Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 029, 34 pages      arXiv:2403.12626      https://doi.org/10.3842/SIGMA.2025.029

On Integrable Nets in General and Concordant Chebyshev Nets in Particular

Michal Marvan
Mathematical Institute in Opava, Silesian University in Opava, Na Rybnívcku 1, 746 01 Opava, Czech Republic

Received March 20, 2024, in final form March 31, 2025; Published online April 28, 2025

Abstract
We consider general 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 reinterpret the result of I.S. Krasil'shchik and M. Marvan (see Section 2, Case 2 in [Acta Appl. Math. 56 (1999), 217-230]) as a curve net satisfying an $\mathbb R$-linear relation between the Schief curvature of the net and the Gauss curvature of the supporting surface. In the special case when the curvatures are proportional (concordant nets), we find a correspondence to pairs of pseudospherical surfaces of equal negative constant Gaussian curvatures. Conversely, we also show that two generic pseudospherical surfaces of equal negative constant Gaussian curvatures induce a concordant Chebyshev net. The construction generalises the well-known correspondence between pairs of curves and translation surfaces.

Key words: integrable surface; integrable curve net; differential invariant; pseudospherical surface; Chebyshev net; concordant net.

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