The Klein-Gordon equation is
globally solvable on Lorentzian manifolds which have no imprisoned causal geodesics
and which satisfy a certain convexity condition: for each compact subset K there
exists a compact subset K′ such that any causal geodesic segment with both
endpoints in K lies in K′. This collection of Lorentzian manifolds includes many
which are not globally hyperbolic. In any such manifold, the causal convex hull of a
compact set is compact. When a curvature condition is satisfied, causally
related points can be joined by at least one causal geodesic. A large class of
these manifolds which fail to be globally hyperbolic may be constructed
using warped products. The construction is independent of the warping
function.
