Abstract

Previous discoveries of the first author (1984-88) on so-called hyperbolic football manifolds and our recent works (2016-17) on locally extremal ball packing and covering hyperbolic space ${H}^3$ with congruent balls had led us to the idea that our ``experience space in small size'' could be of hyperbolic structure. In this paper we construct a new infinite series of oriented 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 timely also for crystallographic applications. Mathematical novelties are foreseen as well, for future investigations.

Keywords: Hyperbolic space form; cobweb manifold; fullerene and nanotube.

MSC: 57M07, 57M60,52C17

Pages:  257--272     

Volume  72 ,  Issue  3 ,  2020