$\varphi_t$ is a one-parameter group of diffeomorphisms generated by a vector field $V$ on $M$,
$$ g_{ij}=\varphi_t^*g_{ij}(x,0) ~~~ \frac{\partial g_{ij}}{\partial t}=-2R_{ij} $$
How to show that $-2Ric=\mathcal{L}_vg$ ? The $\mathcal{L}_vg$ is Lie derivative
I only have a little knowledge of Riemannian geometry and PDE.But I must read some paper about Ricci flow .What should do? What book I should read before read Ricci flow? Thanks for any helping .
You already have it! I assume $v$ in your Lie derivative is the vector field tangent to the orbits of $\phi_t$. If that is the case then $v=\partial/\partial_t$ and the Lie derivative of $g_{ab}$ is just $\partial_t g_{ab}$.
I do not know about Ricci flow in particular, but I suggest you get a good grounding in differential and Riemannian geometry first. For a general introduction to differential geometry I would look at Tu’s Introduction to Manifolds or Lee’s Smooth manifolds. For Riemannian geometry therea re good books by John Lee again or Do Carmo.