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