## 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.

### MSC:

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

Zbl 0496.57005
Full Text: