TAILIEUCHUNG - Đề tài " Isometric actions of simple Lie groups on pseudoRiemannian manifolds "

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 to M and G, respectively, where G carries any bi-invariant metric, then we have n0 ≤ m0 . We study G-actions that satisfy the condition n0 = m0 . | Annals of Mathematics Isometric actions of simple Lie groups on pseudoRiemannian manifolds By Raul Quiroga-Barranco Annals of Mathematics 164 2006 941 969 Isometric actions of simple Lie groups on pseudoRiemannian manifolds By Raul Quiroga-Barranco Abstract 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 to M and G respectively where G carries any bi-invariant metric then we have n0 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 r as considered in Zimmer s program with G normal in L Theorem A . If G has finite center and ranks G 2 then we prove that we can choose M for which L is semisimple and r 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. 1. Introduction In this work G will denote a connected noncompact simple Lie group and M a connected smooth manifold which is assumed to be compact unless otherwise stated. Moreover we will assume that G acts smoothly faithfully and preserving a finite measure on M . We will assume that these conditions are satisfied unless stated otherwise. There are several known examples of such actions that also preserve some geometric structure and all of them are essentially of an algebraic nature see Zim3 and FK . Some of such examples are constructed from homomorphisms G L into Lie groups L that admit a cocompact lattice r. For such setup the G-action is then the one by left translations on K L r where K is some .

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