On an integrable class of Chebyshev nets

Autor:	Marvan, Michal
Rok vydání:	2024
Předmět:	Mathematics - Differential Geometry Nonlinear Sciences - Exactly Solvable and Integrable Systems 53A05 53A55
Druh dokumentu:	Working Paper
Popis:	We consider integrable curve nets in Euclidean space as a particular integrable geometry invariant with respect to rigid motions and net-preserving reparameterisations. For the purpose of their description, we first give an overview of the most important second-order invariants and relations among them. As a particular integrable example, we study curve nets satisfying an $\mathbb R$-linear relation between the Schief curvature of the net and the Gauss curvature of the supporting surface. Starting with an $\mathfrak{so}(3)$-valued zero-curvature representation, associated with an elliptic spectral curve, we reveal two cases when the curve degenerates. In one of these cases, when the curvatures are proportional (concordant nets), we find a correspondence to pairs of pseudospherical surfaces of equal negative constant Gaussian curvatures. The construction generalises the well-known correspondence between translation surfaces and pairs of curves. Comment: 26 pages, 6 figures
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2403.12626 Zobrazit plný text záznamu View this record from Arxiv