Recent results on oscillator spacetimes (Q1624771)

In this article some recent results on the so-called oscillator algebra, and its various generalizations, are collected together. The four-dimensional \textit{oscillator algebra} by definition is a real four-dimensional Lie algebra spanned by the generators \(\{H,P,Q,E\}\) with the only non-trivial commutators \([H,P]=-Q\), \([H,Q]=P\) and \([P,Q]=E\) between them. Its name comes from the fact that it admits a representation \(\rho\) on \(C^2({\mathbb R})\) by putting \(\rho (H):=\frac{1}{2}(x^2-\frac{\partial^2}{\partial x^2})\) (``Hamiltonian''), \(\rho (P):=\frac{\partial}{\partial x}\) (``momentum''), \(\rho (Q):=x\) (``position'') and \(\rho (E):=1\) thereby it is associated with a one-dimensional quantum mechanical oscillator problem. The oscillator algebra is the Lie algebra of a corresponding (non-compact) four-dimensional Lie group \(G\), the \textit{oscillator group}, possessing a bi-invariant Lorentzian metric \(g\) such that \((G,g)\) is a Lorentzian homogeneous space. The article contains a survey on the geometric properties of \((G,g)\) (and its various generalizations), such as the explicit description of its Christoffel symbols, curvature tensor, Killing fields, various tensor collineation vector fields and finally the realization of \((G,g)\) as a Ricci soliton. Recall that a pseudo-Riemannian manifold \((M,g)\) is called a \textit{Ricci soliton} if there exists a vector field \(X\) along \(M\) and a real number \(\Lambda\) such that \(L_Xg+\mathrm{Ric}_g=\Lambda g\). For the entire collection see [Zbl 1394.53001].