KILLING FIELDS IN COMPACT LORENTZ 3-MANIFOLDS

1 J. DIFFERENTIAL GEOMETRY. 43 (1996) 859-894. KILLING FIELDS IN COMPACT LORENTZ . 3-MANIFOLDS . ABDELGHANI ZEGHIB. Abstract Here we classify flows on COMPACT 3-MANIFOLDS that preserve smooth LORENTZ metrics. 1. Introduction The geodesic and horocyclic flows on the unit tangent bundle of a hyperbolic surface are well known by their beautiful, but very different properties. Nevertheless, these two flows with antagonistic dynamics are unified by the LORENTZ geometry. By this, we mean that both of them are KILLING FIELDS for LORENTZ structures. The purpose of this paper is to show that LORENTZ geometry not only unifies but also characterize them. That is, the nontrivial ( , nonequicontinous) KILLING FIELDS for LORENTZ metrics in dimension three, are all "derived from" geodesic or horocyclic flows. Algebraically, the unit tangent bundle of the 2-hyperbolic space is identified with the group PSX(2,R). The fundamental group of a hy- perbolic surface is thus identified with a discrete subgroup in PSX(2,R), and its unit tangent bundle with \ PSX(2,R).

2 A one- parameter group {/'} in PSX(2,R) determines on \ PSX(2,R) a right translation flow x > Txf1. The geodesic (resp. horocyclic). flow corresponds to the hyperbolic (resp. parabolic) one-parameter group: g = f e j ^ j (resp. h = (Q A) . In fact any noncom- pact one-parameter of P5L(2,R) is conjugate to {gat} or {hat} for some real . If a one-parameter group is COMPACT , it is conjugate to fcos( t) -sm{ t)\\. \\sm{ t) cos{ t) Jj Received July 20, 1994, and, in revised form, January 4, 1995. 859. 860 ABDELGHANI ZEGHIB. The KILLING form on the Lie algebra of PSL(2,R) determines a bi- invariant LORENTZ metric. It thus passes to a LORENTZ structure on the left quotients \ PSX(2,R), which is preserved by the right transla- tion flows. The deformations of the LORENTZ structure and those of the right translation flows were independently discovered by W. Goldman [6] and E. Ghys [4]. They are constructed in the following way.}}}

3 Ob- serve first that the flow determined by {/*} is preserved by the group G = PSX(2,R) x R, where the left factor acts by the left transla- tion and the R factor by the right translation by /*, , the flow itself. Therefore in order to get a flow which looks like that deter- mined by {/*}, and in particular preserving the LORENTZ structure, we just need a geometric structure modeled on (G,PSL(2,R)). One may imagine that this does not produce new flows. E. Ghys was the first to see the contrary. For this let us call them Ghys flows (so also the geodesic and horocyclic flows are now trivial Ghys flows). Now a de- formation ' of in G is given by a homomorphism c : > i?, so that P = Graph c = {(7,0(7)) G PSX(2,R) x R}. Thus an element 7' = (7,0(7)) acts by x > xf~ci" \. Next, for cocompact we know that small deformations of are realised by deformations of the geometric structure, and so small coho- mological classes in Hom( , R) generate Ghys flows.

4 Other trivial (in a dynamical sense) examples of isometric flows of LORENTZ manifolds , that we shall call equicontinuous flows, are those which in fact preserve Riemannian metrics. They are easy to understand (see section 2). Our principal result, is that, a nontrivial isometric flow on a LORENTZ COMPACT 3-manifold is conjugate (as a flow) to the suspension of a hyperbolic linear automorphism or to a Ghys flow. More precisely, we have: Theorem 1. Let (M, (/>*) be a smooth flow preserving a smooth LORENTZ structure on a COMPACT 3-manifold M. Suppose that is not equicontinuous. Then up to a rescale of a constant multiple of the pa- at rameter t {that is replacing by for some constant a), and up to finite covers, the flow is smoothly isomorphic (as a flow) to one of the following : i) The suspension with return time 1 of a toral hyperbolic linear diffeomorphism. ii) A Ghys flow on the complete LORENTZ space form M of constant curvature.}

5 Moreover there is C PSX(2,R) isomorphic to the fundamental group of a COMPACT surface and a homomorphism (not necessarily small) c : > R, so that for f a hyperbolic or parabolic one-parameter group, the manifold M is the quotient KILLING FIELDS IN COMPACT LORENTZ 3-MANIFOLDS 861. of PSX(2,R) by = Graph(c) = {(7,0(7))} acting as x >. xf~c^ \ In particular V acts freely properly discontinuously on PSL(2,R). Let us now give some comments about Theorem 1. 1) By "up to finite covers", we mean by taking a quotient of a finite cover of our manifold. It is sometimes necessary to start by passing to a finite cover as in the case of the geodesic and horocyclic flows of orbifolds. 2) Note that the suspension flows may be considered in some sense as a "limit" of geodesic flows. Indeed, such flows are obtained in an algebraic way, as above, with the group SOL instead of PSX(2,R), and at the Lie algebra level 50/ is a limit of algebras isomorphic to that of PSX(2,R).

6 3) Singularities. Note that we do not assume the flows are non- singular. But it follows from our result that this is the case if they are nonequicontinuous. 4) Regularity. To simplify, we assume here the metric C . It then follows that the isometric flow itself is C . In fact the proof of Theorem 1 uses the existence of the curvature tensor for the metric and second derivatives for the flow. Hence the metric and the flow must be C2. Our proof also uses somewheres Sard's theorem applied to functions derived (algebraically) from the curvature. Thus, they must be C 3 (since the dimension is 3) and so the metric must be C 5 . Nervertheless we may avoid this use of Sard's theorem and so the C2 hypothesis for the metric and the flow are enough. Note on the other hand that, by Kanai's construction in [10], a volume preserving Anosov flow on a 3-manifold preserves a C - LORENTZ metric, 1. which may be C ( the geodesic flows of negatively curved surfaces).

7 Nevertheless, the metric should be C , if it is just C 2 , and thus the flow is as in Theorem 1 above (this is the Ghys classification of Anosov flows with smooth stable and unstable distributions (see 2) ). 5) Isometry groups. One may easily deduce from our results (see Theorem 2 below) that if the isometry group of a COMPACT LORENTZ 3- manifold is not discrete ( , noncountable), then it has finitely many components, and further the identity component, if noncompact, is iso- morphic to R or PSX(2, R) (all this, up to finite covers). In fact, our motivation in studying isometric flows on COMPACT LORENTZ manifolds was an attempt to understand the isometry groups of such manifolds , following the point of view of Gromov's theory on rigid transformation groups [7] (although we do not use here results from this theory). For known results on this field , one usually works with some hypothesis deal- 862 ABDELGHANI ZEGHIB.

8 Ing with the algebraic structure of the isometry group ( , it contains SX(2,R), [18], [7]), or with the topology of the manifold (for instance it is simply connected as in [3]), or with the geometry of the underlying manifold, as in [9] where author assumes it to be 3-dimensional and of constant curvature. Let us now give more informations about the invariant LORENTZ metric and the di ferentiable structure of the flow. These may be extracted from the proof of Theorem 1 or deduced from Theorem 1 itself. Theorem 2. ) In the suspension case, all the invariant LORENTZ metrics are flat and obtained from an initial metric by multiplying along the direction of the flow and its orthogonal by some constants. 2) The KILLING metric, up to a multiplicative constant, is the only LORENTZ metric of constant {negative) curvature invariant by a Ghys flow. In particular, an isomorphism between two Ghys flows is an isom- etry between their geometric structures.}

9 3) In the case of Ghys flows, the invariant metrics correspond to the left invariant metrics on PSX(2,R), determined by the scalar products on the Lie algebra which are furthermore Ad(ft)-invariant, and so in particular these metrics are locally homogenous. Up to a multiplicative factor, the space of such scalar products is 1-dimensional. ) In the case ofhyberbolic one-parameter groups, these metrics cor- respond to the multiplication of the KILLING metric by different constants along the flow and its orthogonal . ) In the case of parabolic one-parameter groups, the 1-dimensional space of the scalar products {up to multiplicative factors) which are Ad{h )-invariant gives only 3 isometry types of metrics on PSX(2,R). Nevertheless in M, these metrics are not {globally) isometric unless M. is homogenous. 4) In the case of suspension flows or Ghys flows with f hyperbolic, the flow is {everywhere) spacelike.}}}}

10 In the case of Ghys flows, with f . parabolic, the flow is {everywhere) lightlike. Theorem 3. Up to finite covers, a Ghys flow is smoothly orbit equiv- alent to a geodesic or horocyclic flow on a surface of constant curvature. 2. Preliminaries-steps of the proofs A LORENTZ scalar product on a vector space is a nondegenerate sym- n+1. metric bilinear form <, > of signature + ..+, R endowed with 2. the quadratic form dxl+dx\ + ..+dx n. A vector u is called spacelike, timelike or lightlike respectively, according to that < u,u > is >. KILLING FIELDS IN COMPACT LORENTZ 3-MANIFOLDS 863. 0, < 0, or = 0. Sometimes (perhaps for physical reasons) a LORENTZ scalar product is defined to have a signature + .. Nevertheless the types must not depend on the convention of the signature, and may be defined (when the dimension is at least 3) in the following way. The set of lightlike vectors is a cone, called the light cone.}