A Counterexample to Guillemin's Zollfrei Conjecture

TL;DR

This paper constructs specific Zollfrei Lorentzian metrics on certain circle bundles and discusses a partial resolution of Guillemin's Zollfrei Conjecture under a hyperbolicity assumption.

Contribution

It provides a counterexample to Guillemin's Zollfrei Conjecture and proves a weaker version assuming global hyperbolicity.

Findings

Constructed Zollfrei Lorentzian metrics on circle bundles

Disproved Guillemin's Zollfrei Conjecture in general

Proved a weaker conjecture under hyperbolicity

Abstract

We construct Zollfrei Lorentzian metrics on every nontrivial orientable circle bundle over a orientable closed surface. Further we prove a weaker version of Guillemin's conjecture assuming global hyperbolicity of the universal cover.