Lie applicable surfaces and curved flats

TL;DR

This paper explores the relationship between curved flats in Lie sphere geometry and Lie applicable surfaces, revealing a correspondence with Demoulin families and Darboux transformations.

Contribution

It establishes a novel connection between curved flats and Lie applicable surfaces via Demoulin families and Darboux transformations in Lie sphere geometry.

Findings

Curved flats correspond to pairs of Demoulin families.

Lie applicable surfaces are related through Darboux transformations.

The study advances understanding of geometric structures in Lie sphere geometry.

Abstract

We investigate curved flats in Lie sphere geometry. We show that in this setting curved flats are in one-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces related by Darboux transformation.