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

• 

  The Hantschze-Wendt Manifold

• 

  The Quarter-Twist Cube

• 

  The $\frac 1 6$-Twist Hexagonal Prism

• 

  The $\frac 1 3$-Twist Hexagonal Prism

Proof

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