Crystalline solids are descibed by a material manifold endowed
with a certain structure which we call crystalline. This is characterized by
a canonical 1-form, the integral of which on a closed curve in the material manifold
represents, in the continuum limit, the sum of the Burgers vectors of all the dislocation lines
enclosed by the curve. In the case that the dislocation distribution is uniform, the material manifold
becomes a Lie group upon the choice of an identity element. In this talk crystalline solids
with uniform distributions of the two elementary kinds of dislocations, edge and screw dislocations,
shall be considered. These correspond to the two simplest non-Abelian Lie groups, the affine group
and the Heisenberg group respectively. The statics of a crystalline solid are described in terms of a
mapping from the material domain into Euclidean space. The equilibrium configurations correspond
to mappings which minimize a certain energy integral. The static problem is solved in the case of
a small density of dislocations.