This talk revolves around the question of how close is one Riemannian manifold to being isometrically immersible in another.
We associate with every mapping $f:(M,g) \to (N,h)$ a measure of distortion - an average distance of $df$ from being an isometry. Reshetnyak's theorem states that a sequence of mappings between Euclidean domains whose distortion tends to zero has a subsequence converging to an isometry.
I will present a generalization of Reshetnyak’s theorem to the general Riemannian setting.