Variational status of a class of fully nonlinear curvature prescription problems

Prescribing, by conformal transformation, the k th-elementary symmetric polynomial of the Schouten tensor σ k (P) to be constant is a generalisation of the Yamabe problem. On compact Riemannian n -manifolds we show that, for 3 ≤ k ≤ n , this prescription equation is an Euler–Lagrange equation of some action if and only if the structure is locally conformally flat.