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Let $\tilde{M}$ be a simply connected Lorentzian manifold and suppose that $\tilde{M}$ admits some Riemannian metric.

Question: What can be said about the relation between the geodesic completeness of $\tilde{M}$ as a Lorentzian manifold and the metric completion as a metric space via the Riemannian metric? In particular, if $\tilde{M}$ is already metrically complete, is it then also Lorentzian geodesically complete?

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  • $\begingroup$ You may find some interesting results in dimension two contained in this book by Tilla Weinstein. $\endgroup$
    – Neal
    Feb 16 '16 at 1:49
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As every manifold admits a complete Riemannian metric (see here), you cannot deduce anything about the Lorentzian metric.

Even if every Riemannian metric on $\bar{M}$ is complete (so that $\bar{M}$ is necessarily compact), the Lorentzian metric may not be complete: the Clifton-Pohl torus is an example of a compact Lorentzian manifold which is not geodesically complete. This shows that the Hopf-Rinow Theorem does not generalise to Lorentzian manifolds.

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  • $\begingroup$ Yes I am well aware of these results. I am currently trying to understand that completeness is guaranteed when $M$ has constant sectional curvature and is compact. The proof is rather complicated, and my question was conceptually relevant. $\endgroup$
    – Vertex
    Feb 16 '16 at 20:13

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