The surfaces capable of division into Infinitesimal Squares by their Curves of Curvature

TL;DR

This paper explores the classical characterization of isothermic surfaces as those divisible into infinitesimal squares by curvature lines, aiming to bridge the gap between discrete and smooth differential geometry.

Contribution

It provides a rigorous modern differential geometric interpretation of the classical characterization of isothermic surfaces.

Findings

Classical characterization of isothermic surfaces is formalized in modern terms

Connections between discrete and smooth theories are clarified

Framework for understanding curvature line divisions in differential geometry

Abstract

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 definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory better, it is described how to give the classical characterization a rigorous meaning in the sense of modern differential geometry.