Close geodesics on regular tetrahedra in hyperbolic space

TL;DR

This paper establishes necessary conditions for simple closed geodesics on regular tetrahedra in hyperbolic space and explicitly classifies three types of such geodesics, each unique up to rigid motion.

Contribution

It introduces necessary conditions for geodesics and explicitly describes three classes of simple closed geodesics on hyperbolic tetrahedra, with uniqueness up to rigid motion.

Findings

Identified necessary conditions for simple closed geodesics.

Explicitly described three classes of geodesics: 2-homogeneous, 3-homogeneous, and (3,2)-homogeneous.

Proved uniqueness of each class up to rigid motion.

Abstract

In this paper we present a necessary conditions, that simple close geodesics on regular tetrahedra in the 3-dimensional hyperbolic space must satisfy. Furthermore, we explicitly describe three classes of simple closed geodesics on regular tetrahedra in the hyperbolic 3-space. These are so-called 2-homogeneous, 3-homogeneous and (3,2)-homogeneous geodesics. Up to a rigid motion of a tetrahedron there exists a unique geodesic in each class.