Geodesic loops on tetrahedra in spaces of constant sectional curvature

TL;DR

This paper investigates the existence and properties of geodesic loops on tetrahedra in spaces of constant curvature, revealing new results in spherical and hyperbolic geometries that extend prior Euclidean findings.

Contribution

It provides the first comprehensive analysis of geodesic loops on tetrahedra in spherical and hyperbolic spaces, including existence, non-existence, and classification results.

Findings

No simple geodesic loops on certain spherical tetrahedra.

Existence of three simple geodesic loops on tetrahedra with large angles.

In hyperbolic space, existence of loops of specific types for all regular tetrahedra.

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 $T$.