How to compute the equality 2 ? I think it's to use normal coordinate,then $\Gamma_{ij}^k=0$, then $\nabla_h\Gamma_{ij}^k=0$, then $R_{ijk}^h=0$. Whether I am right ?
$\delta$ is variation, $\partial_tg_{ij}=-2R_{ij}$ .
Below picture is from 200th page of this paper
