  Elasticity between manifolds concerns the study of the "most isometric" way
of immersing one Riemannian manifold $(M,g)$ (the elastic body) into another
$(S,s)$ (ambient space). We study the limit problem of weak-incompatibility: We
obtain, via $\Gamma$-convergence, a limit model for a sequence of bodies
$(M,g_\varepsilon)$ when $g_\varepsilon \to s$, and relate the minimum of the
limit energy to a linearized curvature discrepancy between $g_\varepsilon$ and
$s$, using recent results of Kupferman and Leder. The main technical challenge
is obtaining the correct notion of displacement for manifold-valued
configurations, using Sobolev truncations and parallel transport. We further
show that the associated compactness result is obtained if $(S,s)$ satisfies a
quantitative rigidity property, analogous to the Friesecke--James--M\"uller
rigidity estimate in Euclidean space, and show that this property holds when
$(S,s)$ is a round sphere.
