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Journal of Lie Theory 30 (2020), No. 2, 565--586
Copyright Heldermann Verlag 2020

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

Raul Quiroga-Barranco
Centro de Investigación en Matemáticas, Guanajuato 36023, Mexico

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/Γ (Γ 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), G2(2) or a non-compact simple Lie group of type F4 over R. We also survey the techniques and results related to this work.

Keywords: Pseudo-Riemannian manifolds, exceptional Lie groups, rigidity results.

MSC: 53C50, 53C24, 20G41, 57S20.

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