The Lorentzian oscillator group as a geodesic orbit space

Abstract

We prove that the four-dimensional oscillator group Os endowed with any of its usual left-invariant Lorentzian metrics, is a Lorentzian geodesic (so in particular null-geodesic) orbit space with some of its homogeneous descriptions, corresponding to certain homogeneous Lorentzian structures. Each time that Os is endowed with a suitable metric and an appropriate homogeneous Lorentzian structure, it is a candidate for constructing solutions in eleven-dimensional supergravity with at least twenty-four of the thirty-two possible supersymmetries.