Principal configurations and umbilicity of submanifolds in R-N

Abstract

We consider the principal configurations associated to smooth vector fields nu normal to a manifold M immersed into a euclidean space and give conditions on the number of principal directions shared by a set of k normal vector fields in order to guaranty the umbilicity of M with respect to some normal field nu. Provided that the unibilic curvature is constant, this will imply that ill is hyperspherical. We deduce some results concerning binormal fields and asymptotic directions for manifolds of codimension 2. Moreover, in the case of a surface M in R-N, we conclude that if N > 4

it is always possible to find some normal field with respect to which M is umbilic and provide a geometrical characterization of such fields.