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
- $\frac 1 6$-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
The Hantschze-Wendt Manifold
The Quarter-Twist Cube
The $\tfrac 1 6$-Twist Hexagonal Prism
The $\tfrac 1 3$-Twist Hexagonal Prism
Proof
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