Conformal isosystolic inequality of Bieberbach 3-manifolds

Details Conformal isosystolic inequality of Bieberbach 3-manifolds Conformal isosystolic inequality of Bieberbach 3-manifolds

2010-07-06

arxiv.org/abs/1007.0877v1

Mathematics

Differential Geometry

9 pages, no 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 $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum, over the set of all Riemannian metrics, is known to be finite for a large class of manifolds, including aspherical manifolds. We study a singular metric $g_0$ which has a better systolic ratio than all flat metrics on $3$-dimensional non-orientable Bieberbach manifolds (introduced in [El-La08]), and prove that it is extremal in its conformal class.

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