Geometrical Structure of Two-Dimensional Crystals with Non-Constant Dislocation Density

We outline mathematical methods which seem to be necessary in order to discuss crystal structures with non-constant dislocation density tensor (ddt) in some generality. It is known that, if the ddt is constant (in space), then material points can be identified with elements of a certain Lie group, with group operation determined in terms of the ddt-the dimension of the Lie group equals that of the ambient space in which the body resides, in that case. When the ddt is non-constant, there is also a relevant Lie group (given technical assumptions), but the dimension of the group is strictly greater than that of the ambient space. The group acts on the set of material points, and there is a non-trivial isotropy group associated with the group action. We introduce and discuss the requisite mathematical apparatus in the context of Davini's model of defective crystals, and focus on a particular case where the ddt is such that a three dimensional Lie group acts on a two dimensional crystal state-this allows us to construct corresponding discrete structures too.