Principal configurations and umbilicity of submanifolds in \(\mathbb R^N\) (Q1772522)

In this paper, the authors consider the principal configurations associated to smooth vector fields \(v\) 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 \(v\). Provided that the umbilic curvature is constant, this will imply that \(M\) is hyperspherical. They deduce some results concerning binormal fields and asymptotic directions for manifolds of codimension two. Moreover, in the case of surfaces \(M\) in \(\mathbb{R}^n\), they conclude that if \(n> 4\), it is always possible to find some normal field with respect to which \(M\) is umbilic and provides a geometrical characterization of such fields.