Géométrie systolique et métriques polyèdrales sur les 3-variétés 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 (systole)n/volume. Its supremum, over the set of all Riemannian metrics, is known to be finite for a large class of manifolds, including the K(π, 1). We study the optimal systolic ratio of compact, 3-dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric (using constructions of polyhedral metrics).