Let $\varphi_t$ be a one-parameter group of diffeomorphisms generated by a vector field $V$ on $M$, and the metric is given by $$ g_{ij}(x,t)=\varphi_t^*g_{ij}(x,0) $$
How to show that $\partial_tg=\mathcal{L}_Vg$ ? $\mathcal{L}_Vg$ is Lie derivative of $g$.
Besides, 1,Why a vector field can generate a one-parameter group?
$~~~~~~~~~~~~~$2,How to express the $\varphi_t^*g_{ij}(x,0)$ under the local coordinate?