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 $(p,q)$.