Isometric actions of Heisenberg groups on compact

Lorentz manifolds

Karin Melnick^{1} to appear in Geometriae Dedicata

We prove results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. We give a general construction, leading to a new example, of codimension-one actions—those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. The main result is a classification of codimension-one actions, under the assumption they are real-analytic.

1 Introduction

Connected isometry groups of compact connected Lorentz manifolds have been classified by Adams and Stuck and independently by Zeghib.

Theorem 1.1 (Adams and Stuck [AS1], [AS2]; Zeghib [Ze1], [Ze2]) The identity component of the isometry group of a compact connected Lorentz manifold M is isomorphic to K × R^{k} × S, where

• K is compact

• S, if nontrivial, is one of three types:

1. a finite cover of P SL_{2}(R)

2. a group locally isomorphic to a Heisenberg group H_{n}

3. a quotient S_{λ}/Λ, where S_{λ}, for each λ ∈ Q^{n}, is isomorphic to S^{1} ⋉ H_{n}, and Λ is a lattice in the center of H_{n}

After this classification of isometry groups, the question arises, which manifolds admit these isometry groups. The groups P SL_{2}(R) and S_{λ} admit bi-invariant Lorentz metrics, so their quotients by cocompact lattices are compact homogeneous Lorentz manifolds. In [Ze1], Zeghib shows that these are essentially the only homogeneous compact Lorentz manifolds with noncompact isometry group.

1Department of Mathematics, University of Chicago, Chicago, IL

For an arbitrary Lorentz manifold M, if type 1 in Theorem 1.1 occurs, then Gromov’s splitting theorem ([G], 5.3.D, 5.4.A) says that the universal cover of M is isometric to a warped product L ×_{f} S, where L is a Riemannian manifold, f is a smooth function L → R^{+}, and S carries the bi-invariant Lorentz metric coming from the Killing form. For type 3, Zeghib has proved a similar splitting theorem ([Ze1] 1.7). We consider here the question of which compact manifolds have isometric actions of groups of type 2. In this case, the restriction of the metric to the orbits is degenerate, and the orbits cannot be the fibers of a warped product. Zeghib [Ze2] and Adams have constructed deformations of the S_{λ}-invariant metric on compact quotients S_{λ}/Γ for which some Heisenberg group H_{k} acts isometrically, but no non-nilpotent group does. This paper begins with a proof that Adams’ deformation on S_{λ}/Γ has connected isometry group locally isomorphic to the nilradical H_{n} of S_{λ} (Proposition 3.2). We give a general construction of spaces diffeomorphic to R × H_{n} admitting compact quotients with isometric H_{n}-action (Propositions 3.3 and 3.6). There exist such compact quotients not finitely covered by any S_{λ}/Γ. Our purpose in this paper is to address classification of isometric Heisenberg actions on compact Lorentz manifolds, with particular focus on codimensionone actions—those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. Our main results are in Section 4. We classify codimension-one actions, under the assumption they are real-analytic. Statement of Results. Recall that the (2n + 1)-dimensional Heisenberg group, denoted H_{n}, is the simply-connected Lie group with Lie algebra

𝔥_{n} =< X_{1}, …, X_{n}, Y_{1}, …, Y_{n}, Z | [X_{i}, Y_{i}] = Z, i = 1, …, n >

Given a vector space splitting 𝔥_{n} = 𝔭 ⊕ RZ_{0}, where Z_{0} is a generator of the center, there is a non-degenerate symplectic form ω_{0} on 𝔭 defined by

[X, Y] = ω_{0}(X, Y)Z_{0}

We first describe a necessary condition to have a codimension-one Heisenberg action. If H_{n} acts isometrically on a Lorentz manifold M of dimension 2n+2, then the above splitting determines a vector field W on M transverse to H_{n}-orbits (by conditions (M1) - (M3) in §3.3.1). In order for H_{n} to act isometrically, it is necessary that, for any Killing fields X and Y in 𝔭 and Z ∈ RZ_{0},

< [W, X], Y >_{x}= ω_{0}(X, Y)

and

[W, Z](x) = 0

for all x ∈ M. The symmetry of the metric implies

ω_{0}(X, [W, Y]) + ω_{0}([W, X], Y) = 0

for all X, Y ∈ 𝔭. It follows that the bracket with W is a derivation of 𝔥. It must also be infinitesimally definite (see §3.3), a condition corresponding to positivedefiniteness of the metric. The flow along W from a point x_{0} of M induces a path φ_{t} in Aut(𝔥). Conversely, given such a path φ_{t} in Aut(𝔥) for which the path

φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t}

of derivatives annihilates Z_{0} and is infinitesimally definite, the construction in Section 3.3 gives an H_{n}-invariant Lorentz metric on an H_{n}-space diffeomorphic to R×H_{n}. Provided this path satisfies a compatibility condition with a lattice in H_{n}, the metric descends to a compact quotient M with an isometric H_{n}-action. We have the following classification (see §3.3 for relevant definitions). Theorem 4.9 There is a bijective correspondence between

(a) universal covers of compact Lorentz manifolds with real-analytic codimensionone isometric H_{n}-actions, up to equivariant isometry

(b) real-analytic, lattice-compatible, metric-defining paths Φ_{t} in Aut(H_{n}), up to equivalence of metric-defining paths. The starting point for showing that all codimension-one actions are equivalent to one arising from our construction is Gromov’s centralizer theorem ([G] 5.2.A2). Given a compact real-analytic Lorentz manifold M with isometric H_{n}-action, this theorem gives another copy of 𝔥_{n} acting infinitesimally isometrically on the universal cover M̃, commuting with the lifted 𝔥_{n}-action. One of our intermediate results is the following theorem, which describes the possible diffeomorphism types of compact (2n+2)-dimensional manifolds M with real-analytic isometric H_{n}-action. Theorem 4.6 There is a diffeomorphism of the universal cover M̃ with R×H_{n} carrying H_{n}-orbits to H_{n}-factors. There is a lattice L < H_{n} and a homomorphism Ψ : Z → Aut(H) such that π_{1}(M) ≅ Z ⋉_{Ψ} L. The lattice L acts along the H_{n}-factors of M̃, and Z acts by translations along R.

Acknowledgements: I thank my advisor Benson Farb; Abdelghani Zeghib for telling me a key fact for the proof of Proposition 3.2; and Scot Adams for discussing with me his construction of Heisenberg actions. I thank the referee for several helpful comments.

2 Definitions

Let H_{n} be the (2n+1)-dimensional Heisenberg group defined in the introduction. A warped Heisenberg group is a solvable extension of H_{n} with Lie algebra

𝔰_{n} =< W, 𝔥 | [W, X_{i}] = Y_{i}, [W, Y_{i}] = −X_{i}, i = 1, …, n >

Note that Ad(e^{tW}), with respect to the basis {Z, X_{1}, Y_{1}, …, X_{n}, Y_{n}, W }, is

1

cos t − sin t
sin t cos t
...
cos t − sin t
sin t cos t

1

The group S_{n} is the group isomorphic to S^{1} ⋉ H_{n} with this Lie algebra of rightinvariant vector fields, where h ↦ e^{−it}he^{it} is the automorphism of H_{n} with derivative Ad(e^{tW}). In fact, for any λ ∈ R^{n}, there is a warped Heisenberg group with Lie algebra

𝔰_{λ} =< W, 𝔥 | [X_{i}, Y_{i}] = λ_{i}Z, [W, X_{i}] = λ_{i}Y_{i}, [W, Y_{i}] = −λ_{i}X_{i}, i = 1, …, n >

Two such Lie algebras 𝔰_{λ} and 𝔰_{μ} are isomorphic if and only if the sets {λ_{1}, …, λ_{n}} and {aμ_{1}, …, aμ_{n}} are equal for some a ∈ R ([AS1] 3.2). Let λ ∈ Q^{n}. In this case, there is a group S_{λ} isomorphic to S^{1} ⋉ H_{n} with the Lie algebra 𝔰_{λ}. Here h ↦ e^{−it}he^{it} has derivative Ad(e^{tNW}), where N is the least common multiple of the denominators of the λ_{i}. Any cocompact lattice in H_{λ} is also cocompact in S_{λ}. A cocompact lattice Γ in S_{λ} intersects H_{λ} in a uniform lattice and projects modulo H_{λ} to a finite subgroup of S^{1} (see [Rag] 3.3). Also, a cocompact lattice in H_{λ} intersects the center Z(H_{λ}) = Z(S_{λ}) in a cocompact subgroup (see [Rag] 2.3).

For any λ, the Lie algebra 𝔰_{λ} admits a bi-invariant inner product of signature (1, 2n + 1) in which the vectors X_{1}, …, X_{n}, Y_{1}, …, Y_{n} are orthonormal; the vectors Z and W are both isotropic and orthogonal to X_{i} and Y_{i} for i = 1, …, n; and < W, Z >= 1. This inner product gives rise to a bi-invariant Lorentz metric on S_{λ}. If Γ is a lattice in S_{λ}, the metric descends to the quotient S_{λ}/Γ, for which the isometry group has identity component S_{λ}/Λ, where Λ = Γ ∩ Z(S_{λ}). This last fact can be proved by the methods of Propositon 3.2 below. Note that the center Z(S_{λ}) acts here via a compact quotient isomorphic to S^{1}.

3 Construction of H-actions

Section 3.1 contains a brief account of Zeghib’s method for deforming the homogeneous metric on compact manifolds S_{n}/Γ, and Section 3.2 treats Adams’ deformation of such metrics. Proposition 3.2 says that for Adams’ deformation, the resulting connected isometry group is H_{n}/Λ, where Λ is a lattice in Z(H_{n}). In Section 3.3 a more general construction of codimension-one Heisenberg actions, leading to a new example, is given.

3.1 Perturbing the holonomy for S/Γ

In ([Ze2] 4.1.2) Zeghib constructs metrics on compact quotients of S_{n} with connected isometry groups containing R^{k} or H_{k}, k < n. Given Γ a lattice in S_{n} and a homomorphism ρ : Γ → S_{n}, the group Γ_{ρ} = {(ρ(γ), γ) : γ ∈ Γ} is a discrete subgroup of Isom(S). It is possible to choose ρ so that Γ_{ρ} still acts freely and properly on S. Then the connected component of the isometry group of S/Γ_{ρ} is the centralizer of Γ_{ρ}. One possibility for the Zariski closure Zar(ρ(Γ)) is an (n−k)-dimensional abelian subgroup of H, where k < n, having centralizer in S_{n} isomorphic to R^{n−k} × H_{k}. If the centralizer of Γ_{ρ} is non-abelian, then ρ(Z(Γ)) = 1. In this case, Z(S_{n}) ∩ Γ_{ρ} = Z(Γ), so the center of the isometry group of S_{n}/Γ_{ρ} is one-dimensional and acts via a compact quotient.

3.2 Breaking the symmetry on S/Γ

Assume for simplicity that λ = (1, …, 1), and denote H_{n} and S_{n} simply by H and S, respectively. Also assume Γ ⊆ H. Let m be the pullback of a positive bump function at 1 on S^{1} = H\S to S. This function is left-H-invariant, right-Γ-invariant, and not invariant under left translation by any s ∉ H. Let θ be the bi-invariant metric on S, and denote by mθ this metric rescaled by m. Then mθ descends to a metric on S/Γ, and H acts here by isometries. Represent elements of the semi-direct product S by ordered pairs (t, h), 0 ≤ t < 2π. Since Killing fields on S/Γ coming from the H-action do not have constant norm in mθ, it is not immediate that the orbits of one-parameter subgroups in H are geodesics. The fact that they are geodesic can be seen using results of Zeghib’s [Ze3] paper on totally geodesic lightlike foliations in Lorentz manifolds. A lightlike subspace of a Lorentz inner-product space is one for which the restriction of the inner product is degenerate. In this case, the kernel is onedimensional, and the inner product restricted to any complementary subspace is definite. A lightlike submanifold N of a Lorentz manifold M is one for which T_{x}N is a lightlike subspace of T_{x}M for all x ∈ N. In Section 1.1 of [Ze3], Zeghib shows that a codimension-one lightlike submanifold N of a compact Lorentz manifold is totally geodesic if and only if the kernel distribution on N is transversally Riemannian—the flow generated by a vector field tangent to this distribution preserves the degenerate Riemannian metric on N. The following lemma is key in classifying codimension-one actions.

Lemma 3.1 A linear transformation that preserves a Lorentz inner product and restricts to the identity on a codimension-one lightlike subspace is the identity.

Proof: Suppose V is a Lorentz inner-product space and f ∈ O(V) is the identity on U ⊂ V as above. Let Z, X_{1}, …, X_{n} be a basis for U such that Z spans the kernel of U and X_{1}, …, X_{n} are orthonormal. The orthogonal complement to span{X_{1}, …, X_{n}} is Lorentz and contains a unique isotropic vector W with < W, Z >= 1. Clearly, if f fixes U pointwise, then it must fix W. ♢

Proposition 3.2 Isom^{0}(S/Γ, mθ) = H/Λ, for Λ = Z(H) ∩ Γ.

Proof: It is clear from the construction that H acts isometrically on (S/Γ, mθ), and that Λ = Z(H)∩Γ is the kernel of this action. It suffices to show any isometry in the component of the identity lies in H/Λ. Suppose \overline{f} ∈ Isom^{0}(S/Γ, mθ). Note \overline{f}_{∗} is the identity on π_{1}(S/Γ). By post-composing with an isometry in H/Λ if necessary, one may assume \overline{f} takes the Γ-coset of (0, e) to the coset of (T, e) for some T. Let f be the lift of \overline{f} to S mapping (0, e) to (T, e). Note that f is equivariant with respect to right translation by Γ.

Each H-fiber H_{t} = {t} × H is lightlike with kernel distribution generated by the Killing field corresponding to a generator of the center of H. Thus this distribution is transversally Riemannian and every H_{t} is totally geodesic. In this case, H_{t} inherits a connection from S. Let X^{∗} ∈ 𝔥, and let X be the Killing field on S coming from left translation by e^{tX^{∗}}. The squared norm < X, X >= mθ(X, X) is constant along H_{t}, so for any vector Y tangent to H_{t},

Y < X, X >= 0 = 2 < ∇_{Y} X, X >

Pick a point p ∈ H_{t} and extend Y along the curve e^{tX^{∗}}p such that [X, Y] = 0. Since the flow along X is an isometry, Y also satisfies X < X, Y >= 0. The connection is torsion-free, so at p,

< ∇_{X}X, Y > + < X, ∇_{X}Y > = < ∇_{X}X, Y > + < X, ∇_{Y} X > + < X, [X, Y] >
= < ∇_{X}X, Y >
= 0

The tangent vector Y to H_{t} was arbitrary, so an integral curve for X is geodesic in the induced connection on H_{t}, and hence in S. Thus, all the Killing fields in S coming from left translation by elements of H integrate to geodesics. Identify the tangent space to an H-fiber T_{(t,e)}H_{t} with 𝔥 by evaluating Killing fields at (t, e). At each (t, e) there are two restricted exponential maps T_{(t,e)}H_{t} → H_{t}, one coming from the exponential map in H and one from the connection. These must agree; denote the map by exp_{t}. Since the Lie group exponential on a simply connected nilpotent Lie group is a diffeomorphism (see [Kn] 1.104,

[Rag] 1.9), exp_{t} maps T_{(t,e)}H_{t} diffeomorphically onto the fiber H_{t}. Because Γ ∩ Z(H) is a lattice in Z(H), the inverse image exp_{0}^{−1}(Γ ∩ Z(H)) spans the center 𝔷 ⊂ T_{(0,e)}H_{0}. Then any subspace of 𝔥 containing exp_{0}^{−1}(Γ) is a subalgebra. The corresponding closed connected subgroup of H must be cocompact, so it must equal H. Thus exp_{0}^{−1}(Γ) contains a basis of 𝔥, and an isometry is determined along H_{0} by its derivative on this subset. Now Γ-equivariance implies that f_{∗(0,e)} : T_{(0,e)}H_{0} → T_{(T,e)}H_{T} maps exp_{0}^{−1}(0, γ) to exp_{T}^{−1}(T, γ) for all γ ∈ Γ. The element of 𝔥 evaluating to exp_{T}^{−1}(T, γ) at (T, e) is Ad(e^{−T W})(exp_{0}^{−1}(γ). Thus f_{∗(0,e)} induces Ad(e^{−T W}) on 𝔥, and f maps H_{0} to H_{T} by (0, h) ↦ (T, h). But f is an isometry, so m(0) = m(T). Hence 0 = T, and f_{∗(0,e)} restricts to the identity on T_{(0,e)}H_{0}. By the lemma 3.1, f_{∗(0,e)} ≡ Id. An isometry of a connected pseudo-Riemannian manifold fixing a point and a frame is trivial (see [Ko] II.1.3). ♢

3.3 More general codimension-one actions

3.3.1 Construction of R ∗_{Φ} H

Fix a vector space splitting 𝔥 = 𝔭 ⊕ RZ_{0}, where Z_{0} ≠ 0 belongs to the center 𝔷. Let ω_{0} be the non-degenerate symplectic form on 𝔭 determined by Z_{0}:

[X, Y] = ω_{0}(X, Y)Z_{0}

Call an endomorphism ν of 𝔭 infinitesimally definite if ω_{0}(X, ν(X)) > 0 for all X ∈ 𝔭. Note that an infinitesimally definite endomorphism is invertible with ω_{0}(ν^{−1}(X), X) > 0 for all X. Further, ν is infinitesimally symplectic, meaning

ω_{0}(ν(X), Y) + ω_{0}(X, ν(Y)) = 0

if and only if ν^{−1} is. An endomorphism δ of 𝔥 annihilating Z_{0} is infinitesimally symplectic on 𝔥/𝔷 if and only if it is a derivation. Let M_{𝔭} be the set of derivations δ of 𝔥 preserving 𝔭 such that δ(Z_{0}) = 0 and ν = δ|_{𝔭} is infinitesimally definite. Note that M_{𝔭} is not a subalgebra of Der(𝔥) and does not depend on the choice of Z_{0}. Call a path φ : R → Aut(𝔥) metric-defining if for some 𝔭

φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t} ∈ M_{𝔭} for all t

Call a path Φ : R → Aut(H) metric-defining if the path φ_{t} = D_{e}(Φ_{t}) in Aut(𝔥) is metric-defining. Given a path Φ_{t} in Aut(H), define an H-space R ∗_{Φ} H diffeomorphic to R × H by

h.(t, g) = (t, Φ_{t}(h)g)

This H-action is free, so there exist at each x linear injections

f_{x} : 𝔥 → T_{x}(R ∗_{Φ} H)
f_{x}(Y) = \frac{∂}{∂t}|_{0} (e^{tY} .x)

Suppose that Φ_{t} is metric-defining, and let φ_{t} = D_{e}(Φ_{t}). There is 𝔭 such that

φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t} ∈ M_{𝔭} for all t

Note that 𝔭 is determined by φ_{t} because it is exactly the image of the derivation above. Denote by ν_{t} the restriction of this derivation to 𝔭. Let W be the vector field \frac{∂}{∂t} on R ∗_{Φ} H. Define a Lorentz metric <, >_{x} on R ∗_{Φ} H in which W satisfies at each x:

< W (x), W (x) >_{x}= 0 (M1)
< W (x), f_{x}(Z_{0}) >_{x}= 1 (M2)
W (x) ⊥ f_{x}(𝔭)  (M3)

Further, make f_{x}(Z_{0}) isotropic and f_{x}(Z_{0}) ⊥ f_{x}(𝔭) everywhere. Last, for X, Y ∈ 𝔭, let

< f_{x}(X), f_{x}(Y) >_{x}= ω_{0}(ν_{t(x)}^{−1}(X), Y)

where t(x) is the R-coordinate of x. Since ν_{t(x)} is the restriction of a derivation of 𝔥, both ν_{t(x)} and ν_{t(x)}^{−1} are infinitesimally symplectic. It follows that the form <, >_{x} is symmetric. Because ν_{t(x)} is infinitesimally definite, the form is positive definite on 𝔭. Note that this metric on R ∗_{Φ} H is associated to the path Φ_{t} and the generator Z_{0}. In this section, the notation (R ∗_{Φ} H, Z_{0}) will stand for the Lorentz manifold thus determined.

Proposition 3.3 The metric on R ∗_{Φ} H is H-invariant.

Proof: The H-action is isometric if and only if, for each K ∈ 𝔥, x ∈ R ∗_{Φ} H, and vector fields X, Y near x,

0 = \frac{∂}{∂t}|_{0} < e_{∗}^{tK}X, e_{∗}^{tK}Y >_{e^{tK}x}
= < [f_{x}(K), X], Y >_{x} + < X, [f_{x}(K), Y] >_{x} +f_{x}(K) < X, Y >

Since the above sum depends on X and Y only for their values at x, it suffices to check vanishing for X(x) and Y (x) ranging over a basis of the tangent space at x. For X ∈ 𝔥, denote the vector field given by f_{x}(X) simply by X^{∗}. Since X corresponds to a right-invariant vector field on H, and H acts on the left, X ↦ X^{∗} is a Lie algebra homomorphism. Take Y_{1}, …, Y_{2n} a basis for 𝔭. If X and Y each coincide near x with elements of {Z_{0}^{∗}, Y_{1}^{∗}, …, Y_{2n}^{∗}, W }, then K^{∗}(x) < X, Y >= 0 for any K ∈ 𝔥, because any such inner product is constant along H-orbits. Now the equation is

< [K^{∗}, X], Y >_{x} + < X, [K^{∗}, Y] >_{x}= 0

for all K ∈ 𝔥 and X, Y ∈ {Z_{0}^{∗}, Y_{1}^{∗}, …, Y_{2n}^{∗}, W }. If X and Y both come from 𝔥, then both terms above vanish because Z_{0}^{∗}(x) ⊥ f_{x}(𝔥).

It remains to check vanishing when X = W. Let the coordinates of x be t(x) = t and p(x) = p. The flow along K^{∗} is

e^{rK}.x = (t, Φ_{t}(e^{rK}) · p)
= (t, e^{rφ_{t}(K)} · p)

The derivative on W (x) is

e_{∗}^{rK}(W (x)) = (\frac{∂}{∂t}, r (\frac{∂φ_{s}}{∂s} )_{t} (K) · (e^{rφ_{t}(K)} · p))

where X · h, for X ∈ 𝔥 and h ∈ H, denotes the image of X under the derivative of right translation by h. Finally, the bracket

[K^{∗}, W](x) = \frac{∂}{∂r} |_{0} e_{∗}^{−rK}(W (e^{rK}.x))
= lim_{r→0} \frac{1}{r}(e_{∗}^{−rK}(W (e^{rK}.x)) − W (x))
= (0, lim_{r→0} \frac{1}{r}(−r (\frac{∂φ_{s}}{∂s} )_{t} (K) · p)
= (0, − (\frac{∂φ_{s}}{∂s} )_{t} (K) · p)
= −(φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t} (K))^{∗}(x)

If K = Z_{0}, then

[Z_{0}^{∗}, W] = −(φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t})(Z_{0})^{∗} = 0

If K ∈ 𝔭, then we have

< [K^{∗}, W], Y >_{x} + < W, [K^{∗}, Y] >_{x}=< −ν_{t}(K)^{∗}, Y >_{x} + < W, [K^{∗}, Y] >_{x}

When Y = Z_{0}^{∗}, then both terms vanish because ν_{t}(K)^{∗} ∈ f_{x}(𝔭), which is orthogonal to Z_{0}^{∗}(x), and [K^{∗}, Z_{0}^{∗}] = 0. When Y = Y^{∗} for Y ∈ 𝔭, then we have

< −ν_{t}(K)^{∗}, Y^{∗} >_{x} + < W, [K^{∗}, Y^{∗}] >_{x} = < −ν_{t}(K)^{∗}, Y^{∗} >_{x} + < W, ω_{0}(K, Y)Z_{0}^{∗} >_{x}
= − < ν_{t}(K)^{∗}, Y ∗ >_{x} +ω_{0}(K, Y)
= 0

since < X^{∗}, Y^{∗} >_{x}= ω_{0}(ν_{t}^{−1}X, Y) for all X ∈ 𝔭. ♢ 3.3.2 Equivalence

Next we consider when the spaces associated to pairs (Φ_{t}, Z_{0}) and (Φ′_{t}, Z′_{0}) are H-equivariantly isometric. Suppose given a generator Z_{0} ∈ 𝔷 and a path Φ_{t}, metric-defining with respect to a complementary subspace 𝔭. Let φ_{t} = D_{e}(Φ_{t}) and ν_{t} be as usual. Suppose 𝔭′ is a different complement to 𝔷. There is a canonical isomorphism a : 𝔭 → 𝔭′ inducing the identity on 𝔥/𝔷. This isomorphism has the form

a(X) = X + α(X)Z_{0}

for some functional α on 𝔭. Let A ∈ 𝔭 be such that ω_{0}(A, X) = α(X) for all X ∈ 𝔭. Define

ν′_{t} = a ∘ ν_{t} ∘ a^{−1}

and let φ′_{t} and Φ′_{t} be corresponding metric-defining paths. Let γ be a path in 𝔥 satisfying

γ̇(t) = (\frac{∂φ′_{s}}{∂s} )_{t} (A) − \frac{1}{2}α(ν′_{t}(A))Z_{0}

Then

θ : (t, h) ↦ (t, Φ′_{t} ∘ Φ_{t}^{−1}(h) · e^{γ(t)})

is an H-equivariant isometry between (R ∗_{Φ} H, Z_{0}) and (R ∗_{Φ′} H, Z_{0}). The H-equivariance is clear. By equivariance, it suffices to check isometry at (t, e) for all t; also, Killing fields are carried to Killing fields. Denote by <, > and <, >′ the metrics on the domain and target, respectively. The restriction of θ_{∗x} to f_{x}(𝔥) is isometric: for any X, Y ∈ 𝔭,

< X^{∗}, Y^{∗} >_{x} = ω_{0}(ν_{t(x)}^{−1}X, Y)
= ω_{0}((a ∘ ν_{t(x)}^{−1} ∘ a^{−1})(a(X)), a(Y))
= ω_{0}((ν′_{t(x)})^{−1}(a(X)), a(Y))
= < a(X)^{∗}, a(Y)^{∗} >′_{θ(x)}
= < X^{∗}, Y^{∗} >′_{θ(x)}

Denote by W and W′ the vector fields \frac{∂}{∂t} on the domain and target, respectively.

θ_{∗(t,e)}(W) = (\frac{∂}{∂t}, γ̇(t) · e^{γ(t)})
= W′ + (φ′_{t})^{−1}(γ̇(t))^{∗}
= W′ + ν′_{t}(A)^{∗} − \frac{1}{2}α(ν′_{t}(A))Z_{0}^{∗}

using that φ′_{t} fixes Z_{0}. Now it is easy to check θ_{∗(t,e)}(W) is isotropic, orthogonal to f_{θ(0,e)}(𝔭), and has inner product 1 with Z_{0}^{∗}. Finally, note that for arbitrary d ∈ R, one may set ν′_{t+d} = aν_{t}a^{−1}, and then

(t, h) ↦ (t + d, Φ′_{t+d} ∘ Φ_{t}^{−1}(h) · e^{γ(t)})

is an H-equivariant isometry of (R ∗_{Φ} H, Z_{0}) with (R ∗_{Φ′} H, Z_{0}), where Φ′_{t} is a metric-defining path arising from ν′_{t}, and γ(t) is defined in an obvious way. Next suppose that a metric-defining path Φ_{t} and a generator Z_{0} are given, and define φ_{t} and ν_{t} as usual. Given c ≠ 0 and d ∈ R, let

cν′_{ct+d} = ν_{t}

Let φ′_{t} and Φ′_{t} be metric-defining paths corresponding to ν′_{t}. Note that Φ′_{ct+d} = Φ_{t}. Then

(t, h) → (ct + d, Φ′_{ct+d} ∘ Φ_{t}^{−1}(h)) = (ct + d, h)

is an H-equivariant isometry between (R∗_{Φ} H, Z_{0}) and (R∗_{Φ′} H, cZ_{0}). We leave the verification to the reader. These are the only ways to obtain H-equivariantly isometric spaces.

Proposition 3.4 Suppose there is an H-equivariant isometry

θ : (R ∗_{Φ} H, Z_{0}) → (R ∗_{Φ′} H, Z′_{0})

Let ν_{t} and ν′_{t} be the restricted derivations of 𝔥 associated to Φ_{t} and Φ′_{t}, respectively. Let 𝔭 and 𝔭′ be their respective images, and a the canonical isomorphism 𝔭 → 𝔭′. Then there exist c ≠ 0 and d ∈ R such that Z′_{0} = cZ_{0}, and cν′_{ct+d} = a ∘ ν_{t} ∘ a^{−1}.

Proof: Since θ preserves H-orbits, it has the form

θ : (t, h) ↦ (H(t), K(t, h))

By H-equivariance, θ_{∗} carries each Killing field X^{∗} on the domain to X^{∗} on the target. Let W and W′ be the vector fields \frac{∂}{∂t} on the domain and target, respectively. Let t(x) = t. Since < W, Z_{0}^{∗} >= 1 and Z_{0}^{∗} ⊥ f_{y}(𝔥) for all y,

< θ_{∗}(W), Z_{0}^{∗} >_{θ(x)} = < Ḣ(t)W′, Z_{0}^{∗} >_{θ(x)} = 1

There must exist c ≠ 0 such that Z′_{0} = cZ_{0} and Ḣ(t) = c for all t. Then H(t) = ct + d for some d. Denote by ω_{0} and ω′_{0} the symplectic forms defined by Z_{0} and Z′_{0}, respectively. For X, Y ∈ 𝔭,

< [W, X^{∗}], Y^{∗} >_{x}= ω_{0}(ν_{t}^{−1}X, Y)

which must equal

< [θ_{∗}(W), X^{∗}], Y^{∗} >_{θ(x)} = < [θ_{∗}(W), a(X)^{∗}], a(Y)^{∗} >_{θ(x)}
= ω′_{0}((ν′_{ct+d})^{−1}(a(X)), a(Y))
= \frac{1}{c}ω_{0}((a^{−1} ∘ (ν′_{ct+d})^{−1} ∘ a)(X), Y)

Since ω_{0} is non-degenerate on 𝔭, we have

\frac{1}{c}a^{−1} ∘ (ν′_{ct+d})^{−1} ∘ a = ν_{t}^{−1},

so

cν′_{ct+d} = a ∘ ν_{t} ∘ a^{−1}

♢ We will say that two metric-defining paths Φ_{t} and Φ′_{t} are equivalent if there are c ≠ 0 and d ∈ R such that cν′_{ct+d} = a ∘ ν_{t} ∘ a^{−1}, where a is the canonical isomorphism as above.

3.3.3 Compact quotients

Suppose that Φ_{t} is Z-equivariant: Φ_{q+t} = Φ_{q}Φ_{t} for all t ∈ R, q ∈ Z; in particular, Φ restricts to a homomorphism on Z. Say that Φ is lattice-compatible if it is Z-equivariant and there is some lattice L of H preserved by Φ(q) for all q ∈ Z. Then the semi-direct product Γ = Z ⋉_{Φ} L acts on R ∗_{Φ} H by

(t, g).(q, λ) = (t + q, Φ_{q}(g) · λ)

The action is written on the right to emphasize that it commutes with the Haction. In particular, it preserves the Killing fields X^{∗} for each X ∈ 𝔥. Thus H acts on the compact quotient M = (R ∗_{Φ} H)/Γ.

Lemma 3.5 If the paths Φ_{t} and φ_{t} are Z-equivariant, then ν_{t} is Z-invariant.

Proof: Let q ∈ Z. The endomorphism ν_{q+t} is the restriction to 𝔭 of

φ_{q+t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{q+t} = φ_{q+t}^{−1} ∘ (\frac{∂φ_{q+s}}{∂s} )_{t}
= (φ_{q} ∘ φ_{t})^{−1} ∘ (\frac{∂(φ_{q} ∘ φ_{s})}{∂s})_{t}
= φ_{t}^{−1} ∘ φ_{q}^{−1} ∘ φ_{q} ∘ (\frac{∂φ_{s}}{∂s} )_{t}
= φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t}

which restricts to ν_{t} on 𝔭. ♢

Proposition 3.6 The Γ-action is isometric, so the metric descends to M.

Proof: The vector fields W and Z_{0}^{∗} are Γ-invariant, so Γ is isometric restricted to their span at any x. Also, f_{x}(𝔭) is a Γ-invariant distribution, so it suffices to check that for each γ ∈ Γ and X, Y ∈ 𝔭,

< γ_{∗}(X^{∗}(x)), γ_{∗}(Y^{∗}(x)) >_{γx}=< X^{∗}, Y^{∗} >_{x}

at all x ∈ M. Denote by t(x) the R-coordinate of x. For any γ ∈ Γ,

< γ_{∗}(X^{∗}(x)), γ_{∗}(Y^{∗}(x)) >_{γx} = < X^{∗}(γx), Y^{∗}(γx) >_{γx}
= ω_{0}(ν_{t(γx)}^{−1}(X), Y)
= ω_{0}(ν_{t(x)}^{−1}(X), Y)
= < X^{∗}, Y^{∗} >_{x}

using Z-equivariance of Φ_{t} and Lemma 3.5. ♢

3.3.4 Examples

Example. Adams’ deformation. Let W, X_{1}, …, X_{n}, Y_{1}, …, Y_{n}, Z_{0} be the basis for 𝔰 and θ be the bi-invariant Lorentz metric on S̃ from §2. Let 𝔭 be the span of X_{1}, …, Y_{n}. Let m be the positive function on S^{1} from §3.2. Denote again by m the lift of this function to S̃ ≅ R ⋉ H. Let

H(t) = ∫_{0}^{t} m(s)ds

Note that H is invertible. Let φ_{H(t)} be the automorphism of 𝔥 fixing Z_{0} and acting on each subspace span{X_{i}, Y_{i}} by

[ \begin{matrix}cos t & − sin t \\ sin t & cos t \end{matrix} ]

Note that φ_{H(t)} = Ad(e^{tW}). Now (\frac{∂φ_{s}}{∂s} )_{H(t)} annihilates Z_{0} and acts on each span{X_{i}, Y_{i}} by

m(t)^{−1} [ \begin{matrix}− sin t & − cos t \\ cos t & − sin t \end{matrix} ]

Let ν_{H(t)} be the restriction to 𝔭 of φ_{H(t)}^{−1}∘(\frac{∂φ_{s}}{∂s} )_{H(t)}; it acts on each span\{X_{i}, Y_{i}\} by

[ \begin{matrix}& −m(t)^{−1} \\ m(t)^{−1} \end{matrix} ]

These are infinitesimally definite derivations for each t, so φ is a metric-defining path. Let Φ_{H(t)} be the automorphism of H with derivative φ_{H(t)}. Let ρ be the H-invariant Lorentz metric R ∗_{Φ} H associated with Z_{0}. The map

F : (S̃, mθ) → (R ∗_{Φ} H, ρ)
F : (e^{tW} , h) ↦ (H(t), Φ_{H(t)}(e^{tW} he^{−tW})) = (H(t), h)

is an H-equivariant isometry; we leave the verifications to the reader. Let α = H(2π). Periodicity of m implies H(2nπ + t) = nα + H(t). The automorphisms Φ_{nα} are trivial for each n ∈ Z. To see Φ is (αZ)-equivariant

Φ_{nα+H(t)} = Φ_{H(2nπ+t)}
= Φ_{H(t)}
= Φ_{nα} ∘ Φ_{H(t)}

For any lattice L in H, the group (αZ) ⋉_{Φ} L acts isometrically on R ∗_{Φ} H, commuting with the H-action. Example. Infinite monodromy. Let W, X_{1}, …, Y_{n}, Z_{0} be as in §2 with [X_{i}, Y_{i}] = Z_{0}, and 𝔭 = span{X_{1}, …, Y_{n}}. Let a be the automorphism of 𝔥 fixing Z_{0} and acting on each span{X_{i}, Y_{i}} by

[ \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ]

Let b fix Z_{0} and act on each span{X_{i}, Y_{i}} by

[ 1  1_{\frac{−1+\sqrt{5}}{2} \frac{−1−\sqrt{5}}{2}} ]

The conjugate b^{−1}ab is diagonal. Let V fix Z_{0} and act on each span{X_{i}, Y_{i}} by

b · [ \begin{matrix}ln(\frac{3+\sqrt{5}}{2}) & 0 \\ 0 & ln(\frac{3−\sqrt{5}}{2}) \end{matrix} ] · b^{−1}

so e^{V} = a. Define a path of automorphisms φ_{t} = e^{tV}, so ν_{t} = V |_{𝔭} for all t. Obviously ν_{t} is infinitesimally symplectic; one may compute it is infinitesimally definite, so φ_{t} is metric-defining. Let Φ_{t} be the path of automorphisms of H with derivative φ_{t}. Since Φ is actually a homomorphism from R to Aut(H), it is in particular Z-equivariant. Let A = Φ_{1}, the automorphism with derivative a. Let Λ be the Z-span of X_{1}, …, Y_{n}, \frac{1}{2}Z_{0} in 𝔥. The Campbell-Hausdorff formula (see [H] II.1.8) for an order-two nilpotent group reads

e^{X} · e^{Y} = e^{X+Y + \frac{1}{2} [X,Y]}

The exponential of Λ is a lattice subgroup L in H. Now a^{n}(Λ) ⊂ Λ implies A^{n}(L) ⊆ L for all n ∈ Z. The group Z ⋉_{A} L acts isometrically and cocompactly on R ∗_{Φ} H endowed with the Lorentz metric constructed above. The quotient manifold M admits an isometric H-action. The quotient of M by this action fibers over S^{1}, and the monodromy of this fiber bundle is the infinite group < a >. Any quotient of a warped Heisenberg group S by a lattice Γ has finite monodromy, so M has no finite cover diffeomorphic to such a quotient. Results of §4.3 say that all codimension-one H-actions arise from the construction in this section, assuming the actions are real-analytic.

4 Toward classification of H-actions

We consider real-analytic actions of arbitrary codimension in Section 4.1; we prove a refinement of a result of Gromov to show that if the action of the center factors through a compact quotient, then almost every orbit map is proper for the lifted H-action on the universal cover. In the second section, techniques of Gromov and Zimmer are combined with Lemma 3.1 to show in the codimensionone case that the action on the universal cover is proper and free. Theorem 4.6 determines the diffeomorphism type of compact Lorentz manifolds admitting codimension-one, real-analytic, isometric H_{n}-actions. Finally, we show that all such actions arise from the construction in Section 3.3; a complete classification follows. The following two important properties of isometric actions of nilpotent groups are proved by Adams and Stuck in [AS1]. They will be used extensively below.

Theorem 4.1 (Adams and Stuck [AS1] 6.8) If a Heisenberg group H acts locally faithfully on a compact Lorentz manifold by isometries, then the H-action is locally free.

Theorem 4.2 (Adams and Stuck [AS1] 7.4) For H as above, the pullback to 𝔥 of the metric from any tangent space is Ad H-invariant and lightlike with kernel 𝔷(𝔥).

Recall that the action of a group G on a manifold M is proper if for all compact subsets A ⊆ M, the set G_{A} := {g ∈ G | gA ∩ A ≠ ∅} is compact. A group acting properly on a compact manifold is compact. Orbits for a proper action are closed; moreover, orbits of any closed subgroup of a group acting properly are closed. A map f : X → Y is proper if f^{−1}(K) is compact for all compact subsets K of Y.

4.1 Higher codimension: if Z(H) factors through S^{1}

Suppose the action of the center Z(H) on M factors through a proper action. In [Ze2] 4.3, Zeghib asks whether this factoring occurs for all isometric actions of Heisenberg groups on compact Lorentz manifolds. Results of the next section imply that it does in the codimension-one case. Let Γ = π_{1}(M). Let \overline{Λ} be the kernel of the H-action on M. This discrete normal subgroup must be central and, by assumption, cocompact in Z(H). The group of lifts of the H-action to M̃ is an extension of H/\overline{Λ} by Γ with identity component a covering group of H/\overline{Λ}. In other words, there is some subgroup Λ ⊆ \overline{Λ} such that the identity component of the group of lifts is H/Λ, and (H/Λ) ∩ Γ ≅ \overline{Λ}/Λ. At this point, it is necessary to assume that the Lorentz metric on M is realanalytic, in order to apply the following theorem of Gromov. For the case in which H is simple, Zimmer gives a more detailed proof in [Zi] Corollary 4.3.

Theorem 4.3 ([G] 5.2.A2) Let M be a compact real-analytic Lorentz manifold with an analytic locally free action of a connected Lie group H by isometries.

Assume that the Zariski closure of Ad H in Aut 𝔥 has no proper normal, algebraic, cocompact subgroup. Let 𝔠 be the Lie algebra of global Killing fields on the universal cover M̃ commuting with 𝔥. For almost every x ∈ M̃ and every Y ∈ 𝔥, there is C ∈ 𝔠 for which C(x) = Y (x).

This theorem in fact applies whenever H acts locally freely and real-analytically preserving a finite measure and a real-analytic rigid geometric structure of algebraic type on M. A real-analytic isometric Heisenberg action satisfies the hypotheses of the theorem. Next we show that almost every orbit on M̃ is proper. This result is an extension of a theorem of Gromov ([G] 6.1.A) to the case of Z(H) non-compact, and is seemingly alluded to in [G] 6.1.C(b). Gromov’s result appears with proof for the case of H simple in Zimmer’s paper [Zi].

Theorem 4.4 Let M be a compact real-analytic Lorentz manifold with an analytic locally faithful action of the Heisenberg group H such that the kernel of the action contains a cocompact subgroup of Z(H). Let H/Λ be the identity component of the group of lifts to M̃. Then for almost every x ∈ M̃, the orbit map H/Λ → (H/Λ).x ⊆ M̃ is proper.

Proof: By Theorem 4.1, the H-action on M is locally free. Let T^{O}(M) and T^{O}(M̃) be the subbundles of T (M) and T (M̃), respectively, tangent to Horbits. Let F^{O}(M) and F^{O}(M̃) be the corresponding bundles of frames. These bundles are smoothly trivializable using the maps

f_{x}(X) = \frac{∂}{∂t}|_{0} e^{tX}.x

for X ∈ 𝔥. These maps satisfy the equivariance relation

f_{gx} ∘ Adg^{−1} = g_{∗x} ∘ f_{x}

Therefore, the H-action in either trivialization is h(x, Y) = (hx, Adh^{−1}(Y)). Since Ad : H/\overline{Λ} → Ad(H) is proper, H/\overline{Λ} is proper on F^{O}(M). Let 𝔠 be as in Theorem 4.3 above. The group of isometries of M̃ leaving 𝔠 invariant acts on M̃ × 𝔠 by g.(x, C) = (g.x, g_{∗}C), such that the evaluation map M̃×𝔠 → T (M̃) is equivariant. In particular, H/Λ and Γ have commuting actions on M̃×𝔠, and these actions agree on the intersection \overline{Λ}/Λ. Let q : M̃×𝔠 → T (M) be the evaluation map composed with the quotient T (M̃) → T (M). Note that q is equivariant with respect to H/Λ → H/\overline{Λ}.

Suppose x satisfies the conclusion of Theorem 4.3, and suppose that h.x ∈ K for some h in H/Λ and some compact set K in M̃. The goal is to find a compact subset A ⊂ H/Λ containing all such h. For this purpose, we may assume (H/Λ).x ∩ K is dense in K. For the chosen x, there are C_{1}, …, C_{m} ∈ 𝔠 such that C_{1}(x), …, C_{m}(x) are a framing of T_{x}^{O}(M̃). Since each C_{i} is (H/Λ)-invariant, C_{1}(y), …, C_{m}(y) are a framing of T_{y}^{O}(M̃) for all y in the orbit of x. Now suppose y = lim(h_{n}.x) for a sequence h_{n} ∈ H/Λ. Each C_{i} is smooth, and T^{O}(M̃) is a closed subbundle of T M̃, so C_{i}(y) is also in T^{O}(M̃) for i = 1, …, m. Suppose C_{i}(y) = 0 for some i. Now C_{i}(y) = lim C_{i}(h_{n}.x) = lim h_{n∗}C_{i}(x). Let C ∈ 𝔥 be the image of C_{i}(x) in the trivialization of T^{O}(M̃); then lim Ad(h_{n}^{−1})C = 0. Since Ad(H) ⊂ SL(𝔥), there is some C′ ∈ 𝔥 with Ad(h_{n}^{−1})C′ → ∞. But this C′ also corresponds to some vector field in 𝔠, which is bounded on K, a contradiction. Therefore, each C_{i} is nonzero at all points of K, and C_{1}(y), …, C_{m}(y) are a framing of T_{y}^{O}(M̃) for all y ∈ K. For each y ∈ K, the images q(y, C_{1}), …, q(y, C_{m}) form a frame on T_{y}^{O}(M). The images K × {C_{1}}, …, K × {C_{m}} form a compact subset B of F^{O}(M). Since h(x, C_{i}) ∈ K × \{C_{i}\} for all i, the image of h in H/\overline{Λ} carries the frame (q(x, C_{1}), …, q(x, C_{m})) into B. Since H/\overline{Λ} is proper on F^{O}(M), the images of all such h are contained in a compact subset \overline{A} of H/\overline{Λ}. Let A be a compact subset of H/Λ mapping onto \overline{A}, and write h = aλ with λ ∈ \overline{Λ}/Λ and a ∈ A. Now h.x ∈ K implies λ.x ∈ A^{−1}K. Since \overline{Λ}/Λ is proper on M̃, the set of all possible λ is finite. Thus the subset of H/Λ carrying x into a fixed compact set is compact, and the orbit map for x is proper. ♢ Remark: The conclusion of Theorem 4.4 actually holds for any unimodular H to which 4.3 applies, provided the Z(H)-action factors through a torus.

4.2 Codimension-one: complete classification

Throughout this section, M is a compact real-analytic Lorentz manifold with a real-analytic locally faithful action of the Heisenberg group H = H_{n} such that dimM = 2n + 2. In this case, Gromov’s centralizer theorem plus the basic fact of Lemma 3.1 can be used to show properness of H/Λ on M̃. Two main results are that the quotient of M by the H-action is a fiber bundle over S^{1}, and M̃ is π_{1}(M)-equivariantly diffeomorphic to R × H. We conclude by showing that all of these actions arise from the construction given in Section 3.3.

Theorem 4.5 Let H/Λ be the identity component of the H-action lifted to M̃. Then H/Λ preserves a framing of T (M̃); in particular, the H/Λ-action is free and proper.

Proof: Let 𝔠 be the centralizer of 𝔥 on M̃, as in the previous section, and let x satisfy the conclusion of Theorem 4.3. Any C in the kernel of the evaluation map 𝔠 → T_{x}(M̃) integrates to a (local) isometry fixing x with derivative fixing f_{x}(𝔥) pointwise. Such an isometry fixes a point and a frame by Lemma 3.1, and so is trivial (see [Ko] II.1.3). Thus the evaluation map on 𝔠 is injective, and by Theorem 4.3, the dimension of 𝔠 is either 2n + 2 or 2n + 1. In the first case, 𝔠(x) = T_{x}(M̃) for all x ∈ M̃, and the evaluation map M̃ × 𝔠 → T M̃ defines an H/Λ-equivariant map M̃ × F(𝔠) → F(M̃), where M̃ × F(𝔠) is the bundle of frames on M̃ × 𝔠. The trivial H/Λ-map M̃ → M̃ × {ω}, for any ω ∈ F(𝔠), gives an H/Λ-equivariant section M̃ → F(M̃). In the second case, the set of x such that f_{x}(𝔥) = 𝔠(x) is closed and has full measure, so it must be all of M̃. Fix a generator Z_{0} for 𝔷(𝔥), and note that Z_{0} ∈ 𝔠. Let M̃×F(𝔠)_{0} ⊂ M̃×F(𝔠) be the subbundle of frames with first element the Killing field for Z_{0}. In any such frame, the remaining vectors evaluated at any x ∈ M̃ span a spacelike subspace 𝔭 by Theorem 4.2. Take W to be the unique transverse lightlike vector field that everywhere has inner product one with Z_{0}^{∗} and is orthogonal to 𝔭. An element of F(𝔠)_{0} thus determines a frame on T_{x}(M̃), and there is a well-defined map M̃×F(𝔠)_{0} → F(M̃) that is smooth and H/Λ-equivariant. As above, composition with M̃ → M̃×{ω} for any ω ∈ F(𝔠)_{0} yields an H/Λ-equivariant section. ♢

Theorem 4.6 There is a diffeomorphism of the universal cover M̃ with R×H_{n} carrying H_{n}-orbits to H_{n}-factors. There is a lattice L < H_{n} and a homomorphism Ψ : Z → Aut(H) such that π_{1}(M) ≅ Z ⋉_{Ψ} L. The lattice L acts along the H_{n}-factors of M̃, and Z acts by translations along R.

Corollary 4.7 The quotient H_{n}\M is a fiber bundle over S^{1}, giving rise to a real-analytic map S^{1} → P D(𝔥/𝔷), the space of positive-definite inner products on 𝔥/𝔷.

Proof: of corollary If H\M̃ ≅ R, then H\M ≅ H\M̃/Γ ≅ R/Z ≅ S^{1}. By Theorem 4.2, the pullback of the metric from each H-orbit to 𝔥 is lightlike and descends to a positive-definite inner product on 𝔥/𝔷. ♢ Proof: of theorem We will denote by 𝔠 the Lie algebra of the centralizer of 𝔥 on M̃ and by Γ the fundamental group π_{1}(M). step 1: H acts faithfully. Let Λ be the kernel of the action on M̃, as in the previous theorem. Since H/Λ acts properly and freely by Theorem 4.1, M̃ is a principal bundle over a connected 1-manifold. The fibers are connected, so the quotient is simply connected, hence, diffeomorphic to R. From the long exact sequence of homotopy groups for this fiber bundle, one computes that the fibers must be simply connected. Thus Λ is trivial, and H acts faithfully. step 2: C acts properly and freely. Pick Z_{0} ∈ 𝔷(𝔥) and the rest of a basis Z_{0}, Y_{1}, …, Y_{2n} for 𝔥. Let 𝔭 = span{Y_{1}, …, Y_{2n}}. Let W be the transverse vector field defined with respect to Z_{0} and 𝔭 by (M1)

- (M3). The Killing fields Z_{0}^{∗}, Y_{1}^{∗}, …, Y_{2n}^{∗} together with W form a framing on M̃ that descends to M. This framing of M̃ is also invariant by the flow along any Killing field in 𝔠. It follows that each Killing field in 𝔠 is complete. Let C be the centralizer in Isom(M̃) of H. The Lie algebra of C is 𝔠, and Γ lies in C. Because it preserves a framing on M̃, the group C acts properly and freely. step 3: if dim𝔠 = 2n + 2, then 𝔠 ≅ 𝔰. Suppose dim 𝔠 = 2n + 2, so C^{0}-orbits are open. They are closed because C acts properly; therefore, C acts transitively. Since the action is free, C = C^{0}, and Γ is a lattice in C. Pick any x_{0}, and let G be the set of elements g of C such that gx_{0} = hx_{0} for some h ∈ H. Since H acts freely on M̃, there is a well-defined map θ_{0} : G → H satisfying θ_{0}(g)g.x_{0} = x_{0}. This map is injective because C acts freely, and it respects multiplication because H and C commute. Therefore θ_{0} is an isomorphism from G to H. The orbit map C → C.x_{0} ⊂ M̃ intertwines the G × G-action on C by left and right translation, respectively, with the G × G-action on M̃ by the given G-action and θ_{0} ∘ ι, respectively, where ι is inversion in G. The orbit map also yields an identification of T_{x_{0}} M̃ with 𝔠. Denote by <, > the pullback of the Lorentz metric at x_{0} to 𝔠. The adjoint represention of G on 𝔠 is equivalent to the G-action by θ(g)_{∗}g_{∗} on T_{x_{0}} M̃. Thus <, > is infinitesimally 𝔤-invariant—that is, for any Y ∈ 𝔤 and X, W ∈ 𝔠,

< [Y, X], W > + < X, [Y, W] >= 0

Let W be a vector complementary to 𝔤 in 𝔠. For any Y, Y′ in 𝔤,

< [W, Y], Y′ > + < Y, [W, Y′] > = − < [Y, W], Y′ > − < Y, [Y′, W] >
= < W, [Y, Y′] > + < [Y′, Y], W >
= 0

For any Y ∈ 𝔤,

< [W, Y], W > = − < [Y, W], W >
= < W, [Y, W] >
= 0

Since 𝔤 and W together span 𝔠, it follows that <, > is infinitesimally W -invariant. Then <, > is AdC-invariant, and there is an isometric C × C-action on M̃ extending the C-action. This action descends to a transitive isometric C-action on M. By the classification of compact homogeneous Lorentz manifolds [Ze1], M̃ is equivariantly isometric to S̃_{λ} for some λ ∈ Q^{n}, and Γ ≅ Z ⋉_{Ψ} L for L a lattice in H_{n}. The homomorphism Ψ(q) = Ad(e^{qW}) and has finite cyclic image. Assume now that dim𝔠 = 2n + 1. step 4 : lattice in H. Let C′ be the closed subgroup generated by C^{0} and Γ. Properness of C on M̃ from step 2 implies that C^{0}-orbits have closed image in M̃/Γ = M. These are diffeomorphic to C^{0}/(C^{0} ∩ Γ), so L = C^{0} ∩ Γ is a lattice in C^{0}. The set of x on which f_{x}(𝔥) and 𝔠(x) coincide is closed and full measure, so it is all of M̃. Then C^{0}-orbits coincide with H-orbits. As in step 3, choosing x_{0} determines an isomorphism θ_{0} : C^{0} → H. Then θ_{0}(L) is a lattice in H. step 5: tranvserse foliation. Let Z_{0}, Y_{1}, …, Y_{2n} be the basis for 𝔥 from step 2; let 𝔭 be the span of Y_{1}, …, Y_{2n}. Let W be the vector field on M̃ satisfying (M1) - (M3) with respect to 𝔭 and Z_{0}; it is a C-invariant vector field transverse to H-orbits. Since W descends to M, it is complete. Let x_{0} be as in step 4, and let W be an integral curve for W with W(0) = x_{0}. Let m be a right-invariant Riemannian metric on C^{0}. Define a Riemannian metric on each C^{0}-orbit by identifying Killing fields with right-invariant vector fields on C^{0}. Now define a Riemannian metric on M̃ by making W norm 1 and orthogonal to C^{0}-orbits. The product metric on R × C^{0} arising from \frac{∂}{∂t} and m is complete. The map (t, g) ↦ g.W(t) is a local isometry, so it is a Riemannian covering R × C^{0} → M̃ (see [BH], [CG], [G] 5.3.D, [Ze1] §5.5 for origins and more ambitious applications of this idea). Also, C^{0}-orbits equal H-orbits, so applying θ_{0} ∘ ι, where ι is inversion, on C^{0} gives a diffeomorphism of M̃ with R × H carrying H-orbits to factors {t} × H. Since C preserves both the vector field W and the foliation by H-orbits, the C-action, and, in particular, the Γaction, respect the product structure induced on M̃. Denote by t(x) and p(x) the resulting coordinate functions to R and H, respectively. step 6: splitting of C. Let Q ⊂ C be the subgroup preserving W. Note that c ∈ Q ⇔ c.x_{0} ∈ W. In fact, because any c ∈ C commutes with flow along W, any c ∈ Q acts on W by translation by t(c.x_{0}). The subgroup Q contains the translation by t(c.x_{0}) for all c ∈ C because there is always some g ∈ C^{0} with gc.x_{0} ∈ W. Since C acts freely, it is clear that each element can be written as a product in Q · C^{0}. Define

Ψ : Q → Aut(C^{0})
Ψ(q)(g) = q^{−1}gq

Note that Ψ is conjugate by θ_{0} into Aut(H). We have C ≅ Q ⋉_{Ψ} C^{0}. step 7: splitting of Γ. The quotient C^{0}\M̃ can be identified with W ≅ R, and Γ/L acts faithfully on it with quotient C′\M̃ ≅ S^{1}. Thus the projection of Γ to Q is the group of translations by αZ for some α ≠ 0. By the splitting of C from the previous step, Γ is isomorphic to a semi-direct product Z ⋉_{Ψ} L. ♢ Remark 1: Let θ_{0} : C^{0} → H be as in step 5. Under the diffeomorphism M̃ ≅ R × H, the action of q ∈ Z is

q.x = (t(x) + qα, (θ_{0} ∘ Ψ_{q} ∘ θ_{0}^{−1})(p(x))

Remark 2: The group C inherits a real-analytic structure from M̃ for which it acts real-analytically. The isomorphism θ_{0} is real-analytic. The vector field W is real-analytic, as are the curve W and the diffeomorphism M̃ → R × H. In the remaining statements, we will say, given a metric-defining path Φ_{t}, that a space X is equivariantly isometric to R ∗_{Φ} H if there is some Z_{0} ∈ 𝔷 for which X is equivariantly isometric to R ∗_{Φ} H in the metric determined by Φ_{t} and Z_{0}.

Theorem 4.8 The universal cover M̃ is H_{n}-equivariantly isometric to R∗_{Φ}H_{n} for a real-analytic, metric-defining, lattice compatible path Φ : R → Aut(H_{n}). The restriction of Φ to Z coincides with the homomorphism Ψ of the previous theorem.

Proof: Let 𝔠 and C be as in step 2 of the previous proof. step 1: homogeneous case. If dim𝔠 = 2n + 2, then M is homogeneous by step 3 in the previous proof. By the results of [Ze1], there is λ ∈ Q^{n} for which C is isometric to S̃_{λ} endowed with the unique bi-invariant metric, and M is equivariantly isometric to C/Γ for Γ a lattice subgroup. The group S̃_{λ} is H-equivariantly isometric to R ∗_{Φ} H, where R parametrizes e^{tW} and Φ_{t}(h) = e^{−tW} he^{tW}. The path Φ_{t} is a homomorphism, so equivariance is clear. Note Φ restricted to Z equals Ψ from step 3 of the previous theorem. Now assume dim𝔠 = 2n + 1. step 2: metric-defining path. Let 𝔥 = 𝔭 ⊕ RZ_{0} be the decomposition in the proof of Theorem 4.6 and W the transverse vector field with integral curve W corresponding to this decomposition as above. Let θ_{t} : C^{0} → H be the isomorphism defined by θ_{t}(g)g.W(t) = W(t); these form a real-analytic path of real-analytic isomorphisms. Let Φ_{t} = θ_{0} ∘θ_{t}^{−1}; these form a real-analytic path. Denote by (t, p) the coordinates of the diffeomorphism M̃ → R × H. The C^{0}-action in these coordinates is g.x = (t(x), θ_{0}(g^{−1}) · p(x)). It follows that h.x = (t(x), Φ_{t(x)}(h)·p(x)), so M̃ is H-equivariantly diffeomorphic to R ∗_{Φ} H. Fix a point x and let t(x) = t and p(x) = p. The vector field W is carried to \frac{∂}{∂t} on R ∗_{Φ} H, and Killing fields are preserved. Both \frac{∂}{∂t} and W have properties (M1)-(M3) with respect to 𝔭 and Z_{0}. Also, Z_{0}^{∗} is isotropic and orthogonal to f_{x}(𝔭) in both metrics. It remains to show that Φ_{t} is metric-defining and that our map is isometric restricted to f_{x}(𝔭). As in the proof of Proposition 3.3, for X ∈ 𝔥, and X^{∗} the corresponding Killing field,

[W, X^{∗}](x) = (φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t} (X))^{∗}(x)

where φ_{t} = D_{e}(Φ_{t}). Since Z_{0}^{∗} ∈ 𝔠,

[W, Z_{0}^{∗}](x) = 0

Also as in Proposition 3.3, the fact that H acts isometrically implies that for all K ∈ 𝔥 and x ∈ M̃,

< [K^{∗}, W], Y >_{x} + < W, [K^{∗}, Y] >_{x}= 0

where Y is any vector field such that < W, Y > is constant along H-orbits. Taking K ∈ 𝔭 and Y = W gives

2 < [K^{∗}, W], W >_{x}= 0

Since [K^{∗}, W](x) belongs to f_{x}(𝔥), it is also orthogonal to Z_{0}^{∗}(x). Thus [K^{∗}, W](x) always belongs to f_{x}(𝔭). It follows that φ_{t}^{−1} ∘ (\frac{∂φ_{s}}{∂s} )_{t} annihilates Z_{0} and preserves 𝔭 for all t. Let ν_{t} be the restriction of this endomorphism to 𝔭. Take Y = Y^{∗} and K^{∗} both Killing fields from 𝔭. Recall that X ↦ X^{∗} is a Lie algebra homomorphism. Then

< [K^{∗}, W], Y^{∗} >_{x} + < W, [K^{∗}, Y^{∗}] >_{x}=< [K^{∗}, W], Y^{∗} >_{x} +ω_{0}(K, Y)

because [K, Y] = ω_{0}(K, Y) and < W, Z_{0}^{∗} >= 1 everywhere. Thus

< [W, K^{∗}], Y^{∗} >_{x}=< ν_{t}(K)^{∗}, Y^{∗} >_{x}= ω_{0}(K, Y)

Because ω_{0} is non-degenerate, the kernel of ν_{t} is trivial. We obtain

< K^{∗}, Y^{∗} >_{x}= ω_{0}(ν_{t}^{−1}K, Y)

Now symmetry of the metric implies ν_{t}^{−1} and ν_{t} are infinitesimally symplectic, and definiteness implies ν_{t} is infinitesimally definite for all t. Thus Φ_{t} is metricdefining, and the map M̃ → R ∗_{Φ} H is isometric. step 3: equivariance of Φ Recall Γ = π_{1}(M) ⊂ C and the splitting Γ ≅ Z⋉_{Ψ} L from step 6 of the previous proof. We relate Φ_{qα} and Ψ_{q} for q in the projection of Γ to Z. Applying remark 1 above,

(q ∘ h).x_{0} = (qα, (θ_{0} ∘ Ψ_{q} ∘ θ_{0}^{−1})(h))

while

(h ∘ q).x_{0} = (qα, Φ_{qα}(h))

Because H and Γ commute, we conclude Φ_{qα} = θ_{0} ∘ Ψ_{q} ∘ θ_{0}^{−1}. Next, we show equivariance. Again applying remark 1 for arbitrary x with coordinates t(x) = t and p(x) = p,

(h ∘ q).x = (t + qα, Φ_{qα+t}(h) · Φ_{qα}(p))

This must equal

(q ∘ h).x = (t + qα, Φ_{qα}(Φ_{t}(h) · p)) = (t + qα, (Φ_{qα} ∘ Φ_{t})(h) · Φ_{qα}(p))

Therefore, Φ_{qα+t} = Φ_{qα} ∘ Φ_{t}. Take a new metric-defining path Φ′_{t} = Φ_{αt}. As in Section 3.3.2, there is an H-equivariant isometry R ∗_{Φ} H → R ∗_{Φ′} H, and now Φ′ is lattice-compatible, with Φ′_{q} = θ_{0} ∘ Ψ_{q} ∘ θ_{0}^{−1} for all q ∈ Z. ♢

Theorem 4.9 (Classification). There is a bijective correspondence between (a) universal covers of compact Lorentz manifolds with real-analytic codimensionone isometric H_{n}-actions, up to equivariant isometry

(b) real-analytic, lattice-compatible, metric-defining paths Φ_{t} in Aut(H_{n}), up to equivalence of metric-defining paths.

Proof: Let M be a compact Lorentz manifold as in (a). By Theorem 4.8, M̃ is H-equivariantly isometric to some R ∗_{Φ} H. If M̃ is H-equivariantly isometric to R ∗_{Φ′} H for another path Φ′ (arising from a different choice of splitting 𝔥 = 𝔭′ ⊕ RZ′_{0}), then Proposition 3.4 implies Φ and Φ′ are equivalent. Given a path Φ_{t} as in (b), let L be a lattice such that Φ(q) preserves L for all q ∈ Z, and let Γ = Z ⋉_{Φ} L. Then (R ∗_{Φ} H)/Γ is a compact Lorentz manifold as in (a) by Propositions 3.3 and 3.6. If Φ′_{t} is an equivalent metric-defining latticecompatible path, then R ∗_{Φ} H and R ∗_{Φ′} H are H-equivariantly isometric from Section 3.3.2. ♢

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