The vectorial Ribaucour transformation for submanifolds and applications

• Autores: Marcos Dajczer , Luis A. Florit, Ruy Tojeiro
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 10, 2007, págs. 4977-4997
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04211-0
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• Resumen

  • 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 mby means of a commuting family of mHessian matrices on an open subset of Euclidean space Rn. 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.