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.
Bibliographical notePublisher Copyright:
© American Institute of Mathematical Sciences
- Differential geometry
- Piola identity
- Riemannian geometry