Automorphisms of finite Einstein geometries (Q1299034)

Problems concerning the characterization of automorphisms of 3 special finite geometries are presented. The geometries are the finite analogues of Lorentz-Minkowski geometry (LMG), de Sitter's world (DSW) and Einstein's cylinder universe (ECU). Real LMG is the geometry in \(M := {\mathbb R}^n\) endowed with the distance function \[ d: M\times M \rightarrow {\mathbb R}, \quad (x, y) \rightarrow (x_1 - y_1)^2 + \dots + (x_{n-1} - y_{n-1})^2 - (x_n - y_n)^2. \] Real DSW is the geometry of \(M := Q^n \subset {\mathbb R}^{n+1}\), where \(Q^n\) is the regular hyperquadric given by the equation \(x_1^2 + \dots + x_n^2 - x_{n+1}^2 = 1\). In this case the distance function is given by \[ d: M\times M \rightarrow {\mathbb R}, \quad (x, y) \rightarrow x_1 y_1 + \dots + x_n y_n - x_{n+1} y_{n+1}. \] Real ECU is the geometry of \(M := Z^n \subset {\mathbb R}^{n+1}\), where \(Z^n\) is the quadratic hypercylinder given by \(x_1^2 + \dots + x_n^2 = 1\). Here the distance function is defined via \(d: M\times M \rightarrow {\mathbb R}^2\), \((x, y) \rightarrow (x_1 y_1 + \dots + x_n y_n, (x_{n+1} - y_{n+1})^2)\). The finite counterparts of LMG, DSW and ECU are obtained by replacing the field \({\mathbb R}\) of real numbers by a finite associative ring \(R\) or more special by a Galois field. Mappings \(f: M \rightarrow M\) satisfying the following condition \[ \exists k: \forall x, y \in M: \;d(x,y) = k \Longrightarrow d(f(x), f(y)) = k \tag{*} \] are studied. In the cases LMG, DSW \(k\) is an element in \(R\setminus \{0\}\) and in the case ECU \(k\) is in \(R\times R\). For instance in the real case of DSW (\(R = {\mathbb R}\)) a mapping satisfying (*) is already distance preserving (a Lorentz transformation). The paper gives an overview about results concerning such mappings in the finite cases of LMG, DSW and ECU. Especially the case where \(R\) is a Galois field and \(n = 2\) is treated.