Constrained elastic curves and surfaces with spherical curvature lines

TL;DR

This paper characterizes surfaces with spherical curvature lines using Lie sphere geometry, showing they can be generated from initial data involving elastic curves and providing conditions for Lie applicability, including for constant mean curvature surfaces.

Contribution

It introduces a novel Lie sphere geometric framework for constructing and characterizing surfaces with spherical curvature lines, linking elastic curves to Lie applicable surfaces.

Findings

Surfaces with spherical curvature lines can be generated from initial data involving Lie sphere transformations and Legendre curves.

A Lie applicable surface with one spherical curvature line is generated by a constrained elastic curve.

Constrained elastic curves are characterized via polynomial conserved quantities in Lie sphere geometry.

Abstract

In this paper we consider surfaces with one or two families of spherical curvature lines. We show that every surface with a family of spherical curvature lines can locally be generated by a pair of initial data: a suitable curve of Lie sphere transformations and a spherical Legendre curve. We then provide conditions on the initial data for which such a surface is Lie applicable, an integrable class of surfaces that includes cmc and pseudospherical surfaces. In particular we show that a Lie applicable surface with exactly one family of spherical curvature lines must be generated by the lift of a constrained elastic curve in some space form. In view of this goal, we give a Lie sphere geometric characterisation of constrained elastic curves via polynomial conserved quantities of a certain family of connections.