Abstract
In this thesis we study the systolic geometry of Bieberbach manifolds. The \emph{systole} of a compact non simply connected Riemannian manifold $(M^n,g)$ is the smallest length of a non-contractible closed curve; the \emph{systolic ratio} is the quotient $(\mathrm{systole})^n/\mathrm{volume}$. M. Gromov proved that if $M^n$ is essential, there exists a positive constant $c(M)$ such that for any metric $g$ on $M^n$ we have: $Vol(M,g) \geq c(M) Sys(M,g)^n$. All compact surfaces (except $S^2$) are essential, and the theorem of Gromov is a generalisation of the same results for the torus $T^2$ (C. Loewner), for the projective plane (M. Pu) and for the Klein bottle (C. Bavard). The constant $c(M)$ is well known in the case of these manifolds, but in higher dimension we don't have much information. We study the optimal systolic ratio of $3$-dimensional Bieberbach manifolds that are not homeomorphic to a torus, and prove that it cannot be realized by a flat metric.