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A geometric perspective on the Piola identity in Riemannian settings

  • * Corresponding author: Asaf Shachar

    * Corresponding author: Asaf Shachar

This research was partially funded 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

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  • 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.

    Mathematics Subject Classification: Primary: 53Zxx; Secondary: 74Bxx.


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  • Figure 1.  Illustration of the geometric setting of the Euclidean Piola identity

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