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 nowhere 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 nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Spanish MEC/FEDER

Spanish MEC-FEDER[MTM2007-60731]

Regional Junta Andalucia[P06-FQM-01951]

Regional Junta Andalucia

Spanish MEC

Spanish MEC[MTM2007-64504]

Capes, Brasil[BEX 1509-08-0]

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)