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


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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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.


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, London (2003)zbMATHGoogle Scholar
  2. 2.
    Agostiniani, V., Lucantonio, A., Lučić, D.: Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets, preprint, 2017Google Scholar
  3. 3.
    Aharoni, H., Kolinski, J.M., Moshe, M., Meirzada, I., Sharon, E.: Internal stresses lead to net forces and torques on extended elastic bodies. Phys. Rev. Lett. 117, 124101 (2016)ADSCrossRefGoogle Scholar
  4. 4.
    Ball, J.M.: A version of the fundamental theorem of Young measures. Proceedings of ``Partial differential equations and continuum models of phase transitions'' Lecture Notes in Physics (Eds. M. Rascle, D. Serre, M. Slemrod), vol. 344, 3–16 1989Google Scholar
  5. 5.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bhattacharya, K., Lewicka, M., Schäffner, M.: Plates with incompatible prestrain. Arch. Rational Mech. Anal. 221(1), 143–181 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Calabi, E., Hartman, P.: On the smoothness of isometries. Duke Math. J. 37(4), 741–750 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ciarlet, P.G.: Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity. Elsevier, Amsterdam, 1988Google Scholar
  9. 9.
    Ciarlet, P.G., Mardare, S.: Nonlinear Korn inequalities in \(\mathbb{R}^n\) and immersions in \(W^{2, p}, p > n\), considered as functions of their metric tensors in \(W^{1, p}\). J. Math. Pures Appl. 105, 873–906 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cohen, M.: Equivalence of intrinsic and extrinsic metrics of embedded manifolds,
  11. 11.
    Croke, C.B.: Rigidity theorems in Riemannian geometry. Geometric Methods in Inverse Problems and PDE Control (Eds. C.B. Croke, I. Lasiecka, G. Uhlmann, M. Vogelius). Springer, New York, 2004Google Scholar
  12. 12.
    Convent, A., van Schaftingen, J.: Intrinsic colocal weak derivatives and Sobolev spaces between manifolds. Ann. Sci. Norm. Super. Pisa Cl. Sci. 16(1), 97–128 2016Google Scholar
  13. 13.
    Danescu, A., Chevalier, C., Grenet, G., Regreny, Ph, Letartre, X., Leclercq, J.L.: Spherical curves design for micro-origami using intrinsic stress relaxation. Appl. Phys. Lett. 102(12), 123111 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, revised edn. CRC Press, Boca Raton (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Efrati, E.: Non-Euclidean ribbons. J. Elast. 119(1), 251–261 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Eells, J., Lemaire, L.: Selected Topics in Harmonic Maps. American Mathematical Society, Providence (1983)CrossRefzbMATHGoogle Scholar
  17. 17.
    Efrati, E., Sharon, E., Kupferman, R.: Buckling transition and boundary layer in non-Euclidean plates. PRE 80, 016602 (2009)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Efrati, E., Sharon, E., Kupferman, R.: The metric description of elasticity in residually stressed soft materials. Soft Matter 8, 8187 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  20. 20.
    Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Commun. Pure Appl. Math. 55, 1461–1506 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gromov, M.: Partial Differential Relations. Springer, Berin-Heidelberg (1986)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hajłasz, P.: Sobolev mappings between manifolds and metric spaces, pp. 185–222. Sobolev Spaces in Mathematics I. Springer, New York (2009)zbMATHGoogle Scholar
  23. 23.
    Hartman, P.: On isometries and on a theorem of liouville. Math. Z. 69, 202–210 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Heinonen, J.: Lectures on Lipschitz analysis, Jyväskylän Yliopistopaino, 2005Google Scholar
  25. 25.
    Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  26. 26.
    Hélein, F., Wood, J.C.: Harmonic maps. Handbook of global analysis. (Eds. D. Krupka, D. Saunders) Elsevier, Amsterdam 417–492, 2008Google Scholar
  27. 27.
    James, R.D., Kinderlehrer, D.: Theory of diffusionless phase transformations. Proceedings of ``Partial differential equations and continuum models of phase transitions'', Lecture Notes in Physics (Eds. M. Rascle, D. Serre, M. Slemrod), vol. 344, 51–84, 1989Google Scholar
  28. 28.
    Klein, Y., Efrati, E., Sharon, E.: Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315, 1116–1120 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kupferman, R., Maor, C.: The emergence of torsion in the continuum limit of distributed dislocations. J. Geom. Mech. 7(3), 361–387 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kupferman, R., Maor, C.: Limits of elastic models of converging Riemannian manifolds. Calc. Var. PDEs 55, 1–22 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kupferman, R., Maor, C.: Riemannian surfaces with torsion as homogenization limits of locally-Euclidean surfaces with dislocation-type singularities. Proc. R. Soc. Edinb. A 146(04), 741–768 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kohn, R.V., O'Brien, E.: On the bending and twisting of rods with misfit. J. Elast. 130(1), 115–143 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kupferman, R., Olami, E., Segev, R.: Continuum dynamics on manifolds: application to non-Euclidean elasticity. J. Elast. 128, 61–84 (2017)CrossRefzbMATHGoogle Scholar
  34. 34.
    Klein, Y., Venkataramani, S., Sharon, E.: Experimental study of shape transitions and energy scaling in thin non-euclidean plates. PRL 106, 118303 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    Kupferman, R., Shamai, Y.: Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space. Isr. J. Math. 190(1), 135–156 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Liouville, J.: Théoréme sur l'équation \(dx^2+dy^2+dz^2 = \lambda (d\alpha ^2+d\beta ^2 + d\gamma ^2)\). J. Math, Pures Appl (1850)Google Scholar
  37. 37.
    Lieb, E.H., Loss, M.: Analysis, 2nd edn. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  38. 38.
    Lorent, A.: On functions whose symmetric part of gradient agree and a generalization of Reshetnyak's compactness theorem. Calc. Var. PDEs 48(3), 625–665 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lorent, A.: Rigidity of pairs of quasiregular mappings whose symmetric part of gradient are close. Ann. Inst. H. Poincaré Anal. Nonlinear 33(1), 23–65 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lewicka, M., Pakzad, M.R.: Scaling laws for non-Euclidean plates and the \(W^{2,2}\) isometric immersions of Riemannian metrics. ESAIM Control Optim. Calc. Var. 17, 1158–1173 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lewicka, M., Raoult, A., Ricciotti, D.: Plates with incompatible prestrain of higher order. Ann. Inst. H. Poincaré Anal. Nonlinear 34, 1883–1912 2017CrossRefzbMATHGoogle Scholar
  42. 42.
    Liimatainen, T., Salo, M.: N-harmonic coordinates and the regularity of conformal mappings. Math. Res. Lett. 21(2), 341–361 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Courier Dover Publications, New York (1983)zbMATHGoogle Scholar
  44. 44.
    Maor, C., Shachar, A.: On the role of curvature in the elastic energy of non-Euclidean thin bodies, arXiv:1801.02207
  45. 45.
    Olbermann, H.: Energy scaling law for a single disclination in a thin elastic sheet. Arch. Rat. Mech. Anal. 224(3), 985–1019 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Ozakin, A., Yavari, A.: A geometric theory of thermal stresses. J. Math. Phys. 51, 032902 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Reshetnyak, Yu.G.: Liouville's theorem on conformal mappings for minimal regularity assumptions. Sib. Mat. Zhurnal 8(4), 835–840 (1967)Google Scholar
  48. 48.
    Reshetnyak, Yu.G.: On the stability of conformal mappings in multidimensional spaces. Sib. Mat. Zhurnal 8(1), 91–114 (1967)Google Scholar
  49. 49.
    Reshetnyak, Yu.G.: Differential properties of quasiconformal mappings and conformal mappings of Riemannian spaces. Sib. Mat. Zhurnal 19(5), 1166–1184 (1978)MathSciNetGoogle Scholar
  50. 50.
    Reshetnyak, Yu.G.: Stability Theorems in Geometry and Analysis. Springer, Netherlands (1994)CrossRefzbMATHGoogle Scholar
  51. 51.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 5, 3rd edn, Publish or Perish, 1999Google Scholar
  52. 52.
    Sharon, E., Roman, B., Swinney, H.L.: Geometrically driven wrinkling observed in free plastic sheets and leaves. PRE 75, 046211 (2007)ADSCrossRefGoogle Scholar
  53. 53.
    Taylor, M.: Existence and regularity of isometries. Trans. Am. Math. Soc. 358(6), 2415–2423 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Wehrheim, K.: Uhlenbeck Compactness. European Mathematical Society, Zurich (2004)CrossRefzbMATHGoogle Scholar

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© 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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