Ribaucour transformations revisited

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 definition introduced in this paper, 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 multiplicities bigger than one. We characterize this transformation in terms of differential equations and we study some of its properties. We show that an n-dimensional sphere or hyper-plane can be locally associated by a Ribaucour transformation to any given hypersurface Mn of Rn+1, which admits n orthogonal principal direction vector fields. As an application of Ribaucour transformations, we characterize the Dupin hypersurfaces which have a principal curvature of constant multiplicity one, as a manifold foliated by (n -1)-dimensional Dupin submanifolds associated by Ribaucour transformations.