We show how pairs of isothermic surfaces are given by curved flats in a pseudo Riemannian symmetric space and vice versa. Calapso’s fourth order partial differential equation is derived and, using a… (More)

This book aims to introduce the reader to the geometry of surfaces and submanifolds in the conformal n-sphere. Various models for Möbius geometry are presented: the classical projective model, the… (More)

We study Darboux and Christoffel transforms of isothermic surfaces in Euclidean space. These transformations play a significant role in relation to integrable system theory. Surfaces of constant mean… (More)

Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are… (More)

Using a quaternionic calculus, the Christoffel, Darboux, Goursat, and spectral transformations for discrete isothermic nets are described, with their interrelations. The Darboux and spectral… (More)

Recently there has been much interest in the theory of discrete surfaces in 3-space and its connection with the discretization of soliton equations (see e.g. [4, 11] and references therein). In this… (More)

We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed… (More)

A Clifford algebra model for Möbius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames… (More)

Classically, isothermic surfaces are characterized as those surfaces which are “divisible into infinitesimal squares by their curvature lines”. This characterization is the direct analogue to the… (More)

Many topics in integrable surface geometry may be unified by application of the highly developed theory of harmonic maps of surfaces into (pseudo-)Riemannian symmetric spaces. On the one hand, such… (More)