An extension of the classical Ribaucour transformation
Ruy Tojeiro | Dajczer, Marcos
The classical Ribaucour transformation is extended to submanifolds with arbitrary dimension and codimension of a pseudo-Riemannian space form. Ribaucour transforms of a submanifold are shown to be in correspondence with commuting symmetric Codazzi tensors. A main application is a parameterization, in terms of solutions of a completely integrable linear first order system of PDE's, of all Ribaucour transforms of a given submanifold with constant sectional curvature which have the same constant curvature. In particular, many explicit examples are obtained, some of them are the first of their kind. A permutability result is proved which allows to produce further examples by a simple algebraic procedure once two Ribaucour transforms of a given submanifold are known.