Simple closed geodesics on regular tetrahedra in spherical space

TL;DR

This paper investigates the existence and classification of simple closed geodesics on regular tetrahedra in spherical space, establishing conditions based on face angles and integer pairs, and identifying when such geodesics exist or do not.

Contribution

It provides a complete characterization of simple closed geodesics on spherical tetrahedra depending on face angles and coprime integer pairs, including existence and uniqueness results.

Findings

Existence of simple closed geodesics depends on face angle ranges.

For each coprime pair (p,q), specific angle bounds determine geodesic existence.

Uniqueness of geodesics is established within certain angle intervals.

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 $(p,q)$.