Homogeneous Lorentzian manifolds of a semisimple group

Abstract

We describe the structure of d -dimensional homogeneous Lorentzian G -manifolds M = G / H of a semisimple Lie group G . Due to a result by N. Kowalsky, it is sufficient to consider the case when the group G acts properly, that is the stabilizer H is compact. Then any homogeneous space G / H ̄ with a smaller group H ̄ ⊂ H admits an invariant Lorentzian metric. A homogeneous manifold G / H with a connected compact stabilizer H is called a minimal admissible manifold if it admits an invariant Lorentzian metric, but no homogeneous G -manifold G / H ̃ with a larger connected compact stabilizer H ̃ ⊃ H admits such a metric. We give a description of minimal homogeneous Lorentzian n -dimensional G -manifolds M = G / H of a simple (compact or noncompact) Lie group G . For n ≤ 11 , we obtain a list of all such manifolds M and describe invariant Lorentzian metrics on M .