Weingarten surfaces with moving frames -- a tribute to S.S. Chern and C.L. Terng -- and a duality result

TL;DR

This paper explores Weingarten surfaces using exterior calculus, illustrating methods with quadratic exterior equations, and reveals a duality between isothermic surfaces of constant astigmatism and those satisfying a specific curvature relation.

Contribution

It introduces a novel approach using Maurer-Cartan forms to analyze Weingarten surfaces and establishes a duality between certain classes of isothermic surfaces.

Findings

Weingarten surfaces can be studied via exterior calculus techniques.

Isothermic surfaces of constant astigmatism are dual to surfaces satisfying H + αK = 0.

The methods apply to quadratic exterior equations with constant coefficients.

Abstract

The techniques used in this paper are based on the exterior calculus of Maurer-Cartan forms, and Weingarten surfaces are used to illustrate the methods that apply to quadratic exterior equations with constant coefficients. Isothermic {\it surfaces of constant astigmatism} (non-linear Weingarten surfaces whose difference of principal curvatures is a constant) are shown to represent dual surfaces of isothermic surfaces which satisfy the relation $H + \alpha K = 0$.