A geometric perspective on the Piola identity in Riemannian settings

Raz Kupferman; Asaf Shachar

doi:10.3934/jgm.2019004

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2019, Volume 11, Issue 1: 59-76. Doi: 10.3934/jgm.2019004

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A geometric perspective on the Piola identity in Riemannian settings

Raz Kupferman and
Asaf Shachar^,

Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel

^* Corresponding author: Asaf Shachar
^* Corresponding author: Asaf Shachar

Received: May 2018

Revised: December 2018

Published: January 2019

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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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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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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References

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