Curved Flats and Isothermic Surfaces

TL;DR

This paper explores the relationship between curved flats in pseudo Riemannian symmetric spaces and isothermic surfaces, deriving a key PDE and constructing a M"obius invariant frame for these surfaces.

Contribution

It establishes a novel correspondence between curved flats and isothermic surfaces and derives Calapso's PDE, providing a new framework for their analysis.

Findings

Pairs of isothermic surfaces correspond to curved flats in symmetric spaces

Calapso's PDE is derived and utilized for surface analysis

A M"obius invariant frame for isothermic surfaces is constructed

Abstract

We show how pairs of isothermic surfaces are given by curved flats in a pseudo Riemannian symmetric space and vice versa. Calapso's fourth order partial differential equation is derived and, using a solution of this equation, a M\"obius invariant frame for an isothermic surface is built.