= Thurston geometry
{c}
The eight <Thurston geometries> are the simply connected, three-dimensional homogeneous model spaces
$$
S^3,\quad\mathbb R^3,\quad\mathbb H^3,\quad S^2\times\mathbb R,\quad\mathbb H^2\times\mathbb R,\quad\widetilde{\operatorname{SL}}_2(\mathbb R),\quad\mathrm{Nil},\quad\mathrm{Sol}.
$$
The first three have constant positive, zero and negative <sectional curvature>. The product models and the three remaining Lie-group models allow additional locally homogeneous metrics. A complete geometric structure is a quotient of the entire model by a discrete freely acting <group> of <isometries>.
Compatible model charts define a local geometric structure; to recover a quotient of the entire model one also needs completeness. An open unit <ball> with its usual <Euclidean metric> has compatible Euclidean charts but is incomplete, since a radial <geodesic> reaches its missing boundary in finite length. This distinguishes local <coordinate> geometry from a complete geometric piece in the <geometrization theorem>.
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