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- Title
Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field.
- Authors
Flores, José Luis; Javaloyes, Miguel Ángel; Piccione, Paolo
- Abstract
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.
- Subjects
LORENTZ groups; MANIFOLDS (Mathematics); VECTOR fields; VECTOR analysis; DIFFERENTIAL geometry
- Publication
Mathematische Zeitschrift, 2011, Vol 267, Issue 1/2, p221
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-009-0617-5