Geodesic
Simple closed geodesics on regular tetrahedra in spherical space
Sbornik. Mathematics, Tome 212 (2021) no. 8, pp. 1040-1067

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We prove that there are finitely many simple closed geodesics on regular tetrahedra in spherical space. Also, for any pair of coprime positive integers $(p,q)$, we find constants $\alpha_1$ and $\alpha_2$ depending on $p$ and $q$ and satisfying the inequality $\pi/3\alpha_1\alpha_22\pi/3$, such that a regular spherical tetrahedron with planar angle $\alpha\in(\pi/3, \alpha_1)$ has a unique simple closed geodesic of type $(p,q)$, up to tetrahedron isometry, whilst a regular spherical tetrahedron with planar angle $\alpha\in(\alpha_2, 2\pi/3)$ has no such geodesic. Bibliography: 19 titles.
Keywords: closed geodesics, regular tetrahedron, spherical space. 
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     title = {Simple closed geodesics on regular tetrahedra in spherical space},
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A. A. Borisenko; D. D. Sukhorebska. Simple closed geodesics on regular tetrahedra in spherical space. Sbornik. Mathematics, Tome 212 (2021) no. 8, pp. 1040-1067. http://geodesic.mathdoc.fr/item/SM_2021_212_8_a0/