The Lorentzian oscillator group as a geodesic orbit space (Q2872300)

The authors' abstract reads: ``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 \(d\)-dimensional supergravity with at least 24 of the 32 possible supersymmetries.'' \copyright 2012 American Institute of Physics.NEWLINENEWLINEFor understanding this abstract, it is useful to mention the following facts:NEWLINENEWLINE1: Os is the only four-dimensional non-abelian simply connected solvable Lie group, which admits a bi-invariant Lorentzian metric.NEWLINENEWLINE2: The algebra os related to Os can be defined as follows: with a base set \(\{P, X , Y, Q\}\) the only nonzero brackets are NEWLINE\[NEWLINE [X, Y ] = P, [Q, X] = Y, [Q, Y ] = -X. NEWLINE\]NEWLINE 3: A nontrivial geodesic in the quotient space \(M = G/H\) is said to be a homogeneous geodesic if it is the orbit of a one-parameter subgroup of \(G\).