Some connections between global hyperbolicity and geodesic completeness
Completeness in Riemannian geometry has two aspects: Two points can be joined by a geodesic segment and any geodesic can be infinitely extended. In Lorentzian geometry, these two concepts fall apart: global hyperbolicity and geodesic completeness are independent notions. However, the present article shows relations between these notions under certain curvature condition, namely (a) the strong energy condition Ric(X,X)\(\geq 0\) for any timelike vector X and (b) the corresponding sectional curvature condition that R(.,X)X has nonnegative eigenvalues for any timelike X. In particular it is shown that the future of a suitable spacelike hypersurface S is globally hyperbolic if it is future timelike geodesically complete, and in case (b), also a partial converse is proved. More precisely, S has to be causally complete, i.e. \(J^+(p)\cap J^- (S)\) is compact for any \(p\in J^+(S)\), where \(J^+\) \((J^-)\) denotes the causal future (past), and moreover in case (a), S is maximal. The tools are in case (b) Harris' analogue of the Toponogov triangle theorem and in case (a) maximum principle techniques which have also been used to establish the Lorentzian splitting theorem. In fact, the author obtains a splitting theorem for the future of a suitable maximal spacelike hypersurface.
scientific article; zbMATH DE number 4109575