Hiroshima Mathematical Journal

On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups

Akira Kubo, Kensuke Onda, Yuichiro Taketomi, and Hiroshi Tamaru

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Abstract

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.

Article information

Source
Hiroshima Math. J., Volume 46, Number 3 (2016), 357-374.

Dates
Received: 19 November 2015
Revised: 22 January 2016
First available in Project Euclid: 25 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1487991627

Digital Object Identifier
doi:10.32917/hmj/1487991627

Mathematical Reviews number (MathSciNet)
MR3614303

Zentralblatt MATH identifier
1360.53029

Subjects
Primary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 53B30: Lorentz metrics, indefinite metrics

Keywords
Lie groups left-invariant metrics pseudo-Riemannian metrics Milnor-type theorems

Citation

Kubo, Akira; Onda, Kensuke; Taketomi, Yuichiro; Tamaru, Hiroshi. On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups. Hiroshima Math. J. 46 (2016), no. 3, 357--374. doi:10.32917/hmj/1487991627. https://projecteuclid.org/euclid.hmj/1487991627


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