LORENTZIAN MANIFOLDS ;IgD;TTING A KILLING VECTOR

Lorentzian manifolds admitting a Killing vector field have been studied in the literature from different points of view. Functional analysis, proper actions of Lie groups or Bochner’s technique in Differential Geometry, have been some of the very different tools involved to study them. The purpose of this paper is to review some of their properties, pointing out several techniques and supplying references. We will discuss in the next sections several properties of geodesics, isometry groups and Lorentzian tori, and some classification results. Lorentzian manifolds with few (or none) assumptions on curvature will be considered. So, we will not go through the more specific techniques developed to study space forms. (This topic has been widely studied, especially in dimension 3, and we will make just some comments about it in the last section.) In the remainder of Section 1, some notation and general properties of Killing vector fields on Lorentzian manifolds are introduced. The author acknowledges to Prof. P.E. Ehrlich for bringing to his attention some references. This work has been partially supported by a DGICYT Grant No. PB94-0796.