The projective interpretation of the eight 3-dimensional homogeneous geometries
Let \(X\) be a 3-dimensional simply connected complete Riemannian manifold, and let \(G\) be the group of all isometries of \(X\). The pair \((X,G)\) is called a 3-dimensional (homogeneous) geometry, if \(G\) acts transitively on \(X\) and there is a discrete subgroup \(\Gamma\) of \(G\) such that the quotient of \(X\) modulo \(\Gamma\) is compact. \textit{W. P. Thurston} proved in [Bull. Am. Math. Soc., New. Ser. 6, 357-379 (1982; Zbl 0496.57005)], that any 3-dimensional geometry \((X,G)\) is equivariantly equivalent to one of the eight spaces \(E^3\), \(H^3\), \(S^3\), \(S^2\times R\), \(H^2\times R\), \(\widetilde{\text{SL}}_2\text{R}\), \(\mathbf{Nil}\) and \(\mathbf{Sol}\), with \(G=\text{Isom}(X)\). In Felix Klein's spirit the author constructs in great detail embeddings of these geometries in the 3-dimensional real projective space \(P_3\)R (together with a suitable bilinear form), where \(G\) appears as a subgroup of the collineation group of \(P_3R\). These embeddings are evident if \(X\) is \(E^3\), \(H^3\), or \(S^3\). For most of the other spaces, however, this is much more complicated.
scientific article; zbMATH DE number 1066098
scientific article