On lattice coverings of Nil space by congruent geodesic balls

Details On lattice coverings of Nil space by congruent geodesic balls On lattice coverings of Nil space by congruent geodesic balls

2011-05-10

arxiv.org/abs/1105.1986v1

Mathematics

Metric Geometry

23 pages, 7 figures

Scientific paper

The Nil geometry, which is one of the eight 3-dimensional Thurston geometries, can be derived from {W. Heisenberg}'s famous real matrix group. The aim of this paper to study {\it lattice coverings} in Nil space. We introduce the notion of the density of considered coverings and give upper and lower estimations to it, moreover we formulate a conjecture for the ball arrangement of the least dense lattice-like geodesic ball covering and give its covering density $\Delta\approx 1.42900615$. The homogeneous 3-spaces have a unified interpretation in the projective 3-sphere and in our work we will use this projective model of the Nil geometry.

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