A geometric perspective on the Piola identity in Riemannian settings

TL;DR

This paper extends the Piola identity from Euclidean spaces to Riemannian manifolds, providing two proofs that deepen the understanding of elasticity theory in curved geometries.

Contribution

It introduces a generalized Piola identity for Riemannian manifolds with two distinct proof methods, broadening its applicability in geometric analysis and elasticity.

Findings

Generalized Piola identity for Riemannian manifolds proved

Two different proof approaches established

Enhanced understanding of elasticity in curved spaces

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.