Isometric actions of simple Lie groups on pseudoRiemannian manifolds

• 成果类型: Article
• 署名作者: Quiroga-Barranco, Raul
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2006.164.941
• 发表日期: 2006
• 页码: 941-969
• 关键词: pseudo-riemannian manifolds; fundamental-groups; semisimple groups; EXISTENCE

摘要

Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m(0), n(0) are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n(0) <= m(0). We study G-actions that satisfy the condition n(0) = m(0). With no rank restrictions on G, we prove that M has a finite covering (M) over cap to which the G-action lifts so that (M) over cap is G-equivariantly diffeomorphic to an action on a double coset K\L/Gamma, as considered in Zimmer's program, with G normal in L (Theorem A). If G has finite center and rank(R)(G) >= 2, then we prove that we can choose (M) over cap for which L is semisimple and Gamma is an irreducible lattice (Theorem B). We also prove that our condition n(0) = m(0) 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.