannian when cos t > 0, degenerate when cos t = 0, and Lorentzian when cos t < 0. Obviously, S̃L(2, R) ⊆ Isom^{0}(X). It is straightforward to compute the following values for the Levi-Civita connection ∇:

∇_{A}A = 0 = ∇_{P} P
∇_{K}K = \frac{1}{2} sin^{2} t · K − \frac{1}{2} cos t sin t · T
∇_{T} T = (1 − \frac{1}{2} sin^{2} t) · K + \frac{1}{2} sin t cos t · T
∇_{A}P = K = −∇_{P} A
∇_{P} K = cos t · A = ∇_{K}P − 2A
∇_{A}K = − cos t · P = ∇_{K}A + 2P
∇_{T} A = − sin t = ∇_{A}T
∇_{T} P = sin t = ∇_{P} T
∇_{T} K = −\frac{1}{2} sin t cos t · K − \frac{1}{2} sin^{2} t · T = ∇_{K}T

Let

X_{1} = A  X_{3} = cos \frac{t}{2} · K + sin \frac{t}{2} · T
X_{2} = P  X_{4} = sin \frac{t}{2} · K − cos \frac{t}{2} · T

These vector fields form a Lorentz framing of X—that is, with respect to the basis they form at (x, t), the metric takes the form

ν_{(x,t)} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} )

The sectional curvatures of X with respect to this framing have the following values at (x, t):

S(X_{1}, X_{2}) = −3 cos t − 4  S(X_{3}, X_{4}) = −\frac{1}{2} cos t
S(X_{1}, X_{3}) = cos^{2} \frac{t}{2} = S(X_{2}, X_{3})
S(X_{1}, X_{4}) = − sin^{2} \frac{t}{2} = S(X_{2}, X_{4})

Then the scalar curvature of X at (x, t) is −3 cos t − 8. The important point is that it is a nonconstant function of t. Then any isometric flow must preserve the S̃L(2, R)-fibers of X. Now because S̃L(2, R) ⊆ Isom^{0}(X) acts transitively on each fiber, if the stabilizer of each point were trivial, we could conclude that Isom^{0}(X) ≅ S̃L(2, R). Unfortunately, the stabilizers are not quite trivial: they include SO(2, R), rotating in the plane spanned by A and P. It is thus necessary to perturb the inner products B_{λ} on 𝔰𝔩(2, R) to destroy this symmetry. This can be accomplished, for example, by defining a new inner product B_{ϵ,λ} in which A, P, K are mutually orthogonal, and

B_{ϵ,λ}(P, P) = 1 − ϵ
B_{ϵ,λ}(A, A) = 1
B_{ϵ,λ}(K, K) = λ

Then define a new metric ν′_{(x,t)} on X with B_{ϵ,cos t} in place of B_{cos t}. For the

metric ν′, it is still true that S̃L(2, R) ⊆ Isom^{0}(X). Because ν′ is close to ν for ϵ sufficiently small, the scalar curvature of ν′ is still a nonconstant function of t. Thus any isometric flow on X preserves the S̃L(2, R)-fibers. Let φ^{s} be such a flow. For any fixed t_{0} ∉ (Z + 1/2)π, the flow on the fiber over t_{0} is an isometry of the pulled-back nondegenerate metric on S̃L(2, R). By post-composition with a path in S̃L(2, R), we may assume the restriction of φ^{s} fixes the identity 1. Now the differential φ_{∗1}^{s} must preserve both the inner product B_{ϵ,cos t_{0}} and the Ricci curvature form on 𝔰𝔩(2, R). For example, when cos t_{0} = ϵ − 1, for 0 < ϵ < 1/2, the inner product takes the form, with respect to the basis A, P, K,

( \begin{matrix}1 \\ & 1 − ϵ \\ & & ϵ − 1 \end{matrix} )

and the Ricci curvature is

( \frac{−2}{(1−ϵ)^{3}} ; \frac{−2+5ϵ}{(1−ϵ)^{2}} ; \frac{−2+5ϵ}{(1−ϵ)^{2}} )

For sufficiently small ϵ, the linear isometries of 𝔰𝔩(2, R) preserving both inner products can be identified with

g^{−1}O(2, 1)g ∩ h^{−1}O(3)h

where

g = ( \begin{matrix}1 \\ √ & \overline{1 − ϵ} \\ √ & \overline{1 − ϵ} \end{matrix} )  and  h = ( \sqrt{\frac{2}{(1−ϵ)^{3}}} ; \frac{\sqrt{2−5ϵ}}{1−ϵ} ; \frac{\sqrt{2−5ϵ}}{1−ϵ} )

It is left to the reader to verify that for small positive ϵ, the identity component of this intersection is trivial. We conclude that φ^{s} is trivial when restricted to the fiber over t_{0}. Then φ_{∗(x_{0},t_{0})}^{s} is trivial on the span of A, P, K at (x_{0}, t_{0}). Then it must also fix the orthogonal direction, and so φ_{∗(x_{0},t_{0})}^{s} is trivial for all s. But any isometry of X fixing a point and having trivial derivative at that point is trivial. Finally, φ^{s} is trivial, and Isom^{0}(X) ≅ S̃L(2, R).

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