Preprint A31/2006
RIGIDITY OF BRILLOUIN ZONES

M.M. PEIXOTO | KWAKKEL, F.H. | MARTENS, M.

**Keywords: **
Brillouin zones; Flat torus; Focal decomposition.

In general, topological characteristics of manifolds do not determine the geometry. However, starting in the 1960's, examples have been discovered for which such characteristics do determine the geometry. The manifolds can not be deformed without changing the characteristic. The Mostow rigidity theorems are examples of this phenomenon. The universality observed in one-dimensional dynamics also leads to rigidity results.
The classical Brillouin zones were introduced by Brillouin in the quantum study of wave propagation in crystals. Here we will not go into the physical meaning but interpret the Brillouin zones as characteristics of two-dimensional flat tori. The main results will be rigidity theorems related to Brillouin zones, focal decompositions and torus puzzles.