Lorentzian flows on compact 3-manifolds

We call here briefly "flow" a 1-dimensional foliation. In other words, we consider the orbits of nonsingular flows, i.e. the integral curves of nonsingular vector fields on some manifold M, regardless of their parametrization.

Besides, let us recall that a (pseudo)-Riewmannian metric transverse to some foliation F on a manifold M is a field of nondegenerate symmetric bilinear forms on the normal bundle n(F)=TM/TF of the foliation F, which is invariant by the flow of any vector field tangent to F. If F is the trivial 0-dimensional foliation by points of M, this gives back the usual definition of a (pseudo)-Riemannian metric on M. Whether M admits such a metric or not involves its topology and the index of the metric. Whether (M,F) admits a transverse metric involves moreover the dynamics of the foliation and is hereby generally a difficult question.

On compact 3-manifolds, the flows admitting a transverse Riemannan metric have been classified by Yves Carrière in the eighties. We wish to classify flows with a transverse Lorentzian metric on the same manifolds -- the signature of the metric is then (1,1). The situation is deeply different, e.g. all algebraic Anosov flows are transversely Lorentzian. We give the classification with an assumption of "transverse completeness" and build a new family of flows showing that this completeness is not always satisfied, unlike in the Riemannian case.