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.