Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry

Theorem
Every closed, orientable, path connected 3-dimensional manifold which supports a geometry of zero curvature is homeomorphic to one of the following:


 * $\mathbb{T}^3$
 * Half-Twist Cube
 * Quarter-Twist Cube
 * Hantschze-Wendt Manifold
 * $$\tfrac{1}{6}$$-Twist Hexagonal Prism
 * $$\tfrac{1}{3}$$-Twist Hexagonal Prism

The 3-torus is described on the torus page; the other manifolds can be described using quotient spaces on familiar prisms, with the equivalence relations described below.