Thurston geometry

The eight simply connected three-dimensional homogeneous model geometries are $S^3$, ${\mathbb{R}}^3$, ${\mathbb{H}}^3$, $S^2\times \mathbb{R}$, ${\mathbb{H}}^2\times \mathbb{R}$, ${\widetilde{\mathrm{SL}}}_2(\mathbb{R})$, $\mathrm{Nil}$ and $\mathrm{Sol}$. A geometric structure is a locally homogeneous structure obtained as a quotient of a model by a discrete freely acting isometry group, equivalently with compatible model charts.