  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 a Riemannian manifold
to a sphere (in the spirit of Friesecke--James--M\"{u}ller [20]), 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 [17] to
arbitrary dimensions and co-dimensions.
