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

TL;DR

This paper investigates the existence and properties of periodic geodesics in compact Lorentzian manifolds with a timelike Killing vector field, establishing conditions for their existence and linking to the manifold's topology.

Contribution

It proves the existence of at least one or two periodic geodesics under certain conditions and characterizes manifolds admitting such metrics via circle actions.

Findings

Existence of one timelike non self-intersecting periodic geodesic.

Existence of at least two periodic geodesics if the Killing vector field is nowhere vanishing.

Characterization of manifolds admitting such metrics through free circle actions.

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 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.