Diffeomorphisms and Ricci flow

Question

Let $(M,g)$ be a riemannian manifold. I define a flow of diffeomorphisms $\phi_t$ over $M$ by the formula:

$$g(\frac{\partial \phi_t^*}{\partial t}X,\phi_t^* Y)+g(\phi_t^* X, \frac{\partial \phi_t^*}{\partial t}Y)=-2Ric_{\phi_t^* g}(X,Y)$$

where $Ric_g$ is the Ricci curvature of $g$ and $\phi_t^* g(X,Y)=g(\phi_t^* X,\phi_t^* Y)$.

Is this flow of diffeomorphisms well defined? When have we:

$$\phi_t=e^{tX}$$

where $X$ is a vectors field?