Application: Smooth and discrete isothermic surfaces

Abstract

We already touched on isothermic surfaces and their transformation theory in Chapter 3 (see, in particular, Section 3.3). Here we will elaborate on the transformation theory in more detail, using the just-presented quaternionic model for Mobius differential geometry that seems particularly well adapted to the topic.) To make the current presentation relatively independent from our previous developments, we will start over from the beginning — however, it may be helpful to compare the viewpoint from Section 3.3 with the one elaborated here. In particular, we will extend our definitions and facts from Section 3.3 to higher codimension, enabling the codimension 2 feature provided by the quaternionic model. We already gave some details of the history of isothermic surfaces in Chapter 3. Here we should add that L. Bianchi already gave very comprehensive expositions of the transformation theory of isothermic surfaces, mainly presented in his papers [21] and [20]. There, he does not only introduce the “ T -transformation,” which was independently introduced by P. Calapso [53] and [55], but he also gives an overview of all known transformations, including their interrelations in terms of “permutability theorems.” In fact, there are authors who consider Bianchi's research on transformations and permutability theorems as his most important contribution to geometry (cf., [163]).