Geodesic
Local and Global Rigidity for Isometric Actions of Simple Lie Groups on Pseudo-Riemannian Manifolds
Journal of Lie theory, Tome 30 (2020) no. 2, pp. 565-586
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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.
Classification : 53C50, 53C24, 20G41, 57S20
Mots-clés : Pseudo-Riemannian manifolds, exceptional Lie groups, rigidity results 
@article{JLT_2020_30_2_JLT_2020_30_2_a13,
     author = {R. Quiroga-Barranco},
     title = {Local and {Global} {Rigidity} for {Isometric} {Actions} of {Simple} {Lie} {Groups} on {Pseudo-Riemannian} {Manifolds}},
     journal = {Journal of Lie theory},
     pages = {565--586},
     year = {2020},
     volume = {30},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JLT_2020_30_2_JLT_2020_30_2_a13/}
}
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R. Quiroga-Barranco. Local and Global Rigidity for Isometric Actions of Simple Lie Groups on Pseudo-Riemannian Manifolds. Journal of Lie theory, Tome 30 (2020) no. 2, pp. 565-586. http://geodesic.mathdoc.fr/item/JLT_2020_30_2_JLT_2020_30_2_a13/