Convex Polyhedra Without Simple Closed Geodesics (original) (raw)

In 1905, in his famous article “Sur les lignes géodésiques des surfaces convexes” [1] H. Poincaré posed a question on the existence of at least three geometrically different closed geodesics without self-intersections on any smooth convex two-dimensional surface (2-surface) M homeomorphic to the two-dimensional sphere (2-sphere) S2. Each such geodesic splits the surface M into two domains homeomorphic to 2-discs, and it is naturally to call it simple geodesic. In 1917, G.D. Birkhoff proved [2] the existence of at least one closed simple geodesic on M (in the late 20s he extended the result to the multidimensional case [3]). Nowadays this geodesic is called the (Birkhoff) “equator”. The presence of the Birkhoff equator serves as a basis for proving the existence of infinitely many (non-simple) closed geodesics on the considered surface. This very recent result has been established by V. Bangert [4] and J. Franks [5] in 1991-1992. In 1929 L.Luysternik and L.Shnirel’man gave a proof [6...