The Ribaucour transformation in Lie sphere geometry

We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi’s original Permutability Theorem and its generalisation by Dajczer–Tojeiro. We go on to formulate and prove a higher dimensional version of the Permutability Theorem. It is shown how these theorems descend to the corresponding results for submanifolds in space forms.