Related papers: Simple closed geodesics on regular spherical polyh…
On a regular tetrahedron in spherical space there exist the finite number of simple closed geodesics. For any pair of coprime integers $(p,q)$ it was found the numbers $\alpha_1$ and $\alpha_2$ depending on $p$, $q$ and satisfying the…
In this survey results on the behavior of simple closed geodesics on regular tetrahedra in three-dimensional spaces of constant curvature are presented.
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…
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…
A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le \pi$ surface to each side at every point. In contrast, a geodesic has exactly $\pi$ to each side and so can never pass through a vertex, whereas quasigeodesics…
We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as…
It is well-known that every isosceles tetrahedron (disphenoid) admits infinitely many simple closed geodesics on its surface. They can be naturally enumerated by pairs of co-prime integers $n > m > 1$ with two additional cases $(1,0)$ and…
Pogorelov proved in 1949 that every every convex polyhedron has at least three simple closed quasigeodesics. Whereas a geodesic has exactly pi surface angle to either side at each point, a quasigeodesic has at most pi surface angle to…
We determine (non-necessarily convex) polyhedra having simple dense geodesics.
This article deals with the set of closed geodesics on complete finite type hyperbolic surfaces. For any non-negative integer $k$, we consider the set of closed geodesics that self-intersect at least $k$ times, and investigate those of…
We give existence results for simple closed curves with prescribed geodesic curvature on $S^{2}$, which correspond to periodic orbits of a charge in a magnetic field.
We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.
We study the existence of simple closed geodesics on most (in the sense of Baire category) Alexandrov surfaces with curvature bounded below, compact and without boundary. We show that it depends on both the curvature bound and the topology…
We prove that the geodesic complexity of a regular tetrahedron exceeds its topological complexity by 1 or 2. The proof involves a careful analysis of minimal geodesics on the tetrahedron.
Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern--Brocot tree to explore the recursive…
A closed quasigeodesic on a convex polyhedron is a closed curve that is locally straight outside of the vertices, where it forms an angle at most $\pi$ on both sides. While the existence of a simple closed quasigeodesic on a convex…
We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.
We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of…
In this article, we describe symplectic and complex toric spaces associated to the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.