  The appealing connection between non-Euclidean geometries and defects in
solids is brought forth in this article. Drawing a correspondence between the
nature of a defect and a specific geometric property of the material space not
only illuminates the underlying structure of defects in solids but also
provides an unambiguous way to represent defect densities within a physical
theory. We present a rigorous background of the relevant concepts from
classical differential geometry, as well as illustrations of isolated defects
in a lattice, to motivate the relationship between continuous defect densities
and various tensor fields in differential geometry. We identify the
Riemann-Christoffel tensor (or curvature tensor), the Cartan tensor (or torsion
tensor), and the nonmetricity tensor (obtained from the covariant derivative of
the metric tensor), associated with the material space, with the density of
disclinations, dislocations, and point-defects (vacancies, interstitials,
substitutional), respectively. We end our discussion with remarks on the
elastic stress field associated with defect distribution and on the analogy
between the present theory and the general theory of relativity.
