Vol. 107, No. 1, 1983

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Klein-Gordon solvability and the geometry of geodesics

John Kelly Beem and Phillip E. Parker

Vol. 107 (1983), No. 1, 1–14

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 Ksuch 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.

Mathematical Subject Classification 2000
Primary: 53C50
Received: 1 October 1981
Published: 1 July 1983
John Kelly Beem
Phillip E. Parker