Sur la géométrie systolique des variétés de Bieberbach

Details Sur la géométrie systolique des variétés de Bieberbach Sur la géométrie systolique des variétés de Bieberbach

2008-04-09

arxiv.org/abs/0804.1419v2

Mathematics

Differential Geometry

17 pages, 2 figures, french, to appear in Geom. Dedicata

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

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