The vectorial Ribaucour transformation for submanifolds and applications

Dajczer, M.; Florit, L.; Tojeiro, R.

doi:10.1090/S0002-9947-07-04211-0

The vectorial Ribaucour transformation for submanifolds and applications

Authors: M. Dajczer, L. A. Florit and R. Tojeiro
Journal: Trans. Amer. Math. Soc. 359 (2007), 4977-4997
MSC (2000): Primary 53B25, 58J72
DOI: https://doi.org/10.1090/S0002-9947-07-04211-0
Published electronically: May 7, 2007
MathSciNet review: 2320656
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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 $\mathbb {R}^n$. 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.

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