  Intrinsic nonlinear elasticity deals with the deformations of elastic bodies
as isometric immersions of Riemannian manifolds into the Euclidean spaces (see
Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties
of elastic bodies for the intrinsic approach to nonlinear elasticity. We first
establish a geometric rigidity estimate for mappings from Riemannian manifolds
to spheres (in the spirit of Friesecke-James-M\"{u}ller [23]), which is the
first result of this type for the non-Euclidean case as far as we know. Then we
prove the asymptotic rigidity of elastic membranes under suitable geometric
conditions. Finally, we provide a simplified geometric proof of the continuous
dependence of deformations of elastic bodies on the Cauchy-Green tensors and
second fundamental forms, which extends the Ciarlet-Mardare theorem in [18] to
arbitrary dimensions and co-dimensions.
