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...

We give an explicit construction of any simply connected superconformal surface φ : M 2 → R 4 in Euclidean space in terms of a pair of conjugate minimal surfaces g, h : M 2 → R 4 . That φ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs (g, h) of conjugate minimal surfaces that give rise to images of holomorphic curves by an inversion in R 4 and to images of superminimal surfaces in either a sphere S 4 or a hyperbolic space H 4 by an stereographic projection. We also determine the relation between the pairs (g, h) of conjugate minimal surfaces associated to a superconformal surface and its image by an inversion. In particular, this yields a new transformation for minimal surfaces in R 4 .

Abstract. We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of S n × R, extending the classification of umbilical surfaces in S 2 × R by Souam and Toubiana as well as the local description of umbilical hypersurfaces in S n ×R by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic S 1 ×R or S 2 ×R, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of S n × R and H n × R. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor R is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in S n × R and H n × R having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of S n × R and H n × R.

[Bil ]. A few years after SBRANA, CARTAN (1) published similar results but in the language of envelopes of hyperplanes.Deformable hypersurfaces can be divided into four classes. Submanifolds belonging to the two less interesting ones, namely, surface-like and ruled hypersurfaces, are highly deformable. On the contrary, while hypersurfaces in one of the remaining classes admit, precisely, a continuous one-parameter family of isometric deformations, elements belonging to the other class have a unique one.The main result in the Sbrana-Cartan theory is a parametric classification of all hypersurfaces in the two most important classes. For the first one, the description turns out to be quite satisfactory in the sense that it enables the construction of many explicit examples; cf. p 22 of [Ca1] and w 4 of this paper. For the remaining class the situation is quite different. First of all, the parametric description is cumbersome. One has to search for surfaces for which a pair of Christoffel symbols associated to a conjugate 9 system of coordinates satisfies a certain complicated system of second order partial differential equations. Thus, it is not surprising that no example comes out from this re-(*) Entrata in Redazione il 10 febbraio 1997 e, in versione riveduta, il 4 giugno 1997. Indirizzo degli AA.: M. DAJCZER: IMPA,

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