The projective interpretation of the eight 3-dimensional homogeneous geometries. (English) Zbl 0889.51021

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.
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.


51H25 Geometries with differentiable structure
53C30 Differential geometry of homogeneous manifolds
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)


Zbl 0496.57005
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