2020/01/06 by Raúl Quiroga-Barranco, Quiroga-Barranco, Raul
Mathematics · #20G41 #53C24 #53C50 #57S20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2001.01688
openalex publication_date 2020/01/06 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Let M be a finite volume analytic pseudo-Riemannian manifold that admits an\nisometric G-action with a dense orbit, where G is a connected non-compact\nsimple Lie group. For low-dimensional M, i.e. \dim(M) < 2\dim(G), when the\nnormal bundle to the G-orbits is non-integrable and for suitable conditions,\nwe prove that M has a G-invariant metric which is locally isometric to a\nLie group with a bi-invariant metric (local rigidity theorem). The latter does\nnot require M to be complete as in previous works. We also prove a general\nresult showing that M is, up to a finite covering, of the form H/\Γ\n(\Γ a lattice in the group H) when we assume that M is complete\n(global rigidity theorem). For both the local and the global rigidity theorems\nwe provide cases that imply the rigidity of G-actions for G given by\n\SO0(p,q), G2(2) or a non-compact simple Lie group of type\nF4 over \ℝ. We also survey the techniques and results related to\nthis work.\n