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Every closed orientable three-manifold admits a prime and torus decomposition into pieces modeled on the eight Thurston geometries. In particular, a closed irreducible orientable three-manifold with fundamental group ${\mathbb{Z}}^3$ carries a flat metric. The Bieberbach theorem then identifies it with the three-torus because an abelian crystallographic group has trivial linear holonomy. Simply connected closed three-manifolds are three-spheres by the Poincaré conjecture.

The possible ranks and the corresponding homeomorphism types areThere is no example for $n=2$ or $n\ge 4$. The uniqueness here is topological, not uniqueness of a Riemannian metric.

To see the restrictions, use the prime decomposition of a closed orientable three-manifold. Its fundamental group is the free product of the prime three-manifold groups. A free product of two nontrivial groups is nonabelian, so at most one summand can have nontrivial fundamental group. The remaining simply connected summands are three-spheres by the Poincaré conjecture. If the nontrivial summand is not irreducible, it is $S^2\times S^1$, giving rank one. If it is irreducible and has infinite fundamental group, the sphere theorem for three-manifolds makes it aspherical. Its fundamental group is then a dimension-three Poincare duality group. But ${\mathbb{Z}}^n$ is a duality group of dimension $n$, as seen from the torus classifying space or its Koszul resolution, so $n=3$.

For rank three, the flat case of the geometrization theorem says that a closed irreducible orientable three-manifold with fundamental group ${\mathbb{Z}}^3$ admits a flat metric. Write it as ${\mathbb{R}}^3/\Gamma$. By the Bieberbach theorem, $\Gamma$ contains a full-rank translation lattice $\Lambda$. If $g=(R,v)\in \Gamma$ commutes with translation by each $\lambda \in \Lambda$, then $R\lambda =\lambda$ for every lattice vector, forcing $R=I$. Since $\Gamma$ itself is abelian, it consists entirely of translations. Its quotient is therefore the three-torus. Rank zero follows directly from the Poincaré conjecture, and rank one was already determined by the prime decomposition of a closed orientable three-manifold. Historically, before the Poincare theorem, the uniqueness conclusions were phrased allowing connected sums with homotopy three-spheres; the modern theorem removes that ambiguity.

The six flat types below mean the six closed orientable flat three-manifold types. There are infinitely many individual flat metrics and there are additional noncompact flat Riemannian manifolds. In each closed case the universal cover is Euclidean space and the deck transformation group is a torsion-free crystallographic group. The holonomy groups of closed orientable flat three-manifolds are $1,C_2,C_3,C_4,C_6,C_2\times C_2$.

The first is a flat torus ${\mathbb{R}}^3/\Lambda$, with trivial holonomy. For each cyclic case choose a planar lattice $L$ admitting the relevant rotation $R_m$, and generate the deck transformation group by translations in $L$ and the screwIts $m$th power is unit translation along the axis. A nontrivial holonomy element has nonzero axial displacement modulo integers, so has no fixed point. All transformations preserve orientation. A compact prism fundamental domain gives compactness. Equivalently, the manifold is a torus mapping torus with finite-order monodromy. In planar lattice bases one may takeThe half-turn uses any planar lattice, the quarter-turn a square lattice, and the third- and sixth-turns a hexagonal lattice. These give respectively the half-turn flat three-manifold, third-turn flat three-manifold, quarter-turn flat three-manifold, and sixth-turn flat three-manifold. Their holonomies have orders $2,3,4,6$, and their first Betti numbers are all one, since the only fixed direction of holonomy is the screw axis.

The sixth type is the Hantzsche-Wendt manifold. A concrete group is generated byTheir linear parts are commuting half-turns about perpendicular axes. Moreover ${\alpha }^2$, ${\beta }^2$ and $(\alpha \beta {)}^2$ are translations by $(1,0,0)$, $(0,1,0)$ and $(0,0,-1)$. Their translation subgroup is the unit cubic lattice and their linear quotient is $C_2\times C_2$. In each nontranslation coset, the displacement along the fixed axis is a half-integer, so the action is free. The quotient is compact and orientable. No vector is fixed by both linear parts, so its first Betti number is zero. Bieberbach classification gives precisely these six affine types; the explicit screw descriptions distinguish them without identifying arbitrary finite quotients with holonomy.