Integrable discrete geometry with ruler and compass

Summary

Abstract

Recent results of the geometry of integrable lattices are reviewed in a unified setting. The basic idea is to apply simple geometric constructions in a process of building the lattices from initial-boundary data.

Keywords: Integrable systems, discrete geometry

Introduction

Many of the research papers of the XIX-th century geometers is devoted to studies of special classes of surfaces, like for example:

1. i) surfaces admitting the Chebyshev net parametrization by asymptotic lines (pseudospherical surfaces)

2. ii) surfaces admitting the conjugate net parametrization by geodesic lines (surfaces of Voss)

3. iii) surfaces admitting the orthogonal net parametrization by asymptotic lines (minimal surfaces)

4. iv) surfaces admitting the isothermic net parametrization by curvature lines (isothermic surfaces).

To investigate such surfaces the statements about their geometric properties had been usually expressed in the language of differential equations. This way in the old books of differential geometry one can find, for example, the sine-Gordon equation (pseudospherical surfaces and surfaces of Voss) or the Liouville equation (minimal surfaces) and many others which are now included in the list of the integrable systems. systems. One can find there not only the equations but broad classes of their exact solutions, including N-soliton solutions and even some solutions in terms of the theta functions as well. Also the transformations (of Bianchi, Bäcklund, Darboux, Moutard, Ribaucour, Combescure) between the solutions of these equations or between corresponding surfaces enjoyed in that period a particular attention.