The vectorial Ribaucour transformation for submanifolds and applications

Abstract

Abstract. In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under appropriate conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of this result is the classical permutability of Ribaucour transformations. Our main application is to provide an explicit local construction of an arbitrary Euclidean n-dimensional submanifold with flat normal bundle and codimension m by means of a commuting family of m Hessian matrices on an open subset of Euclidean space R n . Actually, this is a particular case of a more general result. Namely, we obtain a similar local construction of all Euclidean submanifolds carrying a parallel flat normal subbundle, in particular of all those that carry a parallel normal vector field. Finally, we describe all submanifolds carrying a Dupin principal curvature normal vector field with integrable conullity, a concept that has proven to be crucial in the study of reducibility of Dupin submanifolds.An explicit construction of all submanifolds with flat normal bundle of the Euclidean sphere carrying a holonomic net of curvature lines, that is, admitting principal coordinate systems, was given by Ferapontov in [8]. The author points out that his construction "resembles" the vectorial Ribaucour transformation for orthogonal systems developed in [13]. The latter provides a convenient framework for understanding the permutability properties of the classical Ribaucour transformation. This paper grew from an attempt to better understand the connection between those two subjects, as a means of unravelling the geometry behind Ferapontov's construction. This has led us to develop a vectorial Ribaucour transformation for Euclidean submanifolds, extending the transformation in [13] for orthogonal coordinate systems. It turns out that any n-dimensional submanifold with flat normal bundle of R n+m can be locally transformed by a suitable vectorial Ribaucour transformation to the inclusion map of an open subset of an n-dimensional subspace of R n+m . Inverting such a transformation yields the following explicit local construction of an arbitrary n-dimensional submanifold with flat normal bundle of R n+m by means of a commuting family of m Hessian matrices on an open subset of R n . Notice that carrying a principal coordinate system is not required.

Ribaucour coordinates

“…Now we let ξ ∈ Γ(x) and iterate(3.3) to obtain an iterated Ribaucour transform of x into the m-sphere e: with a first Ribaucour transformation of x into the i-th hypersphere e i and its parallel normal field obtained by (3.2) from the j-th parallel normal field of x, ξ i = (t i − e i ) + 1−(ei,ti) (ei,ξ) ξ and t ij − , up to scaling, ξ ij is symmetric in i and j, thus confirming Bianchi's permutability theorem for the particular Ribaucour transformations we use, cf[5, Sect 3] or[9, Cor 16]. Once the circularity claim of Bianchi's permutability theorem is established for x, x i , x j and x ij ,…”

“…For example, it is advantageous to employ this type of coordinates to investigate and construct surfaces of constant mean curvature H = 1 in hyperbolic space, cf [12] or [10, §5.5.27]. These classical Ribaucour coordinates were generalized in more recent work to hypersurfaces, see [8,Cor 2.10], and to submanifolds with flat normal bundle, see [9,Thm 1].…”

The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

“…The following is a straightforward extension for nonflat ambient space forms of a decomposition property of the vectorial Ribaucour transformation proved in [2] for Euclidean submanifolds.…”

Hypersurfaces of space forms carrying a totally geodesic foliation

“…Hence there exists Y ∈ X(L) such thatỸ = jY . From ∇ V jX jY = 0 for X ∈ X(L) such that X, Y L = 0, using (5) and (6) we obtain, respectively,…”

“…We recall that all Euclidean surfaces with flat normal bundle can be given explicitly in terms of a set of solutions of a completely integrable first order linear system of PDEs associated to the vectorial Ribaucour transformation as shown in [7] and [5]. In particular, since these do not satisfy any constraint, there exists an abundance of surfaces of type D.…”