Helmholtz-Hodge Theorems: Unification of Integration and Decomposition Perspectives

TL;DR

This paper generalizes the Helmholtz and Hodge theorems to differential forms on Euclidean and Riemannian manifolds, providing a unified framework for integration and decomposition with explicit term specification.

Contribution

It introduces a Helmholtz-like theorem for differential forms and extends the Hodge decomposition to include explicit term descriptions on Riemannian manifolds.

Findings

Unified integration and decomposition framework for differential forms.

Extended Hodge decomposition with explicit term specification.

Connections between boundary conditions and solutions of differential systems.

Abstract

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_{n}$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the Schlaefli-Janet-Cartan theorem of embedding, are here considered as hypersurfaces in $E_{N}$ with $N\geq n(n+1)/2$. We obtain a Hodge decomposition theorem that includes and goes beyond the original one, since it specifies the terms of the decomposition. We then view the same issue from a perspective of integrability of the system ($d\alpha =\mu ,$ $\delta \alpha =\nu $), relating boundary conditions to solutions of ($d\alpha =0,$ $\delta \alpha =0$), [$\delta $ is what goes by the names of divergence and co-derivative, inappropriate for the Kaehler calculus, with which we obtained the foregoing).