The Classification of S 2 × R Space Groups ∗

The geometrization of 3-manifolds plays an important role in various topological investigations and in the geometry as well. Thurston classified the eight simply connected 3-dimensional maximal homogeneous Riemannian geometries [7], [8]. One of these is S×R, i.e. the direct product of the spherical plane S and the real line R. Our purpose is the classification of the space groups of S×R, i.e. discrete transformation groups which act on S×R with a lattice on R (see Section 3), analogously to that of the classical Euclidean geometry E.