  We present a new definition of defects which is based on a Riemannian
formulation of incompatible elasticity. Defects are viewed as local deviations
of the material's reference metric field, $\bar{\mathfrak{g}}$, from a
Euclidian metric. This definition allows the description of defects in
amorphous materials and the formulation of the elastic problem, using a single
field, $\bar{\mathfrak{g}}$. We provide a multipole expansion of reference
metrics that represent a large family of two-dimensional (2D) localized
defects. The case of a dipole, which corresponds to an edge dislocation is
studied analytically, experimentally and numerically. The quadrupole term,
which is studied analytically, as well as higher multipoles of curvature carry
local deformations. These multipoles are good candidates for fundamental strain
carrying entities in plasticity theories of amorphous materials and for a
continuous modeling of recently developed meta-materials.
