I am trying to prove that that the expression $Q=-\frac{1}{\kappa}\int_{S_\infty} \nabla^i \xi^k \mathrm{d}\sigma_{ik}$ is a conserved quantity for a spacetime with Killing vector $\xi^i$ where $S_\infty$ is the 'boundary at infinity' of the spacelike hypersurface $\Sigma$.
Using the Gauss Theorem, I have written $Q$ as follows:
$$\begin{align*}Q&=\frac{1}{\kappa}\int_{\Sigma} \nabla_c \nabla^c \xi^k \mathrm{d}\sigma_{k}\\&=\frac{1}{\kappa}\int_{\Sigma} R^{kb} \xi_b \mathrm{d}\sigma_{k}\end{align*}$$
Is there any way of seeing why this last expression is constant?
Thanks!