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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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Additional Information

M. Dajczer
Affiliation: IMPA, Estrada Dona Castroina, 110, 22460-320, Rio de Janeiro, Brazil
Email: marcos@impa.br

L. A. Florit
Affiliation: IMPA, Estrada Dona Castroina, 110, 22460-320, Rio de Janeiro, Brazil
Email: luis@impa.br

R. Tojeiro
Affiliation: Universidade Federal de São Carlos, Via Washington Luiz km 235, 13565-905, São Carlos, Brazil
Email: tojeiro@dm.ufscar.br

DOI: https://doi.org/10.1090/S0002-9947-07-04211-0
Received by editor(s): August 3, 2005
Published electronically: May 7, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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