Isothermic submanifolds of symmetric $R$-spaces

TL;DR

This paper generalizes the classical theory of isothermic surfaces from conformal 3-space to submanifolds within symmetric R-spaces, preserving their integrable structure.

Contribution

It extends the classical isothermic surface theory to submanifolds of symmetric R-spaces, broadening the scope of integrable geometric structures.

Findings

Classical isothermic surface theory is successfully generalized.

The integrable structure of isothermic submanifolds is preserved in the new setting.

Provides a framework for studying isothermic submanifolds in symmetric R-spaces.

Abstract

We extend the classical theory of isothermic surfaces in conformal 3-space, due to Bour, Christoffel, Darboux, Bianchi and others, to the more general context of submanifolds of symmetric $R$-spaces with essentially no loss of integrable structure.