# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2022; 59(4): 651-684

Published online July 1, 2022 https://doi.org/10.4134/JKMS.j210460

## The moduli spaces of Lorentzian left-invariant metrics on three-dimensional unimodular simply connected Lie groups

Mohamed Boucetta, Abdelmounaim Chakkar

### Abstract

Let $\mathrm{G}$ be an arbitrary, connected, simply connected and unimodular Lie group of dimension $3$. On the space $\mathfrak{M}(\mathrm{G})$ of left-invariant Lorentzian metrics on $\mathrm{G}$, there exists a natural action of the group ${\rm Aut}(\mathrm{G})$ of automorphisms of $\mathrm{G}$, so it yields an equivalence relation $\backsimeq$ on $\mathfrak{M}(\mathrm{G})$, in the following way: $h_1\backsimeq h_2 \Leftrightarrow h_2=\phi^{*}(h_1) \;\textrm{for some}\; \phi \in {\rm Aut}(\mathrm{G}).$ In this paper a procedure to compute the orbit space ${\rm Aut}(\mathrm{G})/\mathfrak{M}(\mathrm{G})$ (so called moduli space of $\mathfrak{M}(\mathrm{G})$) is given.

Keywords: Moduli space of left-invariant metrics, Lorentzian metrics, 3-dimensional Lie groups, generalized Ricci solitons

MSC numbers: Primary 22E15, 53C50