Supplement on curved flats in the space of point pairs and isothermic surfaces: A quaternionic calculus

Abstract

A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. It introduces a rich algebraic structure and allows the use of global frames while, at the same time, incorporates the classical “geometric” model of Möbius geometry providing geometric clarity. This way, it provides the foundation for the development of new techniques in Möbius differential geometry.