Necessary condition for the existence of a simple closed geodesic on a regular tetrahedron in the spherical space

1Sukhorebska, DD
1B. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv
Dopov. Nac. akad. nauk Ukr. 2020, 10:9-14
Section: Mathematics
Language: English

In the spherical space the curvature of the tetrahedron’s faces equals 1, and the curvature of the whole tetrahedron is concentrated into its vertices and faces. The intrinsic geometry of this tetrahedron depends on the value α of faces angle, where π/3 < α ⩽ 2π/3. The simple (without points of self-intersection) closed geodesic has the type (p,q) on a tetrahedron, if this geodesic has p points on each of two opposite edges of the tetrahedron, q points on each of another two opposite edges, and (p+q) points on each edges of the third pair of opposite one. For any coprime integers (p,q), we present the number αp, q (π/3 < αp, q < 2π/3) such that, on a regular tetrahedron in the spherical space with the faces angle of value α > αp, q, there is no simple closed geodesic of type (p,q)

Keywords: closed geodesic, regular tetrahedron, spherical space

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