Lie applicable surfaces

TL;DR

This paper explores the gauge-theoretic framework for Lie applicable surfaces, linking it to classical surface theories and developing a transformation theory within this context.

Contribution

It provides a detailed account of the gauge-theoretic approach and demonstrates its equivalence to classical notions of $ ext{Omega}$- and $ ext{Omega}_0$-surfaces of Demoulin.

Findings

Establishes the gauge-theoretic approach as equivalent to classical surface theories.

Develops a transformation theory for Lie applicable surfaces.

Bridges modern gauge theory with classical differential geometry.

Abstract

We give a detailed account of the gauge-theoretic approach to Lie applicable surfaces and the resulting transformation theory. In particular, we show that this approach coincides with the classical notion of $\Omega$- and $\Omega_{0}$-surfaces of Demoulin.