Yau conjectured that a particular Lorentzian version of Cheeger and Gromoll's theorem should hold. Proofs in various levels of generality were found by Jost Eschenburg, Gregory Galloway, and Richard Newman. In these results, the role of geodesic completeness is replaced by either the condition of global hyperbolicity or of timelike geodesic completeness. The nonnegativity of Ricci curvature is replaced by the timelike convergence condition that the Ricci curvature is nonnegative in all timelike directions. The geodesic line is required to be timelike. References Notes. Historical articles. Textbooks. Theorems in Riemannian geometry