Reductive homogeneous Lorentzian manifolds
Dmitri Alekseevsky, Ioannis Chrysikos, Anton Galaev
TL;DR
This work classifies reductive homogeneous Lorentzian manifolds $M=G/L$ with totally reducible isotropy by first classifying totally reducible subalgebras of the Lorentz algebra into three types (I, II, III) and showing that such $M$ admit reductive decompositions. It then develops the admissibility framework, including minimal admissibility, and constructs invariant Lorentz metrics via centralizers and extensions from semisimple factors, yielding explicit descriptions for both compact and non-compact semisimple groups. The paper provides detailed realizations: standard homogeneous contact manifolds (Type Ia) and special contact manifolds (Type Ib) for compact $G$, and their non-compact analogues (Type Ia/Ib) with pseudo- and para-3-Sasakian structures; it also classifies minimal admissible spaces in those regimes and analyzes Type II and III stabilizers, showing a product structure with a 2D constant-curvature Lorentzian factor and constraining the possible ambient geometry. Collectively, the results furnish a comprehensive taxonomy of reductive Lorentzian homogeneous spaces with invariant metrics, linking Lorentzian geometry to contact, Wolf, and quaternionic/para-quaternionic symmetric structures and providing explicit models with geometric and algebraic data for applications in differential geometry and mathematical physics.
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.