Let $\widehat{g}_{ij}(x,t)$ be a solution of $$\frac{\partial g_{ij}}{\partial t}=-2R_{ij},$$ and $\varphi_t:M\rightarrow M$ is a family of diffeomorphisms of $M$. Let $$g_{ij}(x,t)=\varphi_t^*\widehat{g}_{ij}(x,t)$$ be the pull-back metric and $y(x,t)=\varphi_t(x)=\{y^1(x,t),...,y^n(x,t)\}$,$\partial_i=\frac{\partial}{\partial x^i},\partial_t=\frac{\partial}{\partial t}.$
How to show that:
1, $g_{ij}(x,t)=\partial_iy^a\partial_jy^b\widehat{g}_{ab}(y,t)$
2, $\partial_t \widehat{g}_{ab}(y,t)=-2\widehat{R}_{ab}(y,t)+ \frac{\partial\widehat{y}_{ab}}{\partial y^r}\frac{\partial y^r}{\partial t}$