Lorentzian 3-manifolds and dynamical systems

— The following article aims at presenting classical aspects of dynamical systems preserving a geometric structure, focusing on 3-dimensional Lorentzian dynamics. The reader won’t find here any new result, but rather an expository approach of classical ones. Our main goal, in particular, is to introduce part of the techniques and arguments used in [7] to obtain the classification of closed 3-dimensional Lorentzian manifolds admitting a non-compact isometry group. Doing so, we will present a self-contained, and somehow shortened proof of Zeghib’s classification [24] of non-equicontinuous Lorentzian isometric flows in dimension 3. 1. An invitation to Lorentzian dynamics 1.1. From dynamical systems to geometry A flow φX , generated by a vector field X, on a closed manifold M is said to be Anosov when it is non-singular, and there exists a φX-invariant splitting TM = E− ⊕RX ⊕E+ such that vectors of E− (resp. of E+) are exponentially contracted (resp. dilated) under φX . Precisely, there exist positive constants c and λ such that ‖DφX(u)‖ 6 ce−λt‖u−‖ for every u− ∈ E−, t > 0, and ‖Dφ−t X (u+)‖ 6 ce−λt‖u+‖ for every u+ ∈ E+, t > 0. Here, the norms are taken with respect to any auxiliary Riemannian metric on M . Examples of Anosov flows are provided by the geodesic flows of a closed, negatively curved, Riemannian manifolds (N,h) (the manifoldM then corresponds to T 1N , the unit tangent bundle of N).