Natural PDE's of Linear Fractional Weingarten surfaces in Euclidean Space

TL;DR

This paper studies the natural partial differential equations (PDEs) associated with linear fractional Weingarten surfaces in Euclidean space, showing their invariance under parallel surface transformations and classifying these PDEs.

Contribution

It proves the invariance of natural principal parameters under parallel surfaces and classifies the natural PDEs for linear fractional Weingarten surfaces.

Findings

Natural principal parameters are preserved under parallel surfaces.

Natural PDEs of a Weingarten surface are shared by its parallel surfaces.

Complete classification of natural PDEs for linear fractional Weingarten surfaces.

Abstract

We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fractional Weingarten surfaces are exactly the surfaces satisfying a linear relation between their three curvatures. Our main result is classification of the natural PDE's of Weingarten surfaces with linear relation between their curvatures.