Guillemin calls a compact Lorentzian -manifold « Zollfrei » if the geodesics flow on the nonzero lightlike vectors induces a fibration by circles (especially all lightlike geodesics are closed). He conjectured that these metric can only exist on -manifolds covered by . I will explain counterexamples on every nontrivial circle bundle over a closed surface. If time permits I will discuss what additional assumptions imply the conjecture and hint at what is the right conjecture in the general case.
