Geodesic loops on tetrahedra in spaces of constant sectional curvature

2023/08/03 by Borisenko, Alexander A., Miquel, Vicente
#52B10 #53C22 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.01699

Abstract

Geodesic loops on polyhedra were studied only for Euclidean space and it was known that there are no simple geodesic loops on regular tetrahedra. Here we prove that: 1) On the spherical space, there are no simple geodesic loops on tetrahedra with internal angles π/3 < α_i<π/2 or regular tetrahedra with α_i=π/2, and there are three simple geodesic loops for each vertex of a tetrahedra with α_i > π/2 and the lengths of the edges alpha_i>π/2. 2) On the hyperbolic space, for every regular tetrahedron T and every pair of coprime numbers (p,q), there is one simple geodesic loop of (p,q) type through every vertex of T.

Related

• Simple closed geodesics on regular tetrahedra in spaces of constant curvature
• Simple closed geodesics on regular spherical polyhedra
• Close geodesics on regular tetrahedra in hyperbolic space
• Geodesic trajectories on regular polyhedra
• Closed simple geodesics on a polyhedron