Polynomial Weingarten surfaces of tubular type

Abstract

This work seeks to contribute to the classification of Weingarten surfaces. More precisely, it fully classifies three families of surfaces (named tubular, cyclic and canal surfaces) in a tridimensional space form (Euclidean, Lorentzian and Hyperbolic spaces) that verify an arbitrary polynomial relation among its Gaussian and mean curvatures. The results obtained provide geometric features of the surface as well as algebraic conditions over the polynomial that defines a surface as Weingarten. Furthermore, results that allow us to investigate Weingarten surfaces only by the polynomial analysis are presented.