Local and Global Rigidity for Isometric Actions of Simple Lie Groups on Pseudo-Riemannian Manifolds

摘要

Let M be a finite volume analytic pseudo-Riemannian manifold that admits an isometric G-action with a dense orbit, where G is a connected non-compact simple Lie group. For low-dimensional M, i.e. dim(M) < 2 dim(G), when the normal bundle to the G-orbits is non-integrable and for suitable conditions, we prove that M has a G-invariant metric which is locally isometric to a Lie group with a bi-invariant metric (local rigidity theorem). The latter does not require M to be complete as in previous works. We also prove a general result showing that M is, up to a finite covering, of the form H/Gamma (Gamma a lattice in the group H) when we assume that M is complete (global rigidity theorem). For both the local and the global rigidity theorems we provide cases that imply the rigidity of G-actions for G given by SO0(p, q), G(2(2)) or a non-compact simple Lie group of type F-4 over R. We also survey the techniques and results related to this work.