Homogeneous Lorentzian structures on the oscillator groups (Q1961708)

The \((2m+1)\)-dimensional Heisenberg group \(H_m\) may be identified with the set \(\mathbb{R}\times \mathbb{C}^m\) equipped with the product \((p_1, z_1) (p_2,z_2)= (p_1+p_2+ {1\over 2}\text{Im}\langle z_1,z_2 \rangle, z_1+ z_2)\). Let \(\lambda= (\lambda_1, \dots, \lambda_m)\) be a sequence of positive real numbers. The oscillator group \(G_m(\lambda)\) is the semi-direct product \(H_m\times \mathbb{R}\), with \(\mathbb{R}\) operating on \(H_m\) by \(r\cdot (p,z_1, \dots,z_m)= (p,e^{i\lambda_1 r}z_1, \dots,e^{i\lambda_m r}z_m)\). The authors consider on \(G_m(\lambda)\) a natural family of left-invariant homogeneous Lorentzian metrics \(g_\varepsilon\), \(-1<\varepsilon<1\). Fix such a metric \(g= g_\varepsilon\), and let \(\nabla\) be the corresponding Levi-Civita connection. A pseudo-Riemannian structure is a tensor field \(S\) of type \((1,2)\) such that the connection \(\widetilde\nabla= \nabla-S\) satisfies \(\widetilde \nabla g=0\), \(\widetilde \nabla R=0\), \(\widetilde\nabla S=0\): here, \(R\) is the curvature tensor field of \(g\). The authors determine all homogeneous pseudo-Riemannian structures on the oscillator group. Further related results are obtained for the group \(G_4(\lambda)\).