Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry

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Theorem

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

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.

Proof