Sur la g\'eom\'etrie systolique des vari\'et\'es de Bieberbach

The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{l}\mathrm{e}{)}^n\mathrm{/}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}$. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including the $K(\pi ,1)$. We study the optimal systolic ratio of compact, 3-dimensional non orientable Bieberbach manifolds, and prove that it cannot be realized by a flat metric.