Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems

TL;DR

This paper explores the classical and integrable geometry of isothermic surfaces across all co-dimensions, highlighting the invariance of transformation theories and introducing Clifford algebra matrices as computational tools.

Contribution

It extends the classical transformation theory of isothermic surfaces to arbitrary co-dimensions and links Darboux transformations with dressing actions using Clifford algebra matrices.

Findings

Classical transformation theory applies in arbitrary co-dimension.

Darboux transformations correspond to dressing actions of simple factors.

Clifford algebra matrices facilitate computations in conformal geometry.

Abstract

We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we identify Darboux transformations with the dressing action of "simple factors" in the sense of Terng and Uhlenbeck. In so doing, we advertise the use of Vahlen's Clifford algebra matrices as an efficient computational tool in conformal geometry.