Let $\varphi_t :M\rightarrow M $ is a family of diffeomorphism. $\widehat{g}_{ij}(x,t)$ is a solution of $$\frac{\partial}{\partial t}g_{ij}=-2R_{ij} ,\ y(x,t)=\varphi_t(x)=\{y^1(x,t),...,y^n(x,t)\},\ g_{ij}(x,t)=\varphi_t^*\widehat{g}_{ij}(x,t)$$
How to show that : $$ \Gamma_{jl}^k(x,t)= \frac{\partial y^a}{\partial x^j} \frac{\partial y^b}{\partial x^l} \frac{\partial x^k}{\partial y^r}\widehat{\Gamma}_{ab}^r(y,t) +\frac{\partial x^k}{\partial y^a} \frac{\partial^2y^a}{\partial x^j\partial x^l} $$
I really can't compute it out , so I really need a detail answer ,so thanks.
Assume that $ f(x)=y$ is a diffeomorphism and $$h:=f^\ast g $$ Define $$ g_{\alpha\beta }:= g(\partial_{y_\alpha},\partial_{y_\beta}) $$ $$ h_{ij}:= h(\partial_{x_i},\partial_{x_j}) $$ $$ f^\alpha_i:= \frac{\partial y^\alpha}{\partial x_i } $$ $$ h^{jk}=f_\zeta^j g^{\eta\zeta } f_\eta^k $$
So $$ h_{ij,l}:= \partial_{x_l} h_{ij} =\partial_{x_l} \{ g_{\alpha\beta } f^\alpha_i f^\beta_j\} $$ $$ = g_{\alpha\beta,\gamma}f^\gamma_lf^\alpha_i f^\beta_j + g_{\alpha\beta } f^\alpha_{il} f^\beta_j +g_{\alpha\beta } f^\alpha_i f^\beta_{jl} $$ That is $$ \Gamma(h)_{ij}^k = \frac{1}{2} h^{kl} (h_{il,j} + h_{jl,i} - h_{ij,l}) $$
$$ = \frac{1}{2}f_\zeta^l g^{\eta\zeta } f_\eta^k \{g_{\alpha\beta,\gamma}f^\gamma_jf^\alpha_i f^\beta_l + g_{\alpha\beta } f^\alpha_{ij} f^\beta_l +g_{\alpha\beta } f^\alpha_i f^\beta_{jl} $$ $$+ g_{\alpha\beta,\gamma}f^\gamma_if^\alpha_j f^\beta_l + g_{\alpha\beta } f^\alpha_{ij} f^\beta_l +g_{\alpha\beta } f^\alpha_j f^\beta_{il}$$ $$ -( g_{\alpha\beta,\gamma}f^\gamma_lf^\alpha_i f^\beta_j + g_{\alpha\beta } f^\alpha_{il} f^\beta_j +g_{\alpha\beta } f^\alpha_i f^\beta_{jl} ) \}$$
$$ = \frac{1}{2}f_\zeta^l g^{\eta\zeta } f_\eta^k \{g_{\alpha\beta,\gamma}f^\gamma_jf^\alpha_i f^\beta_l + 2g_{\alpha\beta } f^\alpha_{ij} f^\beta_l $$ $$+ g_{\gamma\beta,\alpha}f^\alpha_if^\gamma_j f^\beta_l - g_{\alpha\gamma,\beta}f^\beta_lf^\alpha_i f^\gamma_j \}$$
$$ = \Gamma(g)_{\alpha\gamma}^\tau f^\gamma_jf^\alpha_i f_\tau^k + f^\alpha_{ij} f_\alpha^k$$