On the isometric deformation of surfaces via the B\"acklund transformation

TL;DR

This paper explores how Bäcklund transformations can be generalized to produce isometric deformations of various surfaces, extending classical results beyond quadrics.

Contribution

It introduces a general framework for Bäcklund transformations that induce isometric deformations of surfaces, independent of specific surface shapes.

Findings

Bäcklund transformations can be applied broadly to different surface classes.

A general method for isometric deformation via Bäcklund transformations is developed.

The approach is independent of the initial surface shape.

Abstract

In trying to generalize Bianchi's B\"acklund transformation of quadrics to B\"acklund transformations of isometric deformations of other (classes of) surfaces, we investigate basic features of the isometric deformation of surfaces via the B\"acklund transformation with isometric correspondence of leaves of a general nature (independent of the shape of the seed).