A manifold with a complete locally homogeneous metric is said to be geometric, and the >metric is called the geometric structure.

It is known that there are essentially eight 3-dimensional geometries which compact 3-manifolds can possess.

They are H³, E³, S³, H²×ℝ, S²×ℝ,̴ S̴ L₂, Nil, Sol.

(Nil is the geometry of the Heisenberg group with left-invariant metric.)

(Except for H³ and Sol, geometric 3-manifolds are Seifert fibered manifolds. )

The Teichmüller space of a geometric manifold M is the set of all geometric structures on M factored by isotopy.