Homogeneous Lorentzian structures on the oscillator groups
type: Item
instance of: scholarly article
title: Homogeneous Lorentzian structures on the oscillator groups
zbMATH Open document ID: 0954.53029
published in: Archiv der Mathematik
publication date: 2001-02-08T00:00:00Z
full work available at URL: http://hdl.handle.net/10261/15840
review text: 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 \(G_4(\lambda)\).
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 22E43
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1394471
zbMATH Keywords: oscillator group
zbMATH Keywords: homogeneous Lorenzian metric
zbMATH Keywords: homogeneous pseudo-Riemannian structure
author: P. M. Gadea
author: J. A. Oubiña
reviewed by: Hassan Azad
MaRDI profile type: Publication
OpenAlex ID: W2034727122
DOI: 10.1007/S000130050403
description: scientific article; zbMATH DE number 1394471