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

In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses – beside the Hodge decomposition for one-forms – an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss’ theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations. 1 The decomposition theorem In cosmological perturbation theory one can regard the various perturbation amplitudes as time dependent tensor fields on a three-dimensional Riemannian space (M, g) of constant curvatureK (see, e.g., [1]). For skew-symmetric tensor fields (p-forms) there is on arbitrary compact Riemannian manifolds the profound Hodge decomposition into an orthogonal direct sum of exact, coexact, and harmonic forms. No analogous decomposition for symmetric tensor fields, say, is available in general. However, when the space has constant curvature, a symmetric tensor field tij can be decomposed as follows: tij = t (S) ij + t (V ) ij + t (T ) ij , (1)