In this paper we prove some rigidity theorems associated to \(Q\)-curvature analysis on asymptotically Euclidean (AE) manifolds, which are inspired by the analysis of conservation principles within fourth order gravitational theories. A central object in this analysis is a notion of fourth order energy, previously analysed by the authors, which is subject to a positive energy theorem. We show that this energy can be more geometrically rewritten in terms of a fourth order analogue to the Ricci tensor, which we denote by \(J_g\). This allows us to prove that Yamabe positive \(J\)-flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this \(J\)-tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of \(J\) as a fourth order analogue to the Ricci tensor.
