I am trying to calculate the variation of the Gauss-Bonnet-Invariant, which is given by $R_{GB} = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$, with respect to the metric tensor $g^{\mu\nu}$. I already calculated the $R^2$ part from $R_{GB}$ which gives me
$\delta\big(R^2\big) = 2RR_{\mu\nu}\delta g^{\mu\nu}$.
For the variation of the Ricci-Tensor part i get: \begin{align} \delta\Big(R_{\mu\nu}R^{\mu\nu}\Big) &= R_{\mu\nu}\delta R^{\mu\nu}+ R^{\mu\nu}\delta R_{\mu\nu} = R^{\mu\nu}\delta R_{\mu\nu} + R_{\mu\nu}\delta\Big(g^{\mu\rho}g^{\nu\sigma} R_{\rho\sigma}\Big) \\ &= R^{\mu\nu}\delta R_{\mu\nu}+ R_{\mu\nu}\Big(R_{\rho\sigma}g^{\mu\rho}\delta g^{\nu\sigma}+R_{\rho\sigma}g^{\nu\sigma}\delta g^{\mu\rho}+g^{\nu\sigma}g^{\mu\rho}\delta R_{\rho\sigma}\Big)\\ &= 2R^{\mu\nu}\delta R_{\mu\nu}+ 2R_{\mu\nu}R^\rho_\nu\delta g^{\mu\nu} \\ &= 2R_{\mu\nu}R^\rho_\nu\delta g^{\mu\nu} \end{align} Is it right to assume that the $2R^{\mu\nu}\delta R_{\mu\nu}$ part vanishes?
