Geodesic loops on tetrahedra in spaces of constant sectional curvature

Abstract

AbstractGeodesic loops on tetrahedra were studied only for the Euclidean space and it was known that there are no simple geodesic loops on regular tetrahedra. Here we prove that: 1) In the spherical space, there are no simple geodesic loops on tetrahedra with internal angles $$\pi/3 < a_i<\pi/2$$ π / 3 < a i < π / 2 or regular tetrahedra with $$a_i=\pi/2$$ a i = π / 2 , and there are three simple geodesic loops for each vertex of a tetrahedra with $$a_i > \pi/2$$ a i > π / 2 and the lengths of the edges $$a_i>\pi/2$$ a i > π / 2 . 2) We obtain also a new theorem on simple closed geodesics: If the angles $$a_i$$ a i of the faces of a tetraedron satisfy $$\pi/3 < a_i<\pi/2$$ π / 3 < a i < π / 2 and all faces of the tetrahedron are congruent, then there exist at least $$3$$ 3 simple closed geodesics. 3) In the hyperbolic space, for every regular tetrahedron $$T$$ T and every pair of coprime numbers $$(p,q)$$ ( p , q ) , there is one simple geodesic loop of type $$(p,q)$$ ( p , q ) through every vertex of $$T$$ T . The geodesic loops that we have found on the tetrahedra in the hyperbolic space are also quasi-geodesics.