INFINITE SERIES OF COMPACT HYPERBOLIC MANIFOLDS, AS POSSIBLE CRYSTAL STRUCTURES

Abstract

Previous discoveries of the first author (1984-88) on so-called hyperbolicfootball manifolds and our recent works (2016-17) on locally extremal ballpacking and covering hyperbolic space ${H}^3$ with congruent balls had led us tothe idea that our "experience space in small size'' could be of hyperbolic structure.In this paper we construct a new infinite series oforiented hyperbolic space forms so-called cobweb (or tube) manifolds$Cw(2z, 2z, 2z)=Cw(2z)$, $3\le z$ odd, which can describe nanotubes, very probably.So we get a structure of rotational order $z=5,7\dots$, as new phenomena.Although the theoretical basis of compact manifolds of constant curvature(i.e.\ space forms) are well-known (100 years old), we are far from an overview.So our new very natural hyperbolic infinite series $Cw(2z)$ seems to be very timelyalso for crystallographic applications. Mathematicalnovelties are foreseen as well, for future investigations.