What is Globally hyperbolic manifold?

In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold. It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is

which is one of the usual equations representing an hyperbola. But this expression is only true relative to the ordinary origin; this article then outline bases for generalizing the concept to any pair of points in spacetime.This is relevant to Albert Einstein's theory of general relativity, and potentially to other metric gravitational theories.