Surfaces with simple geodesics

Abstract

A regular submanifold in a Euclidean space $R^{N}$ is called a submanifold with simple geodesics if all its geodesics have constant Frenet curvatures in $R^{N}$. A submanifold with congruent simple geodesics is called helical. We prove that a compact surface with simple geodesics is either a rational torus, or the image of the unit sphere under a polynomial map $F:R^{3}\rightarrow R^{N}$ of the special structure. As a corollary, a compact surface $F^{2}\subset R^{N}$ is helical if $F^{2}=\Phi(S^{2})$, where $\Phi=(a_{1}\Phi_{1}, \ldots, a_{m}\Phi_{m})$ and $\Phi_{j}$ the $i-$th eigenmap of the Laplacian of $S^{2}$.