Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

TL;DR

This paper develops a discretised surface theory in projective differential geometry that preserves integrable structures, introduces a canonical frame, and connects discrete Demoulin systems with integrable equations like the discrete Tzitzeica equation.

Contribution

It introduces a canonical frame for discrete surfaces, classifies discrete projective minimal surfaces, and links discrete Demoulin systems to integrable equations with geometric applications.

Findings

Derived a Backlund transformation for discrete Demoulin systems.

Formulated a two-component generalisation of the discrete Tzitzeica equation.

Connected discrete Demoulin surfaces to affine spheres in affine differential geometry.

Abstract

We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Backlund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalisation of the integrable discrete Tzitzeica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretisation of affine spheres in affine differential geometry.