The Bianchi-Darboux transform of L-isothermic surfaces

TL;DR

This paper introduces a Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, providing a method to construct these transforms via integrable systems and establishing a permutability theorem for their iteration.

Contribution

It extends the classical Bianchi-Darboux transformation to L-isothermic surfaces in Laguerre geometry, including a construction method and a permutability property.

Findings

Constructed Bianchi-Darboux transforms using integrable linear differential systems

Established a permutability theorem for iterated transforms

Extended classical transformations to a new geometric setting

Abstract

We study an analogue of the classical Bianchi-Darboux transformation for L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation. We show how to construct the Bianchi-Darboux transforms of an L-isothermic surface by solving an integrable linear differential system. We then establish a permutability theorem for iterated Bianchi-Darboux transforms.