  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.
