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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 367–408 | Cite as

Reshetnyak Rigidity for Riemannian Manifolds

  • Raz Kupferman
  • Cy Maor
  • Asaf ShacharEmail author
Article

Abstract

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map \({f:\mathcal{M} \to \mathcal{N}}\) 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 \({f_n:\mathcal{M} \to \mathcal{N}}\), whose differentials converge in Lp 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.

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Notes

Acknowledgements

We are grateful to Pavel Giterman,Amitai Yuval andYael Karshon for useful discussions. We also thank Deane Yang for suggesting the current form of Lemma 3.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

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

Authors and Affiliations

  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada

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