I'm not sure if this question is too low level for Math Overflow (so feel free to move this to SE if you think it is).


Inspired by [this](https://math.stackexchange.com/questions/1863314/when-is-the-universal-cover-of-a-riemannian-manifold-complete) and [this question](https://math.stackexchange.com/questions/1502865/the-covering-manifold-of-a-complete-riemannian-manifold-is-complete?rq=1) I'm wondering if the following statement is also true for Pseudo-Riemannian manifolds: 
**A Pseudo-Riemannian manifold $(M,g)$ is complete if and only if its universal covering $(\tilde{M},\tilde{g})$ is complete.**


The reason I ask is that in the above links the theorem of Hopf Rinow plays an important role in the proof of this statement and since this theorem is not available for Pseudo-Riemannian manifolds, I'm wondering if the statement also holds for Pseudo-Riemannian manifolds.


If this statement is not true, are there special instances where it holds nevertheless?



$(M,g)$ is complete iff $(\tilde{M},\tilde{g})$ is complete (non-Riemannian version)
Completeness of a pseudo-Riemannian manifold (or a manifold with affine connection, more generally) means precisely the completeness of its geodesic flow on its tangent bundle. But the tangent bundle of any covering space is a covering space of the tangent bundle. The geodesic flow is the flow of the geodesic vector field on the tangent bundle. A vector field on a manifold is complete if and only if its lift to some (hence any) covering space is complete, which is just because the flow lines both lift (as a covering space) and project.


In the Riemannian setting, Hopf-Rinow says that completeness of the flow of the geodesic vector field is precisely completeness as a metric space. But completeness as a metric space is not defined on pseudo-Riemannian manifolds; the automorphism group of the pseudo-Riemannian geometry on Minkowski space doesn't preserve any metric, as the stabilizer of point is not compact. 


