Geodesic completeness in static space-times

Let (H, h) be a Riemannian manifold and assume f: H(0, ) is a smooth function. The Lorentzian warped product (a, b)_f×H,–a<b, with metric ds²=(–f² dt²) h is called a standard static space-time. A study is made of geodesic completeness in standard static space-times. Sufficient conditions on the warping function f: H(0, ) are obtained for (a, b)_f×H ×H to be timelike and null geodesically complete. In the timelike case, the sufficient condition is independent of the completeness of the Riemannian manifold (H, h).