Darya Sukhorebska, B. Verkin Institute for Low Temperature Physics and Engineering of NAS of Ukraine

Title: Simple closed geodesics on tetrahedra in spaces of constant curvature

Abstract: In Euclidean space the Gaussian curvature of the faces of a tetrahedron is zero and the curvature of the tetrahedron concentrated only on its vertices. A complete classification of closed geodesics on a regular tetrahedron in Euclidean three-dimensional space follows from a tiling of Euclidean plane with regular triangles.

However, in spherical or hyperbolic space, the faces of the tetrahedron have Gaussian curvature 1 or -1 respectively. The curvature of a tetrahedron is determined not only by its vertices, but also by its faces. The intrinsic geometry of a tetrahedron depends on the planar angle. Thus the behavior of closed geodesics on a regular tetrahedron in three dimensional spaces of constant curvature k differ depending on the sign of k.

In this talk we present the full classification of simple closed geodesics on regular tetrahedra in spherical and hyperbolic spaces and show the estimates for the number of these geodesics depending on the planar angle of the tetrahedron.

This is joint work with Alexandr A. Borisenko