Rigidity of Brillouin Zones

In general, geometric properties of a manifold are not determined by topological invariants of this manifold. Starting in the 1960’s, however, a number of fascinating results have been proved that show that, under certain conditions, the topology of a manifold can determine its geometry. In this case, one often speaks of rigidity. The prototype rigidity theorem is due to Mostow [1]. This result, also known as the strong rigidity theorem, can be stated as Theorem 1.1 (Mostow’s Rigidity Theorem, 1968). Suppose M and N are closed manifolds of constant sectional curvature −1 with the dimension of M is at least 3. If 1(M) = 1 N), then M and N are isometric. In this thesis we study the rigidity of the focal decomposition of the flat 2-torus, as introduced and studied in [2]. We show that the focal decomposition determines the torus up to conformal equivalence.