An Extension of the Classical Ribaucour Transformation

Abstract

In this paper we extend the notion of Ribaucour transformation from classical surface theory to the theory of holonomic submanifolds of pseudo-Riemannian space forms with arbitrary dimension and codimension, that is, submanifolds with at normal bundle admitting a global system of principal coordinates. Named and extensively used by Bianchi [6], this transformation is a correspondence preserving lines of curvature between the focal surfaces of a two-parameter congruence of spheres.In the process of proving the converse of Guichard's theorem on the deformations of quadrics of revolution, Bianchi gave a positive answer to the question of whether, among the Ribaucour transforms of a surface with constant mean or Gaussian curvature, there exist other surfaces with the same property. One of our main achievements is to solve the same problem for the class of holonomic submanifolds with constant sectional curvature. As Bianchi did in the surface case, we are able to write down a parameterization, in terms of solutions of a completely integrable linear ®rst order system of partial differential equations (PDEs), of all Ribaucour transforms of a given holonomic submanifold with constant sectional curvature that also have the same constant curvature.It is a classical fact that surfaces in R 3 of constant negative curvature and umbilic-free surfaces of constant positive curvature are in correspondence with solutions of the sine-Gordon and elliptic sinh-Gordon equations, respectively. Similar correspondences between n-dimensional submanifolds of constant sectional curvature c T 0 in R 2n À 1 (free of weak-umbilics when c > 0) and solutions of completely integrable systems of partial differential equations, named the generalized sine-Gordon equation if c < 0 and the generalized elliptic sinh-Gordon equation if c > 0, were obtained in [2], [11] and [20], and extended to broader classes of submanifolds in [3], [4], [11], [19] and [20]. Here, we further generalize these results by allowing the codimension of the submanifolds to be arbitrary.The analytical counterpart of Bianchi's result is a method for generating a family of new solutions of the sine-Gordon and elliptic sinh-Gordon equations from a given one. Similarly, our results yield a process for constructing a family of new solutions of the aforementioned systems of PDEs starting from a given one. Moreover, the new solutions are explicitly computed in terms of the former and the solutions of a linear ®rst order system of PDEs.Basically, the same method for a similar class of systems of PDEs was developed by Bianchi as a result of his transformation theory for n-orthogonal systems in manifolds of constant sectional curvature (see [6]). The close relation between his analytical results and ours is due to the fact that some of these n-orthogonal systems can be realized as principal coordinate systems of holonomic submanifolds of constant sectional curvature.We apply our method to produce new parameterized examples of (noncomplete!) n-dimensional submanifolds with constant secti...

On Ribaucour transformations and applications to linear Weingarten surfaces

“…In that case, the system of equations (which appear in [Corro 1997, Corro et al 1999, Dajczer and Tojeiro 2002) is indeed equivalent to the definition. Moreover, for submanifolds which admit principal curvatures with multiplicity bigger than one, in any dimension or codimension, the choice of distinct set of orthonormal principal directions may provide, by solving the system of equations, distinct families of submanifolds associated by Ribaucour transformations (see Remark 2.11).…”

“…In the classical theory and in both (Corro 1997) and (Dajczer and Tojeiro 2002), the definition is characterized essencially by the same integrable system of differential equations, whose solutions provide immersions locally associated by Ribaucour transformations to a given immersion. However, one can show (see Corollary 2.10 and also Corro et al 1999) that this procedure does not always preserve multiplicity of principal curvatures.…”

“…The revised definition was introduced in (Corro and Tenenblat 2002), motivated by a discussion of the classical definition of a Ribaucour transformation for hypersurfaces and some more recent extensions to submanifolds with higher codimension and flat normal bundle (Corro 1997, Dajczer andTojeiro 2002). The new definition provides a precise treatment of the geometric aspect of such transformations preserving lines of curvature and it also extends Ribaucour transformations to submanifolds whose principal curvatures have multiplicity bigger than one.…”

Ribaucour Transformations Revisted

“…More recently, two distinct extensions were considered by Corro [Co] and , for submanifolds, with flat normal bundle and parametrized by lines of curvature. We point out that a version of the definition in [DT1], for nonholonomic submanifolds, was introduced in [DT2] and it was used in [BDPT].…”

Transformations of Flat Lagrangian Immersions and Egoroff Nets

“…Mañas, Alonso, and Medina used dressing methods for multicomponent KP hierarchies in [9,10] to construct dressing actions for Egoroff orthogonal nets. Dajczer-Tojeiro generalized sphere congruence and Ribaucour transformations to submanifolds in space-forms in a series of papers [4,5], and they found Ribaucour transformations for flat Lagrangian submanifolds in C n and CP n .…”