| Russian Mathematical Surveys |
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Brief communications Curvature and isometries of the Lorentzian Lobachevsky plane Yu. L. Sachkovab a Ailamazyan Program Systems Institute of Russian Academy of Sciences b RUDN University
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     Lorentzian geometry is the mathematical foundation of relativity theory [1], [2]. It is different from Riemannian geometry is that information can propagate along curves with velocity vectors lying in a certain acute cone. A natural problem consists in finding Lorentzian length maximizers, which maximize a length-type functional along admissible curves. Thus, it is important to describe length maximizers and the properties of the corresponding Lorentzian distance function, including the curvature and isometries of the space. In this note we investigate left-invariant Lorentzian structures on the unique connected and simply connected non-Abelian two-dimensional Lie group. In our previous paper [3] we presented a description of Lorentzian maximizers for these structures, of the distances, and of spheres. In this note we show that these structures have a constant curvature, so that they are locally isometric to model constant curvature spaces (the Minkowski space ${\mathbb R}_1^{2}$, the de Sitter space ${\mathbb S}^2_1$, or the anti-de Sitter space $\widetilde{{\mathbb H}^2_1}$). In the case of curvature zero we provide an explicit isometric embedding in the two-dimensional Minkowski space. We also consider isometries of these Lorentz structures, both infinitesimal and global ones. 1. The statement of the problemLet $G=\operatorname{Aff}_+({\mathbb R})= \{(x,y) \in {\mathbb R}^2 \mid y > 0\}$ be the group of proper affine functions on the line, which is the Lobachevsky plane in its Poincaré model on the upper half-plane. Let $\mathfrak{g}$ denote the Lie algebra of left-invariant vector fields on the Lie group $G$. Consider the left-invariant frame $X_1=y\,\partial/\partial x$, $X_2=y\,\partial/\partial y$ in $\mathfrak{g}$. A left-invariant Lorentzian structure on $G$ is a non-degenerate quadratic form $g$ of index $(1,1)$ on $\mathfrak{g}$ [1], [2], [4], [5]. The set of such structures is parametrized by the matrices
$$
\begin{equation*}
A=\begin{pmatrix} a&b \\ c&d \end{pmatrix},\qquad |A| > 0,
\end{equation*}
\notag
$$
such that $g(u)=-(au_1+bu_2)^2+(cu_1+du_2)^2$. Let $\alpha$, $\beta$, $\gamma$, and $\delta$ denote the entries of the inverse matrix
$$
\begin{equation*}
A^{-1}=\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}.
\end{equation*}
\notag
$$
2. Curvature Theorem 1. The Levi-Civita connection $D$ of a left-invariant Lorentzian structure $g$ on the group $G=\operatorname{Aff}_+({\mathbb R})$ can be defined by where Theorem 2. A left-invariant Lorentzian structure $g$ on $G=\operatorname{Aff}_+({\mathbb R})$ has constant sectional curvature $K=g(X_1)/|A|^2$. Corollary 1. A left-invariant Lorentzian structure $g$ on $G=\operatorname{Aff}_+({\mathbb R})$ is locally isometric to the Minkowski space ${\mathbb R}^2_1$ (for $K=0$), the de Sitter space ${\mathbb S}^2_1$ (for $K>0$), or the anti-de Sitter space $\widetilde{{\mathbb H}^2_1}$ (for $K<0$). 3. IsometriesWe calculate the Lie algebra $i(G)$ of Killing vector fields (infinitesimal symmetries) for left-invariant Lorentzian structures on $G=\operatorname{Aff}_+({\mathbb R})$. By Theorem 2 such Lorentzian structures have a constant curvature, so $\dim i(G)=3$. Left shifts on $G$ are clearly isometries. They are generated by right-invariant vector fields on $G$: $\widetilde{X}_1(q)=R_{q*}X_1(\operatorname{Id})=\partial/\partial x$ and $\widetilde{X}_2(q)=R_{q*}X_2(\operatorname{Id})=x\,\partial/\partial x+ y\,\partial/\partial y$, where $R_q\colon\overline{q}\mapsto\overline{q}q$ is a right shift on $G$. Therefore, $\widetilde{X}_1$ and $\widetilde{X}_2$ are Killing vector fields. To describe the three-dimensional Lie algebra $i(G)$ it remains to find just one Killing vector field which is linearly independent of $\widetilde{X}_1$ and $\widetilde{X}_2$. Theorem 3. Let $K \ne 0$. Then $i(G)=\operatorname{span}(\widetilde{X}_1,\widetilde{X}_2,X_{\pm})$, where $\pm=\operatorname{sgn} K$ and $X_{\pm}=(y^2+w^2)\,\partial/\partial w+2wy\,\partial/\partial y$ for $w=(x \pm \nu(y-1))/\lambda$, $\lambda=(\alpha\delta-\beta\gamma)/\Delta$, $\nu=(\beta\delta-\alpha\gamma)/\Delta$, and $\Delta=\pm\gamma^2\mp\delta^2$. The commutator table for this Lie algebra is as follows: Theorem 4. Let $K=0$. Then $i(G)=\operatorname{span}(\widetilde{X}_1,\widetilde{X}_2,X_0)$, where $X_0=w\,\partial/\partial w+y(1-y)\,\partial/\partial y$ for $w=(x+g(y-1))/f$, $f=-(\alpha-s_1\beta)/(2\gamma)$, $g=-(\alpha+s_1 \beta)/(2\gamma)$, and $s_1=\operatorname{sgn}\gamma$. The commutator table for this Lie algebra is as follows: The Lie algbera $i(G)$ is isomorphic to the Lie algebra $\mathfrak{sh}(2)$ of the hyperbolic motion group $\operatorname{SH}(2)$ of the plane. Theorem 5. The Lie algebra of complete Killing fields is two-dimensional; it is generated by the vector fields $\widetilde{X}_1$ and $\widetilde{X}_2$. The identity component of the Lie group of isometries of the Lorentz group $\operatorname{Aff}_+({\mathbb R})$ is two dimensional and consists of the left shifts on this group. Theorem 6. Let $K=0$. Then the map $i\colon\operatorname{Aff}_+({\mathbb R})\to \Pi \subset {\mathbb R}^2_1$, where $\Pi=\bigl\{(\widetilde{x},\widetilde{y}) \in {\mathbb R}^2_1 \mid s_1 \widetilde{y}+\widetilde{x} < 1/\gamma \bigr\}$ and $i(x,y)=(\widetilde{x},\widetilde{y})=\bigl(\bigl((y-1)/y-w/\gamma\bigr)/2, s_1\bigl((y-1)/y+w/\gamma\bigr)/2\bigr)$, is an isometry. The author of grateful to L. V. Lokutsievskiy, D. V. Alekseevsky, N. I. Zhukova, and E. Le Donne for useful discussions of the questions under consideration. 
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\Bibitem{Sac24} |
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