On Integrable Nets in General and Concordant Chebyshev Nets in Particular

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Author: Marvan, Michal
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/213185
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On Integrable Nets in General and Concordant Chebyshev Nets in Particular. Michal Marvan. SIGMA 21 (2025), 029, 34 pages

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 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 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.