by Jose G. Vargas
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≥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 (dα=μ, δα=ν), relating boundary conditions to solutions of (dα=0, δα=0), [δ is what goes by the names of divergence and co-derivative, inappropriate for the Kaehler calculus, with which we obtained the foregoing).
Source: Email by J. G. Vargas, 13 May 2014, josegvargas_AT_earthlink.net, http://arxiv.org/abs/1405.2375. Information about update: 11 Dec. 2014.