## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Principal configurations and umbilicity of submanifolds in $\mathbb R^N$

#### 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 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 11, Number 2 (2004), 227-245.

Dates
First available in Project Euclid: 11 June 2004

https://projecteuclid.org/euclid.bbms/1086969314

Digital Object Identifier
doi:10.36045/bbms/1086969314

Mathematical Reviews number (MathSciNet)
MR2080424

Zentralblatt MATH identifier
1074.53005

#### Citation

Moraes, S.M.; Romero-Fuster, M.C.; Sánchez-Bringas, F. Principal configurations and umbilicity of submanifolds in $\mathbb R^N$. Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 2, 227--245. doi:10.36045/bbms/1086969314. https://projecteuclid.org/euclid.bbms/1086969314