The splitting theorem for space-times with strong energy condition (Q1104568)

A proof of the following theorem is given: Let (M,g) be a connected, time oriented, globally hyperbolic Lorentzian manifold which is timelike geodesically complete and satisfies the strong energy condition. Let \(\gamma\) : \({\mathbb{R}}\to M\) be a timelike geodesic which realizes the distance between any two of its points. Then (M,g) is isometric to (\({\mathbb{R}}\times H, -dt^ 2\otimes h)\) where (H,h) is a complete Riemannian manifold, and the first factor is represented by \(\gamma\). This is another version of a Cheeger-Gromoll splitting theorem for Lorentzian manifolds. A related version has been shown by \textit{G. J. Galloway} [Commun. Math. Phys. 96, 423-429 (1984; Zbl 0575.53040)] replacing the assumption of timelike geodesic completeness by the existence of a smooth function whose level sets are compact spacelike Cauchy hypersurfaces. Also a hint to a more recent splitting theorem due to \textit{G. J. Galloway} [The Lorentzian splitting theorem without completeness assumption (Preprint, 1987)] is given by the author.