From a purely mathematical viewpoint, one can say that most recent works in Lorentz geometry, concern group actions on Lorentz manifolds. For instance, the three major themes: space form problem of Lorentz homogeneous spacetimes, the completeness problem, and the classification problem of large isometry groups of Lorentz manifolds, all deal with group actions. However, in the first two cases, actions are "zen" (e.g., proper), and in the last, the action is violent (i.e., with strong dynamics).

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... e will survey recent progress in these themes, but we will focus attention essentially on the last one, that is, on Lorentz dynamics. Example 2.1. The "simplest" non complete Lorentz metric is the Bohl metric on the torus. Endow R 2 − {0} with the metric dxdy x 2 +y 2 . Any line {x = Constant = 0} is an isotropic non-complete geodesic. Completeness Let us notice that for Lorentz manifolds there are many interesting notions of partial completeness: future (or past) completeness, lightlike (timelike, spacelike) completeness. . . Projectivized geodesic foliation. In the complete case, we have a geodesic flow which is a R-action. It is therefore, a kind of a group action related to Lorentz geometry, which however will not be considered in our survey here. No systematic investigation of this dynamics, exist in the literature. Maybe, the mathematical (and psychological) difficulty comes from the noncompactness of the ambient manifold to this flow, even when the basis M is assumed compact. However, the projectivized tangent bundle PT M is compact in this case, and is endowed with a one-dimensional geodesic foliations. This seems to be a most tame object to study (see [29] for the 2-dimensional case). b-Completeness Usual non-completeness, means that some geodesic reaches "infinity" with finite energy. However, completeness does not prevent existence of non-geodesic curves reaches "infinity" by using only "finite energy".