Gauge invariance of the linearized Riemann tensor
= Gauge invariance of the linearized Riemann tensor
Under $h_{\mu\nu}\mapsto h_{\mu\nu}+\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$, every contribution to the linearized Riemann tensor contains three commuting derivatives of $\xi$ and cancels pairwise. Linearized curvature is therefore gauge invariant about flat spacetime.