2008/12/05 by José Luis Flores, Jose Luis Flores, Miguel Ángel Javaloyes +5
Mathematics · Physics and Astronomy · #53C12 #53C22 #53C50 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53C12 #msc:53C22 #msc:53C50
paper · pdf · doi:10.48550/arxiv.0812.1163
11 pages
openalex publication_date 2008/12/05 · arxiv created 2009/02/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.