Notes on transformations in integrable geometry

TL;DR

This paper explains how gauge theory can be used to understand transformations in integrable geometry, focusing on classical examples like surfaces of constant negative curvature and isothermic surfaces.

Contribution

It provides an accessible exposition of the gauge-theoretic approach to transformations in integrable geometry, illustrated through classical examples.

Findings

Clarifies the role of gauge theory in integrable geometry transformations

Connects classical surface theory with modern gauge-theoretic methods

Serves as educational material for understanding integrable surface transformations

Abstract

We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surfaces of constant negative Gauss curvature and isothermic surfaces. These are purely expository notes written to accompany some lectures I gave in Fukuoka in May 2015.