A major difference between elastic manifolds and elastic media is that
in the latter, we must account for the possibility of topological
defects. In the manifolds, the elastic fields were simply generalized
``heights'' of an interface or line, and as such had well-defined
meanings as real-space coordinates. In an elastic medium, the
displacements 
are only defined relative to some background
configuration, or in the continuum limit in terms of their effect upon
the density. We should therefore allow for the possibility of
defects, e.g. vacancies, interstitials, and dislocations in the
lattice. These have the effect of making multivalued.
Configurations of the medium which include topological defects are
said to involve plastic deformations, as opposed to purely
elastic deformations when they are not present.