Let $(M,g)$ is a Riemannian manifold.$g$ evolve under $\partial_tg_{ij}(x,t)=-2R_{ij}(x,t)$.
When I read Shi's derivative estimate, I need the metric $g_{ij}(x,t)$ to be continuous with respect to the parameter $t$, and $\forall t\in (0,T], x\in M$, $g_{ij}(x,t)$ to be smooth in $x$. But I don't know how to show that.
Forgive me so ignorant, I just a beginner of geometry.
Thanks very much for detail answer or hint.