 Other equivalent characterizations of global hyperbolicity make use of the notion of Lorentzian distance formula_7 where the supremum is taken over all the formula_8 causal curves connecting the points (by convention d=0 if there is no such curve).They are

Global hyperbolicity, in the first form given above, was introduced by Leray in order to consider well-posedness of the Cauchy problem for the wave equation on the manifold.In 1970 Geroch proved the equivalence of definitions 1 and 2.