  Almost all available results in elasticity on curved topographies are
obtained within either a small curvature expansion or an empirical covariant
generalization that accounts for screening between Gaussian curvature and
disclinations. In this paper, we present a formulation of elasticity theory in
curved geometries that unifies its underlying geometric and topological content
with the theory of defects. The two different linear approximations widely used
in the literature are shown to arise as systematic expansions in reference and
actual space. Taking the concrete example of a 2D crystal, with and without a
central disclination, constrained on a spherical cap, we compare the exact
results with different approximations and evaluate their range of validity. We
conclude with some general discussion about the universality of non-linear
elasticity.
