## Abstract

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map f: M→ 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→ 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 (J Math Pures Appl 1850) and Reshetnyak (Sib Mat Zhurnal 8(1):91–114, 1967) from Euclidean to Riemannian settings. Finally, we describe applications of these theorems to non-Euclidean elasticity and to convergence notions of manifolds.

Original language | American English |
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Pages (from-to) | 367-408 |

Number of pages | 42 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 231 |

Issue number | 1 |

DOIs | |

State | Published - 22 Jan 2019 |

### Bibliographical note

Funding Information:This research was partially supported by the Israel Science Foundation (Grant No. 1035/17), and by a grant from the Ministry of Science, Technology and Space, Israel and the Russian Foundation for Basic Research, the Russian Federation.

Publisher Copyright:

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.