# Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry

## Theorem

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

- Torus $\mathbb T^3$
- Half-Twist Cube
- Quarter-Twist Cube
- Hantschze-Wendt Manifold
- $\frac 1 6$-Twist Hexagonal Prism
- $\frac 1 3$-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.

The Half-Twist Cube