Properties and discretization of the twelve surfaces of Darboux

Abstract

The present thesis deals with the differential geometric problem of deformations of immersions preserving length in first order. Such deformations are called in-finitesimal isometric deformations. Darboux summarized an immersion and one of its infinitesimal isometric deformations as a tuple and constructed a new pair of immersion and its infinitesimal isometric deformation out of this tuple. After six iteration steps, we end up back at the initial tuple, which gives us a closed Darboux wreath consisting of 12 immersions. These immersions have fascinating geometric and algebraic properties in relation to each other, whose mesmerizing feature finally unfolds as differential triality. We will summarize all this and look at it from a new modern point of view. The theory will be complemented by its discrete counterpart, the discrete differential triality. The aim is to better understand infinitesimal iso-metric deformations in the smooth and discrete cases and to provide an important basis for generalizations.