Bäcklund transformations as integrable discretization. The geometric approach

Abstract

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by Bäcklund 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 Bäcklund 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.

arXiv comment18 pages, 4 figures; dedication added, typos corrected (v2). This article is part of an OCNMP Special Issue in Memory of Professor Decio Levi

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