The systolic constant of orientable Bieberbach 3-manifolds

TL;DR

This paper investigates the maximal systolic ratio of 3D orientable Bieberbach manifolds, showing it cannot be achieved by flat metrics and constructing a special metric with notable geometric features.

Contribution

It proves the supremum of the systolic ratio for these manifolds is not attained by flat metrics and introduces a conformally extremal metric with many systolic geodesics.

Findings

The maximal systolic ratio is not realized by flat metrics.

A constructed metric on type C2 has extremal conformal properties.

The systole in this metric is realized by many geodesics.

Abstract

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact of $3$-dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric. We also highlight a metric that we construct on one type of such manifolds ($C_2$) which has interesting geometric properties : it is extremal in its conformal class and the systole is realized by "very many" geodesics.