  For an infinitesimal deformation of a Riemannian manifold, we prove that the
scalar, vector, and tensor modes in decompositions of perturbations of the
metric tensor, the scalar curvature, the Ricci tensor, and the Einstein tensor
decouple if and only if the manifold is Einstein. Four-dimensional space-time
satisfying the condition of the theorem is homogeneous and isotropic.
Cosmological applications are discussed.
