Dynamical properties of the space of Lorentzian metrics
Abstract
We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic foliations of codimension 1. On the 2torus, we prove that a metric with constant curvature along one of its lightlike foliation is actually flat. This allows us to show that the restriction of the action to the set of nonflat metrics is proper and that on the set of flat metrics of volume 1 the action is ergodic. Finally, we show that, contrarily to the Riemannian case, the space of metrics without isometries is not always open.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2001
 arXiv:
 arXiv:math/0110082
 Bibcode:
 2001math.....10082M
 Keywords:

 Mathematics  Differential Geometry;
 58D17;
 (53C50;
 53C12)
 EPrint:
 19 pages, Latex file, new introduction, to be published in Commentarii Mathematici Helvetici