On Ribaucour transformations and applications to linear Weingarten surfaces

Abstract

We present a revised definition of a Ribaucour transformation for submanifolds of space forms, with flat normal bundle, motivated by the classical definition and by more recent extensions.The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it can be applied to submanifolds whose principal curvatures have multiplicity bigger than one. Ribaucour transformations are applied as a method of obtaining linear Weingarten surfaces contained in Euclidean space, from a given such surface. Examples are included for minimal surfaces, constant mean curvature and linear Weingarten surfaces. The examples show the existence of complete hyperbolic linear Weingarten surfaces in Euclidean space.

The Ribaucour transformation in Lie sphere geometry

“…The notion of Ribaucour transform currently available in the literature [4,12,13,19,26] involves a pair of k-dimensional submanifolds enveloping a congruence of k-spheres so that curvature directions of corresponding normals coincide (that is, corresponding shape operators commute). In view of Proposition 4.2, we propose the following…”

“…A persistent and characteristic feature of integrable submanifold geometries is the existence of transformations of solutions. Examples include the Bäcklund transformations of surfaces of constant Gauss curvature and their generalisations [1,13,26]; Darboux transformations of isothermic surfaces [2,9,14,25]; Eisenhart transformations of Guichard surfaces [19, §92]; Jonas transformations of R-congruences 1 , to name but a few.…”

Channel surfaces in Lie sphere geometry

“…By performing a parallel transformation, one obtains a Ribaucour transform of a cylinder. An explicit parametrisation of this Ribaucour transform is given in Tenenblat (2002). The tangent circles between the Ribaucour pair of curves become tori with the same radii as that of the tubular surfaces.…”

Lie applicable surfaces

“…It was shown in [7] that Definition 2.9 is equivalent to the classical definition of Ribaucour transform [1,14,15,21,36], that is, that the curvature directions of f and f correspond. Suppose that f and f are umbilic-free and let s 1 , s 2 ≤ f denote the curvature sphere congruences of f and let ŝ1 , ŝ2 ≤ f denote the curvature sphere congruences of f .…”