Completeness of Compact Lorentz Manifolds Admitting a Timelike Conformal Killing Vector Field

It is proved that every compact Lorentz manifold admitting a timelike conformai Killing vector field is geodesically complete. So, a recent result by Kamishima in J. Differential Geometry [37 (1993), 569-601] is widely extended. Recently, it has been proved in [2] that a compact Lorentz manifold of constant curvature admitting a timelike Killing vector field is (geodesically) complete. It is natural to think about the importance of the assumption on the curvature in this result. Moreover, we could ask ourselves if there is a more general condition than the existence of a timelike Killing vector field. The answers to these questions are given in the following Theorem. Let (M, g) be a compact Lorentz manifold which admits a timelike conformai Killing vector field K. Then (M, g) is geodesically complete. Proof. We are going to see that any geodesic y : [0, è[-> M, 0 < b < oo, is extendible beyond b. It suffices to show that the vector field / remains in a compact subset of the tangent bundle. As g(y', /) is a constant C, it is enough to see that the projection onto the subbundle Span{A^} maps y' into a compact subset of Span{A^} (here Span{AT} denotes the line bundle {À • KP\A £ R, p £ M}). As K is timelike, the compactness of M implies that Inf \g(K, A')! > 0. Thus, we have only to check that g(K, y') is bounded. Taking into account the fact that Lng = o • g for some function o , we obtain that j-(g(K, y') = ^C-ooy, so (d/dt)g(K, y') and, as a consequence g(K, y') is bounded on [0, b[. Received by the editors October 20, 1993 and, in revised form, January 25, 1994. 1991 Mathematics Subject Classification. Primary 53C50, 53C22.