1. Introduction

Abstract

Since the time of Gauss, parametrized surfaces (x, y) ↦ F(x,y) in differential geometry have been described through a moving frame Ψ(x,y) attached to the surface. One introduces the Gauss-Weingarten equations, which are linear differential equations Ψ_x = UΨ, Ψ_y = VΨ, for the frame, and their compatibility condition U _y - V _x + [U,V] = 0, which represents the Gauss-Codazzi equations. For surfaces in a three-dimensional Euclidean space, the frame Ψ usually lies in the group SO(3) or SU(2).