Helmholtz-Hodge Theorems: Unification of Integration and Decomposition Perspectives

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).