Long geodesics on convex surfaces

We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.Comment: 8 pages, 10 figure

Ramification conjecture and Hirzebruch's property of line arrangements

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on the complex projective plane with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a CAT[0] ramification and prove this in several cases. In the latter case we prove that the ramification is CAT[0] if the metric is non-negatively curved. We deduce that complex line arrangements in the complex projective plane studied by Hirzebruch have aspherical complement.Comment: 19 pages 1 figur