B\"{a}cklund transformations as integrable discretization. The geometric approach

TL;DR

This paper interprets classical and modern geometric results on discrete nets and transformations as integrable discretizations via Bäcklund transformations, unifying continuous and discrete surface theories.

Contribution

It offers a geometric framework linking Bäcklund transformations to integrable discretizations of classical surface nets, including asymptotic, conjugate, and orthogonal nets.

Findings

Discrete analogs of asymptotic nets constructed via Bäcklund transformations.

Discrete conjugate nets derived from classical transformations.

Unified geometric approach to integrable discretizations of surface theories.

Abstract

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by B\"{a}cklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that B\"{a}cklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.