Isometric actions of simple Lie groups on pseudoRiemannian manifolds

• Autores: Raúl Quiroga-Barranco
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 164, Nº 3, 2006, págs. 941-969
• Idioma: inglés
• DOI: 10.4007/annals.2006.164.941
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• Resumen

  • Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m0, n0 are the dimensions of the maximal lightlike subspaces tangent toM and G, respectively, where G carries any bi-invariant metric, then we have n0 ¡Ü m0. We study G-actions that satisfy the condition n0 = m0. With no rank restrictions on G, we prove that M has a finite covering
    M to which the G-action lifts so that
    M is G-equivariantly diffeomorphic to an action on a double coset K\L/¿£, as considered in Zimmer¡¯s program, with G normal in L (Theorem A). If G has finite center and rankR(G) ¡Ý 2, then we prove that we can choose
    M for which L is semisimple and ¿£ is an irreducible lattice (Theorem B). We also prove that our condition n0 = m0 completely characterizes, up to a finite covering, such double coset G-actions (Theorem C). This describes a large family of double coset G-actions and provides a partial positive answer to the conjecture proposed in Zimmer¡¯s program.