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 umbilic curvature is constant, this will imply that $M$ 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 $\mathbb R^N$, 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.
Citation
S.M. Moraes. M.C. Romero-Fuster. F. Sánchez-Bringas. "Principal configurations and umbilicity of submanifolds in $\mathbb R^N$." Bull. Belg. Math. Soc. Simon Stevin 11 (2) 227 - 245, June 2004. https://doi.org/10.36045/bbms/1086969314
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