Group interpretation of the spectral parameter. The case of isothermic surfaces

TL;DR

This paper explores the group interpretation of the spectral parameter for isothermic surfaces, revealing how Lie symmetries relate to the non-removable spectral parameter in different geometric contexts.

Contribution

It provides a detailed analysis of the Lie symmetry structure of Gauss-Codazzi equations for isothermic surfaces and extends the results to higher-dimensional Euclidean spaces using Clifford algebra.

Findings

Lie algebra of symmetries is 4-dimensional for Gauss-Codazzi equations

A 3-dimensional algebra of symmetries leads to a non-removable spectral parameter

Results are generalized to isothermic immersions in multidimensional Euclidean spaces

Abstract

It is well known that in some cases the spectral parameter has a group interpretation. We discuss in detail the case of Gauss-Codazzi equations for isothermic surfaces immersed in $E^3$. The algebra of Lie point symmetries is 4-dimensional and all these symmetries are also symmetries of the Gaus-Weingarten equations (which can be considered as so(3)-valued non-parametric linear problem). In order to obtain a non-removable spectral parameter one has to consider so(4,1)-valued linear problem which has a 3-dimensional algebra of Lie point symmetries. The missing symmetry introduces a non-removable parameter. In the second part of the paper we extend these results on the case of isothermic immersions in arbitrary multidimensional Euclidean spaces. In order to simplify calculations the problem was formulated in terms of a Clifford algebra.