On Integrable Nets in General and Concordant Chebyshev Nets in Particular

TL;DR

The paper develops a unifying framework for integrable curve nets in $\,\mathbb{E}^3$, organizing the theory around differential invariants that include the intersection angle $\omega$, Gauss curvature $K$, mean curvature $H$, and Schief curvature $\sigma$. It identifies integrable Chebyshev nets through the condition $K+\kappa\sigma+\lambda=0$, analyzes degenerate subcases, and proves a constructive correspondence: concordant Chebyshev nets (with $K=\kappa\sigma$) correspond to pairs of pseudospherical surfaces of equal negative curvature, and conversely, generic pairs of pseudospherical surfaces induce concordant Chebyshev nets on their middle surface. A key development is the vector conservation law for concordant nets, enabling the explicit construction of associated pseudospherical surfaces $\mathbf{r}^{\pm}=\mathbf{r}\pm\mathbf{m}/\kappa$ and the middle-surface framework that links nets to surface pairs via the GMC system and Lelieuvre relations. The paper provides detailed geometric proofs, several explicit examples, and a roadmap for further exploration of integrable net classes, including potential extensions to higher dimensions and systematic generation of exact concordant nets for architectural and geometric applications.

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.

Figures (12)

• Figure 1: The space of conditions $\mu K + \kappa \sigma + \lambda = 0$. Antipodal points coincide.
• Figure 2: Colour visualisation of the Gauss map by outward normals.
• Figure 3: A snippet of the middle surface of two pseudospheres.
• Figure 4: Gaussian images of the cuspidal edges.
• Figure 5: Various positions of $R(\phi,\theta)$.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Remark 3.1
• Proposition 4.1
• Remark 4.2