2001/10/08 by Pierre Mounoud, Mounoud, Pierre
Mathematics · #(53C50 #53C12) #58D17 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:58D17
paper · pdf · doi:10.48550/arxiv.math/0110082
19 pages, Latex file, new introduction, to be published in Commentarii Mathematici Helvetici
arxiv created 2003/01/17 · arxiv updated 2009/11/30
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 2-torus, 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 non-flat 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.