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What is the heuristic meaning of the global hyperbolicity condition for semi-Riemannian manifolds?

Also what is the role of this condition in the study of geodesic connectedness?

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    $\begingroup$ The first question is pretty well addressed in the Wikipedia entry en.wikipedia.org/wiki/Globally_hyperbolic_manifold , see especially the remarks. For the second question: the role is very little. There are well known examples (for example de Sitter space) where the manifold is globally hyperbolic but not geodesic-connected. $\endgroup$ – Willie Wong Jan 8 at 20:42

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