Stability of geodesic incompleteness for Robertson-Walker space-times

Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = –dt² fh, where –a<b+, (H, h) is a Riemannian manifold andf: (a, b)(0, ) is a smooth function. We show that ifa>– and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC⁰ perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC¹ perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia.Partially supported by a grant from the Research Council of the Graduate School of the University of Missouri-Columbia and NSF grant No. MCS77-18723(02).