Isometric Surfaces in Isotropic 3-Space

Abstract

While the notion of isometric deformations of surfaces is straightforward for surfaces with Euclidean metric, a corresponding notion in isotropic space has been missing. By making Gauss' Theorema Egregium a necessary condition we develop a sensible notion of isometric surfaces in isotropic space. The well-known examples in Euclidean space, like isometries within the associated family of minimal surfaces, Bour's theorem, and Minding isometries, find their natural analogues in isotropic space. We also include an extensive treatment of infinitesimal flexibility, or infinitesimal deformation, of surfaces. We prove results for the isotropic displacement diagrams in analogy to its well-known counterparts in Euclidean space culminating in the existence of an isotropic Darboux wreath consisting of six surfaces. We show several interesting relations for special parametrizations involving Koenigs and Voss nets of smooth and discrete surfaces within the Darboux wreath and we encounter surfaces of constant Gaussian and mean curvature. At several occasions, we point to connections to statics as the isotropic space is a natural language to describe the Airy stress function.