A geometric perspective on the Piola identity in Riemannian settings

Raz Kupferman; Asaf Shachar

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arXiv · 1805.12365

A geometric perspective on the Piola identity in Riemannian settings

Raz Kupferman·Asaf Shachar

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Abstract

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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Raz Kupferman, Asaf Shachar. 2018-05-31. A geometric perspective on the Piola identity in Riemannian settings. https://arxiv.org/abs/1805.12365

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