Editing Geometrization conjecture (section)

===Euclidean geometry ''E''<sup>3</sup>===
{{Main|Euclidean geometry}}
The point stabilizer is O(3, '''R'''), and the group ''G'' is the 6-dimensional Lie group '''R'''<sup>3</sup> × O(3, '''R'''), with 2 components. Examples are the [[three-torus|3-torus]], and more generally the [[mapping torus]]  of a finite-order [[automorphism]] of the 2-torus; see [[torus bundle]]. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. This geometry can be modeled as a left invariant metric on the [[Bianchi classification|Bianchi groups of type I or VII<sub>0</sub>]]. Finite volume manifolds with this geometry are all compact,  and have the structure of a [[Seifert fiber space]] (sometimes in two ways). The complete list of such manifolds is given in the article on [[Seifert fiber space]]s. Under Ricci flow, manifolds with Euclidean  geometry  remain invariant.