# Reshetnyak rigidity for Riemannian manifolds

## TL;DR

This paper extends classical rigidity theorems to Riemannian manifolds, proving that maps with differentials close to isometries are themselves isometric immersions, with applications in elasticity and manifold convergence.

## Contribution

It provides new, simplified proofs of Reshetnyak-type rigidity theorems in Riemannian geometry, generalizing Euclidean results and introducing convergence results for sequences of maps.

## Key findings

- Lipschitz maps with differentials almost everywhere isometric are isometric immersions.
- Sequences of maps with differentials converging to isometries have subsequences converging to isometric immersions.
- Applications to non-Euclidean elasticity and manifold convergence.

## Abstract

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.