Simple closed geodesics on regular tetrahedra in spherical space
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 $α_1$ and $α_2$ depending on $p$, $q$ and satisfying the inequalities $π/3< α_1 < α_2 < 2π/3$ such that on a regular tetrahedron in spherical space with the faces angle $α\in \left( π/3, α_1 \right)$ 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 $α\in \left( α_2, 2π/3 \right)$ there is no simple closed geodesic of type $(p,q)$.
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