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 language | American English |
|---|---|
| Pages (from-to) | 59-76 |
| Number of pages | 18 |
| Journal | Journal of Geometric Mechanics |
| Volume | 11 |
| Issue number | 1 |
| DOIs | |
| State | Published - 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