Some connections between global hyperbolicity and geodesic completeness
type: Item
instance of: scholarly article
title: Some connections between global hyperbolicity and geodesic completeness
zbMATH Open document ID: 0677.53071
DOI: 10.1016/0393-0440(89)90004-1
author: Gregory J. Galloway
published in: Journal of Geometry and Physics
publication date: 1989-01-01T00:00:00Z
review text: 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.
reviewed by: J.-H. Eschenburg
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 83C75
Mathematics Subject Classification ID: 53C80
zbMATH DE Number: 4109575
zbMATH Keywords: global hyperbolicity
zbMATH Keywords: geodesic completeness
zbMATH Keywords: strong energy condition
zbMATH Keywords: sectional curvature condition
zbMATH Keywords: causally complete
MaRDI profile type: Publication
full work available at URL: 0393-0440(89)90004-1
OpenAlex ID: W1976956830
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description: scientific article; zbMATH DE number 4109575