Submanifolds with a non-degenerate parallel normal vector field in euclidean spaces

Abstract

We consider the class of submanifolds $M$ in an euclidean space $\mathbb{R}^n$ which admit a non-degenerate parallel normal vector field $\nu$. The image of the associated Gauss map $G_{\nu} : M \to S^{n-1}$ defines an immersed hyperspherical submanifold $M^{\nu}$ which has the following property: if $M$ has a contact of Boardman type $\Sigma^{i_1, \dots, i_k}$ with a hyperplane, then $M^{\nu}$ has the same contact type with the translated hyperplane. In particular, for a space curve $\alpha$ in $\mathbb{R}^3$, the spherical curve $\alpha^{\nu}$ has the same fiattenings and we deduce an extension of the Four Vertex Theorem. For an immersed surface $M$ in $\mathbb{R}^4$, it admits a local non-degenerate parallel normal vector field if and only if it is totally semi-umbilic and has non zero gaussian curvature $K$. Moreover, $G_{\nu}$ preserves the inflections and the asymptotic lines between $M$ and $M^{\nu}$. As a consequence, we deduce an extension for this class of surfaces of the classical Loewner and Carathéodory conjectures for umbilic points of analytic immersed surfaces in $\mathbb{R}^3$.