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A compact manifold is called Bieberbach if it carries a flat Riemannianmetric. Bieberbach manifolds satisfy an isosystolic inequality by a general andfundamental result of M. Gromov. In dimension 3, there exist four classes ofnon-orientable Bieberbach manifolds up to an affine diffeomorphism. In thispaper, We prove the existence on each diffeomorphism class of non-orientableBieberbach 3-manifolds of a two-parameter family of singular Riemannian metricsthat are systolically extremal in their conformal class. The proof uses aone-parameter family of singular Riemannian metrics on the Klein bottlediscovered by C. Bavard (\cite{bavard88}): each one of these metrics isextremal in its conformal class.

Bavard's systolically extremal Klein bottles and three dimensional applications