On a new definition of the B\"{a}cklund transformation in the isometric deformation of surfaces

TL;DR

This paper introduces a new definition of the Bäcklund transformation for isometric surface deformations by analyzing a 4D integrable rolling distribution and its reduction to 3D distributions, expanding the theoretical framework.

Contribution

It provides a novel formulation of the Bäcklund transformation in the context of isometric surface deformations through geometric distribution analysis.

Findings

A 4D integrable rolling distribution splits into a 3D family.

New geometric definition of Bäcklund transformation.

Framework excludes developable seed and isotropic leaves.

Abstract

We prove that a generic $4$-dimensional integrable rolling distribution of contact elements with the symmetry of the tangency configuration (excluding developable seed and isotropic developable leaves) splits into an $1$-dimensional family of generic $3$-dimensional integrable rolling distributions of contact elements with the symmetry of the tangency configuration, thus introducing a new definition of the B\"{a}cklund transformation in the isometric deformation of surfaces.