Polynomial Conserved Quantities of Lie Applicable Surfaces

TL;DR

This paper characterizes subclasses of Lie applicable surfaces using polynomial conserved quantities, revealing how transformations of these surfaces are induced by underlying Lie geometry, and introduces a new Bäcklund-type transformation for linear Weingarten surfaces.

Contribution

It introduces a polynomial conserved quantity framework for Lie applicable surfaces and develops a novel Bäcklund-type transformation for linear Weingarten surfaces.

Findings

Polynomial conserved quantities characterize subclasses of Lie applicable surfaces.

Transformations of these surfaces are induced by underlying Lie geometry.

A new Bäcklund-type transformation for linear Weingarten surfaces is proposed.

Abstract

Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and $L$-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new B\"{a}cklund-type transformation for these surfaces.