Constant curvature rotational nets and periodic B\"acklund transforms

TL;DR

This paper explicitly parametrizes discrete pseudospherical surfaces of revolution with constant negative Gaussian curvature and constructs new B"acklund transformations that preserve periodicity, expanding the class of known discrete pseudospherical surfaces.

Contribution

It provides explicit parametrizations of discrete constant negative Gaussian curvature surfaces of revolution and introduces new B"acklund transformations with preserved periodicity.

Findings

Explicit parametrizations of discrete pseudospherical surfaces of revolution.

Construction of B"acklund transformations with explicit formulas.

Conditions for B"acklund transformations to preserve periodicity.

Abstract

After giving explicit parametrizations of discrete constant negative Gaussian curvature surfaces (negative CGC, i.e. discrete pseudospherical surfaces) of revolution, we construct B\"acklund transformations that again will have explicit parametrizations and are new examples of non-rotational discrete pseudospherical surfaces. In the process of doing this, for discrete CGC circular nets, we can provide rotationally invariant families of flat connections and give conditions on them so that the B\"acklund transformations preserve periodicity, that is, have annular topology.