Abstract: Generalizing the idea of a "straight line" to the setting of Riemannian geometry, one arrives at the concept of a geodesic: curves with "zero acceleration", characterized by a certain second-order differential equation. A geodesic is called complete if it its maximal domain of definition is the entire real line, and the given Riemannian manifold itself is called complete if all of its geodesics are complete. A well-known fact is that compact Riemannian manifolds are complete. This whole story makes sense when one considers instead Lorentzian metrics -- the ones used in the mathematical formalism of General Relativity -- but compact Lorentzian manifolds are no longer necessarily complete. A question is raised: when is a compact Lorentzian manifold complete? In this talk we will review and survey some results (both old and recent) providing additional geometric conditions sufficient for Lorentzian completeness.
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