Publication details

Reductive homogeneous Lorentzian manifolds



Year of publication 2022
Type Article in Periodical
Magazine / Source Differential Geometry and its Applications
Keywords Reductive homogeneous Lorentzian manifolds; Lorentz algebra; Totally reducible subalgebras of the; Lorentz algebra; Admissible subgroups; Contact homogeneous manifolds; Wolf spaces
Description We study homogeneous Lorentzian manifolds M = G/L of a connected reductive Lie group Gmodulo 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 Iare 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/Lis an admissible manifold, i.e., an effective homogeneous manifold that admits an invariant Lorentzian metric. Whenever the subgroup Lis 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 Ga nd describe all invariant Lorentzian metrics on them.

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