A note on Ribaucour transformations in Lie sphere geometry

TL;DR

This paper explores Ribaucour transformations of Legendre submanifolds within Lie sphere geometry, providing explicit parametrizations based on a single real function, advancing understanding of geometric transformations.

Contribution

It offers an explicit parametrization of Ribaucour transformations in Lie sphere geometry using a single real function, building on prior theoretical frameworks.

Findings

Explicit parametrization of Ribaucour transformations

Representation of the transformation via a single real function

Enhanced understanding of sphere congruences in Lie geometry

Abstract

Following Burstall and Hertrich-Jeromin we study the Ribaucour transformation of Legendre submanifolds in Lie sphere geometry. We give an explicit parametrization of the resulted Legendre submanifold $\hat{F}$ of a Ribaucour transformation, via a single real function $\tau$ which represents the regular Ribaucour sphere congruence $s$ enveloped by the original Legendre submanifold $F$.