Closed geodesics on compact geodesically complete Lorentz manifolds

Determine whether every compact geodesically complete Lorentz manifold admits a closed geodesic.

Background

The paper establishes that compact geodesically complete pseudo-Riemannian manifolds can fail to admit closed geodesics in signature (2,4), while also proving several positive existence results. In the Lorentzian setting, however, the authors do not resolve whether geodesic completeness prevents such examples. They formulate the question for arbitrary compact geodesically complete Lorentz manifolds.

References

In Lorentzian signature, we do not know whether geodesic completeness forces the existence of closed geodesics. Does every compact geodesically complete Lorentz manifold admit a closed geodesic?

— Closed geodesics on compact semi-Riemannian manifolds  (2610.23552 - Allout et al., 16 Oct 2026) in Section 1, Problem 1.1 (labeled prob:lorentz)

This leaves open the following question. Does there exist a compact pseudo-Riemannian manifold whose geodesic flow preserves the energy of a Riemannian metric but which has no closed geodesics?

— Closed geodesics on compact semi-Riemannian manifolds  (2610.23552 - Allout et al., 16 Oct 2026) in Section 1, Subsection 1.3, “Further discussion”