Simple closed geodesics on regular tetrahedra in spherical space

A. A. Borisenko   ; D. D. Sukhorebska

Sbornik. Mathematics, Tome 212 (2021) no. 8, pp. 1040-1067

Résumé

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 $\pi/3<\alpha_1<\alpha_2<2\pi/3$, 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.

Détail

Keywords: closed geodesics, regular tetrahedron, spherical space.

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