Homogeneous Lorentzian structures on the oscillator groups

Abstract

We obtain all the homogeneous pseudo-Riemannian structures on the oscillator groups equipped with a family of left-invariant Lorentzian metrics. Moreover, in the 4-dimensional case we determine all the corresponding reductive decompositions and groups of isometries.

Four-dimensional naturally reductive pseudo-Riemannian spaces

“…• If (R 2 143 ) 2 =R 2 133R 2 144 then h = span{R(X, Y )|X, Y ∈ m} = span{A, B}. By using (8) we write down the non-vanishing brackets for the Lie algebra g = m + h:…”

“…Therefore, withR 1 133 = 0 andR 3 344 = 0, we have g = m + h, with h = span{A 3 }. By using (8) we obtain that the non-vanishing brackets are…”

“…One of the most celebrated examples of Lorentzian naturally reductive spaces is the oscillator group. We refer to [8] for notation and definitions. This group is defined as G = R × C × R with group structure (p 1 , z 1 , q 1 ) · (p 2 , z 2 , q 2 ) = (p 1 + p 2 + 1 2 Im(z 1 e iq1 z 2 ), z 1 + e iq1 z 2 , q 1 + q 2 ).…”

“…The manifold is symmetric if and only if ε = 0. For ε = 0, the naturally reductive structure tensors D and the curvature operators R are given in [8]. With respect to the orthonormal basis (P ′ , X, Y, Q ′ ), P ′ = (2 − 2ε) −1/2 (P − Q), Q ′ = (2 + 2ε) −1/2 (P + Q), we haveR XY P ′ =R XY Q ′ = 0,…”

The Lorentzian oscillator group as a geodesic orbit space

“…This is the lowest-dimensional member of a family of Lie groups called the oscillator groups (see, for instance, Medina and Revoy, 20 Bromberg and Medina,4 and Ref. 16) but we restrict ourselves in the present paper to the 4-dimensional one and denote it by Os.…”