Periodic Geodesics and Geometry of Compact Lorentzian Manifolds with a Killing Vector Field

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold $M$ admits a Lorentzian metric with a never vanishing Killing vector field which is timelike somewhere if and only if $M$ admits a smooth circle action without fixed points.