## Tsukuba Journal of Mathematics

### Surfaces with simple geodesics

Yury Nikolayevsky

#### 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 $S^{2}$.

#### Article information

Source
Tsukuba J. Math., Volume 24, Number 2 (2000), 233-247.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496164147

Digital Object Identifier
doi:10.21099/tkbjm/1496164147

Mathematical Reviews number (MathSciNet)
MR1818084

Zentralblatt MATH identifier
1023.53006

#### Citation

Nikolayevsky, Yury. Surfaces with simple geodesics. Tsukuba J. Math. 24 (2000), no. 2, 233--247. doi:10.21099/tkbjm/1496164147. https://projecteuclid.org/euclid.tkbjm/1496164147