  We prove two rigidity theorems for maps between Riemannian manifolds. First,
we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian
manifolds, whose differential is almost everywhere an orientation-preserving
isometry, is an isometric immersion. This theorem was previously proved using
regularity theory for conformal maps; we give a new, simple proof, by
generalizing the Piola identity for the cofactor operator. Second, we prove
that if there exists a sequence of mapping $f_n:M\to N$, whose differentials
converge in $L^p$ to the set of orientation-preserving isometries, then there
exists a subsequence converging to an isometric immersion. These results are
generalizations of celebrated rigidity theorems by Liouville (1850) and
Reshetnyak (1967) from Euclidean to Riemannian settings. Finally, we describe
applications of these theorems to non-Euclidean elasticity and to convergence
notions of manifolds.
