Bonnet pairs, isothermic surfaces and the retraction form

TL;DR

This paper revisits classical isothermic surface theory, linking Bonnet pairs to the retraction form, and offers a modern interpretation of Bianchi's work with new insights into their derivatives.

Contribution

It introduces a novel connection between Bonnet pairs' derivatives and the retraction form of isothermic surfaces, advancing the understanding of their geometric structure.

Findings

Derivatives of Bonnet pairs relate to the retraction form.

Provides a modern reinterpretation of classical isothermic surface theory.

Establishes new links between Bonnet pairs and isothermic surfaces.

Abstract

We give a modern account of the classical theory of Bianchi \cite{Bia03} (see also \cite{KamPedPin98}) relating isothermic surfaces to Bonnet pairs. The main novelty is to identify the derivatives of the Bonnet pair with a component of the retraction form of the isothermic surface.