Séance Séminaire

Séminaire Gaston Darboux

A full classification of simple closed geodesics on regular tetrahedra in hyperbolic and spherical spaces.

In Euclidean space, the classification of closed geodesics on a regular tetrahedron follows from a fact that a development of the tetrahedron along the geodesic is contained in a standard triangular tiling of Euclidean plane. Moreover since the faces of a tetrahedron in Euclidean space have zero Gaussian curvature and the curvature of the tetrahedron is concentrated only on its vertices. In hyperbolic or spherical space the curvature of a tetrahedron is determined not only by its vertices, but also by its faces. The intrinsic geometry of the tetrahedron depends on the value of the planar angle. In my talk, I will show how the properties of geodesics change depending on the curvature of an ambient space. Also in general there is no tiling of hyperbolic or spherical plane by regular triangles. I will present new methods that we have developed to find a complete classification of simple closed geodesics on right tetrahedra in hyperbolic and spherical spaces.