Abstract: Riemannian manifolds can be equipped with a natural
metric space structure, thanks to which many results involving
curvature bounds have been extended to metric (measure) spaces. A
crucial ingredient in this process is the Hopf-Rinow Theorem which
equates geodesic completeness with metric completeness. Lorentzian
manifolds, in contrast, do not admit a canonical metric space
structure and geodesic incompleteness is actually a desired
feature. Still, the "best" Lorentzian manifolds mimic the good
properties of complete Riemannian manifolds in many other ways. They
are called globally hyperbolic spacetimes and are also of utmost
importance in General Relativity (well-posedness of the initial value
formulation of the Einstein equations, singularity theorems of Penrose
and Hawking, splitting results etc.). In this talk, we present a
surprising new characterization of global hyperbolicity. We show that
globally hyperbolic spacetimes are precisely those Lorentzian
manifolds for which the null distance is complete. This is joint work
with Leonardo García-Heveling.