the eight Thurston geometries
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In two dimensions there are exactly three uniform model geometries a surface can be built from — the round sphere (positive curvature), the flat plane (zero), and the hyperbolic plane (negative) — and every closed surface wears exactly one. Thurston discovered the three-dimensional analogue: not three but EIGHT model geometries, and unlike surfaces a single 3-manifold may have to be cut into pieces each modeled on a different one. These eight homogeneous geometries are the building blocks of all closed 3-manifolds.
Precisely, a model geometry here is a simply connected homogeneous Riemannian 3-manifold (X, G) where G is the full isometry group, G acts transitively, point stabilizers are compact, and X admits at least one compact quotient by a discrete subgroup (maximality and the compact-quotient condition are what cut the list down to eight). The eight are: the constant-curvature trio S^3, E^3 (Euclidean), and H^3 (hyperbolic); the two product geometries S^2 x R and H^2 x R; and three twisted geometries — the universal cover of SL(2,R)-tilde (which fibers over H^2), Nil (the Heisenberg geometry, fibering over E^2), and Sol (a solvable geometry with the least symmetry). Each is rigid as a local model: a manifold modeled on one of them carries that geometry as a (G,X)-structure.
Their importance is that they organize ALL of 3-manifold topology. Six of the eight (everything except H^3 and Sol) are 'small' Seifert-fibered or otherwise special geometries supporting very restricted manifolds; H^3 is overwhelmingly the generic case, the geometry of 'most' 3-manifolds. The geometrization theorem says every closed orientable 3-manifold canonically decomposes (along spheres and tori) into pieces each admitting exactly one of the eight geometries — so understanding 3-manifolds reduces to understanding these eight models plus how they are glued.
The three-torus T^3 = R^3 / Z^3 is modeled on E^3 (Euclidean); the product S^2 x S^1 is modeled on S^2 x R; a unit tangent bundle of a hyperbolic surface is modeled on SL(2,R)-tilde; and a closed orientable surface bundle over the circle whose monodromy is an Anosov (hyperbolic) map of the torus is modeled on Sol. A knot complement like the figure-eight knot complement, by contrast, is modeled on H^3 — the generic, hyperbolic case.
One manifold per geometry: T^3 (E^3), S^2 x S^1 (S^2 x R), a torus-bundle (Sol), the figure-eight complement (H^3).
There are eight 3-dimensional model geometries in Thurston's sense, but this is NOT the claim that every closed 3-manifold is geometric — most are not. The correct statement (geometrization) is that every such manifold decomposes into geometric pieces; a single manifold can require several of the eight at once.