The closed orientable flat Riemannian manifold with holonomy can be presented by Euclidean screws and . Their squares and the square of their product generate the unit translation lattice. Each nonidentity holonomy coset has a half-integer displacement along its fixed axis, so acts freely. The holonomy has no common fixed vector, giving first Betti number zero and a finite isometry group for every flat metric.
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The Hantzsche–Wendt manifold is a specific type of 3-manifold that serves as an example in the study of topology and geometry. It can be characterized as a compact, orientable, triangulated manifold with non-trivial fundamental group. One main feature of the Hantzsche–Wendt manifold is that it can be constructed from 3-dimensional Euclidean space and is related to the theory of solvable Lie groups.