Wadsley's Theorem in pseudo-Riemannian Geometry

TL;DR

This paper extends Wadsley's theorem to pseudo-Riemannian geometry, showing conditions under which geodesics are closed and characterizing the topology of certain 2-manifolds.

Contribution

It proves Wadsley's theorem for non-lightlike geodesics in pseudo-Riemannian manifolds and characterizes the topology of 2-manifolds with all geodesics closed.

Findings

Pseudo-Riemannian 2-manifolds with all geodesics closed are diffeomorphic to $S^1\times \R$.

Every pseudo-Riemannian 2-manifold with index 1 has a non-closed geodesic.

Wadsley's theorem is extended to foliations by closed non-lightlike geodesics.

Abstract

We prove Wadsley's theorem for foliations by closed non-lightlike geodesics. As an application we show that every pseudo-Riemannian and non-Riemannian 2-mainfold, all of whose time- or spacelike geodesics are closed, is diffeomorphic to $S^1\times \R$. Further we show that every pseudo-Riemannian 2-manifold with index 1 contains a non-closed timelike or spacelike geodesic.