Global hyperbolicity and Cauchy hypersurfaces

Statement

A time-oriented Lorentzian manifold is globally hyperbolic if and only if it admits a Cauchy hypersurface.

Thus the compact-diamond causal condition and the existence of a global surface intersecting every inextendible timelike curve exactly once encode the same class of spacetimes. The theorem is a global statement; neither local Lorentzian coordinates nor local hyperbolicity of a differential operator implies it.

The stronger smooth splitting theorem shows that the Cauchy hypersurfaces can be chosen smooth and spacelike.