Convex polyhedra without simple closed geodesics (Q1407953)

The author proves that in contrast to the case of smooth surfaces the most of convex polyhedra homeomorphic to a two-sphere have no simple-closed geodesics. The idea is as follows. By the Gauss-Bonnet theorem, a closed geodesic splits a smooth two-sphere into two pieces with integral curvatures equal to \(2\pi\). In the polyhedral case the integral curvature is replaced by the sum of defects at vertices lying in the domain where the defect of a vertex equals \(2\pi\) minus the sum of angles of faces at this vertex. In particular, following this analogy the author proves that if for any proper subset of vertices of a tetrahedron the sum of their defects is rationally independent of \(\pi\) then such a tetrahedron has no simple closed geodesics and proves nonexistence results for general combinatorial structures. The author also studies stability of closed geodesics on polyhedra applying such results to the stability problem for closed billiard trajectories.