Geodesic loops on tetrahedra in spaces of constant sectional curvature

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 $\pi/3 < \alpha_i<\pi/2$ or regular tetrahedra with $\alpha_i=\pi/2$, and there are three simple geodesic loops for each vertex of a tetrahedra with $\alpha_i > \pi/2$ and the lengths of the edges $alpha_i>\pi/2$. 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 $T$.