2005/08/12 by L. Florit, Luis A. Florit, Florit, L. +6
Mathematics · Physics and Astronomy · #53B25 #58J72 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #math.DG #msc:53B25 #msc:58J72
paper · pdf · doi:10.48550/arxiv.math/0508217
24 pages
arxiv created 2005/08/12 · openalex publication_date 2005/08/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 an explicit local construction of all Euclidean submanifolds with flat normal bundle. Actually, this is a particular case of a more general result. Namely, we obtain a local explicit 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.