A geometric perspective on the piola identity in Riemannian settings

Raz Kupferman, Asaf Shachar*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

The Piola identity div cof ∇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.

Original languageAmerican English
Pages (from-to)59-76
Number of pages18
JournalJournal of Geometric Mechanics
Volume11
Issue number1
DOIs
StatePublished - Mar 2019

Bibliographical note

Funding Information:
2010 Mathematics Subject Classification. Primary: 53Zxx; Secondary: 74Bxx. Key words and phrases. Differential geometry, Riemannian geometry, Piola identity, Null-Lagrangians, elasticity. 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. ∗ Corresponding author: Asaf Shachar.

Publisher Copyright:
© American Institute of Mathematical Sciences

Keywords

  • Differential geometry
  • Elasticity
  • Null-Lagrangians
  • Piola identity
  • Riemannian geometry

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