The $g_{ij}$ is Riemannian metric ,and is solution of $\frac{\partial }{\partial t}g_{ij}(x,t)=-2R_{ij}(x,t)$. In the below picture , computing the $\frac{\partial}{\partial t}\Gamma_{jl}^h$. I think it should has two parts on the red line, but the part of $\frac{1}{2}\partial_tg^{hm}\{...\}$ is missed. Whether it is because we can choice a suitable coordinate make the $g^{hm}=\delta_{hm}$ so, the $\partial_t g^{hm}=0$?
This picture is from 183th page of the paper
