Curved Spaces and Geometrical Frustration

Abstract

Regular structures are such that no contradiction exists between local and global requirements in which case the global approach (with symmetry groups) reveals to be very powerful. In less regular structures, the local configuration may be viewed in some cases as the discrete analog of a quantity which is the local curvature. Defining an ideal structure where the local configuration can propagate, is then equivalent to finding a new geometry with the appropriate distribution of curvature. If such geometry allows for a global description, this ideal model is again regular and can be studied on its own. The relation between the initial structure and the ideal one is studied under different types of mapping. This point of view is called the “curved space model” of disordered systems and will be discussed here.