Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature

摘要

In cosmological perturbation theory a first major step consists in thedecomposition of the various perturbation amplitudes into scalar, vector andtensor perturbations, which mutually decouple. In performing this decompositionone uses -- beside the Hodge decomposition for one-forms -- an analogousdecomposition of symmetric tensor fields of second rank on Riemannian manifoldswith constant curvature. While the uniqueness of such a decomposition followsfrom Gauss' theorem, a rigorous existence proof is not obvious. In this note weestablish this for smooth tensor fields, by making use of some importantresults for linear elliptic differential equations.