The moduli space of left-invariant pseudo-Riemannian metrics on a given Lie group is defined as the orbit space of a certain isometric action on some pseudo- Riemannian symmetric space. In terms of the moduli space, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics. In this paper, we describe the orbit space of the action of a particular parabolic subgroup, and then apply it to obtain a generalization of Milnor frames for so-called the Lie groups of real hyperbolic spaces, and also for the three-dimensional Heisenberg group. As a corollary we show that all left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.
Akira Kubo. Kensuke Onda. Yuichiro Taketomi. Hiroshi Tamaru. "On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups." Hiroshima Math. J. 46 (3) 357 - 374, November 2016. https://doi.org/10.32917/hmj/1487991627