Pseudo-Riemannian geodesic foliations by circles

TL;DR

This paper studies when pseudo-Riemannian geodesic foliations by circles are generated by circle actions, establishing conditions for their existence and exploring implications for Lorentzian surfaces with closed geodesics.

Contribution

It proves a pseudo-Riemannian analogue of Wadsley's theorem, showing such circle actions exist without lightlike leaves, and applies this to classify Lorentzian surfaces with closed geodesics.

Findings

Existence of circle actions for geodesic foliations without lightlike leaves

Counterexamples in the pseudo-Riemannian case

Lorentzian surfaces with all geodesics closed are finitely covered by S^1×R

Abstract

We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an $S^1$-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley's Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by $S^1\times \R$. It follows that every Lorentzian surface contains a non-closed geodesic.