It is called hyperbolic in analogy with the linear theory of wave propagation, where the future state of a system is specified by initial conditions.
Other equivalent characterizations of global hyperbolicity make use of the notion of Lorentzian distance
They are Global hyperbolicity, in the first form given above, was introduced by Leray[2] in order to consider well-posedness of the Cauchy problem for the wave equation on the manifold.
Definition 3 under the assumption of strong causality and its equivalence to the first two was given by Hawking and Ellis.
Later Hounnonkpe and Minguzzi[6] proved that for quite reasonable spacetimes, more precisely those of dimension larger than three which are non-compact or non-totally vicious, the 'causal' condition can be dropped from definition 3.
It is possible to remedy this problem strengthening the causality condition as in definition 4 proposed by Minguzzi[8] in 2009.
Definition 4 is also robust under perturbations of the metric (which in principle could introduce closed causal curves).
In fact using this version it has been shown that global hyperbolicity is stable under metric perturbations.
It was previously well known that any Cauchy surface of a globally hyperbolic manifold is an embedded three-dimensional
submanifold, any two of which are homeomorphic, and such that the manifold splits topologically as the product of the Cauchy surface and
In view of the initial value formulation for Einstein's equations, global hyperbolicity is seen to be a very natural condition in the context of general relativity, in the sense that given arbitrary initial data, there is a unique maximal globally hyperbolic solution of Einstein's equations.