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 (systole)n/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(π, 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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Mir, C.E., Lafontaine, J. Sur la géométrie systolique des variétés de Bieberbach. Geom Dedicata 136, 95–110 (2008). https://doi.org/10.1007/s10711-008-9276-7
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DOI: https://doi.org/10.1007/s10711-008-9276-7