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

TL;DR

This paper investigates the optimal systolic ratio of 3-dimensional non-orientable Bieberbach manifolds, demonstrating that the maximum ratio cannot be achieved by flat metrics, thus advancing understanding of geometric inequalities in these manifolds.

Contribution

It establishes that the supremum of the systolic ratio for these manifolds is not attained by flat metrics, providing new insights into their geometric properties.

Findings

The supremum of the systolic ratio is not realized by flat metrics.

The study extends systolic geometry to non-orientable Bieberbach manifolds.

Provides bounds and properties of the systolic ratio in 3D non-orientable cases.

Abstract

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 $(\mathrm{systole})^n/\mathrm{volume}$. 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.