Resumen de Isometric actions of simple Lie groups on pseudoRiemannian manifolds

Raúl Quiroga-Barranco

• 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.