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A geometric perspective on the Piola identity in Riemannian settings
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel |
The Piola identity $ \operatorname{div}\; \operatorname{cof} \;\nabla f = 0 $ is a central result in the mathematical theory of elasticity. We prove a generalized version of the Piola identity for mappings between Riemannian manifolds, using two approaches, based on different interpretations of the cofactor of a linear map: one follows the lines of the classical Euclidean derivation and the other is based on a variational interpretation via Null-Lagrangians. In both cases, we first review the Euclidean case before proceeding to the general Riemannian setting.
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doi: 10.1090/cbms/050. |
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R. Kupferman, C. Maor and A. Shachar,
Reshetnyak Rigidity for Riemannian Manifolds, Arch. Rat. Mech. Anal., 231 (2019), 367-408.
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show all references
References:
[1] |
P. Ciarlet, Mathematical Elasticity, Volume 1: Three-Dimensional Elasticity, Elsevier, 1988. Google Scholar |
[2] |
J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, CBMS Regional Conference Series in Mathematics, 50. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983.
doi: 10.1090/cbms/050. |
[3] |
L. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, 1998.
doi: 10.1090/gsm/019. |
[4] |
T. Iwaniec, Null lagrangians: Definitions, examples and applications, Warsaw Lectures, Part 2. Google Scholar |
[5] |
R. Kupferman, C. Maor and A. Shachar,
Reshetnyak Rigidity for Riemannian Manifolds, Arch. Rat. Mech. Anal., 231 (2019), 367-408.
doi: 10.1007/s00205-018-1282-9. |
[6] |
J. Marsden and T. Hughes, Mathematical Foundations of Elasticity, Dover, 1983. Google Scholar |
[7] |
D. Saunders, The Geometry of Jet Bundles, Cambridge University Press, 1989.
doi: 10.1017/CBO9780511526411.![]() ![]() |
[8] |
P. Steinmann, Geometrical Foundations of Continuum Mechanics, Springer, 2015.
doi: 10.1007/978-3-662-46460-1. |

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