The systolic constant of orientable Bieberbach 3-manifolds

Details The systolic constant of orientable Bieberbach 3-manifolds The systolic constant of orientable Bieberbach 3-manifolds

2009-12-19

arxiv.org/abs/0912.3894v1

Mathematics

Differential Geometry

21 pages, 3 figures

Scientific paper

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 Eilenberg-Maclane spaces. 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.

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