Reductive homogeneous Lorentzian manifolds

Abstract

We study homogeneous Lorentzian manifolds M=G/L of a connected reductive Lie group G modulo a connected reductive subgroup L, under the assumption that M is (almost) G-effective and the isotropy representation is totally reducible. We show that the description of such manifolds reduces to the case of semisimple Lie groups G. Moreover, we prove that such a homogeneous space is reductive. We describe all totally reducible subgroups of the Lorentz group and divide them into three types. The subgroups of Type I are compact, while the subgroups of Type II and Type III are non-compact. The explicit description of the corresponding homogeneous Lorentzian spaces of Type II and III (under some mild assumption) is given. We also show that the description of Lorentz homogeneous manifolds M=G/L of Type I reduces to the description of subgroups L such that M=G/L is an admissible manifold, i.e., an effective homogeneous manifold that admits an invariant Lorentzian metric. Whenever the subgroup L is a maximal subgroup with these properties, we call such a manifold minimal admissible. We classify all minimal admissible homogeneous manifolds G/L of a compact semisimple Lie group G and describe all invariant Lorentzian metrics on them.