Simple closed geodesics on regular tetrahedra in spherical space

Abstract

On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics. For any pair of coprime integers $(p,q)$ it was found the numbers ${\alpha }_1$ and ${\alpha }_2$ depending on $p$, $q$ and satisfying the inequalities $\pi /3<{\alpha }_1<{\alpha }_2<2\pi /3$ such that on a regular tetrahedron in spherical space with the faces angle $\alpha \in \left(\pi /3,{\alpha }_1\right)$ there exists unique, up to the rigid motion of the tetrahedron, simple closed geodesic of type $(p,q)$, and on a regular tetrahedron with the faces angle $\alpha \in \left({\alpha }_2,2\pi /3\right)$ there is no simple closed geodesic of type $(p,q)$.

23 pages, 19 figures