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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
54140
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
Received by editor(s):
August 3, 2005
Published electronically:
May 7, 2007
Article copyright:
© Copyright 2007
American Mathematical Society