Misha's comment is not correct. One way to see this is by thinking about a theorem of Bando (see http://www.springerlink.com/content/v0764574t4764138/ if you have access) which says that if $(M,g_t)$ is a solution for Ricci Flow on the interval $[0,T)$, then $g_t$ is real analytic with respect to the normal coordinate charts on $M$ for $t>0$. In particular, if $g_0$ was not real analytic, we cannot extend the flow backwards for any $\epsilon>0$ (the comment was only in charts, but by compacntess if we can do it in charts, we can do it on the whole manifold for some small $\epsilon>0$) because then Bando's theorem would imply that $g$ were real analytic.

The correct statement is just like for the heat equation. We say that $f$ is a solution to the heat equation $$ \frac{\partial f}{\partial t} = \Delta f $$ on $[0,T)$ if the above equation is satisfied for $t>0$ and $\lim_{t\searrow 0} f= f_0$. In particular, there is no "meaning" of the equation at $t=0$, only for $t>0$. Do not get confused by trying to apply ODE intuition to the PDE. Parabolic equations are not like ODE's in the sense that you can just "go in the direction of $\Delta f$".

So, for completeness, here is what it means to be a solution to RF on the interval $[0,T)$ with initial data $g_0$:

The metric $g_t$ is smooth for $t\in (0,T)$ and for such $t$, $g_t$ satisfies $$ \frac{\partial g_t}{\partial t} = -2Ric_{g_t}. $$ You can think of this either in local coordinate charts, as Misha does, or just as a coordinate free equation for the symmetric 2-tensor $\frac{\partial}{\partial t} g_t$.

Furthermore, we require that $g$ is continuous up to $t=0$ (because here we're only interested in solving RF with smooth initial data, if we wanted to start with rough data, we'd require a limit in some sense) and $$ g_{t=0} = g_0 $$ at each point in $M$.

If you're confused, you should read up on the heat equation first. Its exactly the same. In particular, after reading about the heat equation, you should read about the De Turk trick, which transforms the RF into a strongly parabolic equation (i.e. heat-type equation) by fixing the diffeomorphism gauge. A quick google suggests the following chapter http://www.springerlink.com/content/0t673151r72133r7/ as a possible reference. Any book on Ricci Flow should have a good description of this.

Misha's comment could be a bit misleading. In particular, it is not true that the Ricci flow should exist on a slightly bigger interval $(-\epsilon,T)$ with $g(0) = g_0$. One way to see this is by thinking about a theorem of Bando (see http://www.springerlink.com/content/v0764574t4764138/ if you have access) which says that if $(M,g_t)$ is a solution for Ricci Flow on the interval $[0,T)$, then $g_t$ is real analytic with respect to the normal coordinate charts on $M$ for $t>0$. In particular, if $g_0$ was not real analytic, we cannot extend the flow backwards for any $\epsilon>0$ (the comment was only in charts, but by compacntess if we can do it in charts, we can do it on the whole manifold for some small $\epsilon>0$) because then Bando's theorem would imply that $g$ were real analytic.

The correct statement is just like for the heat equation. We say that $f$ is a solution to the heat equation $$ \frac{\partial f}{\partial t} = \Delta f $$ on $[0,T)$ if the above equation is satisfied for $t>0$ and $\lim_{t\searrow 0} f= f_0$. In particular, there is no "meaning" of the equation at $t=0$, only for $t>0$. Do not get confused by trying to apply ODE intuition to the PDE. Parabolic equations are not like ODE's in the sense that you can just "go in the direction of $\Delta f$".

So, for completeness, here is what it means to be a solution to RF on the interval $[0,T)$ with initial data $g_0$:

The metric $g_t$ is smooth for $t\in (0,T)$ and for such $t$, $g_t$ satisfies $$ \frac{\partial g_t}{\partial t} = -2Ric_{g_t}. $$ You can think of this either in local coordinate charts, as Misha does, or just as a coordinate free equation for the symmetric 2-tensor $\frac{\partial}{\partial t} g_t$.

Furthermore, we require that $g$ is continuous up to $t=0$ (because here we're only interested in solving RF with smooth initial data, if we wanted to start with rough data, we'd require a limit in some sense) and $$ g_{t=0} = g_0 $$ at each point in $M$.

If you're confused, you should read up on the heat equation first. Its exactly the same. In particular, after reading about the heat equation, you should read about the De Turk trick, which transforms the RF into a strongly parabolic equation (i.e. heat-type equation) by fixing the diffeomorphism gauge. A quick google suggests the following chapter http://www.springerlink.com/content/0t673151r72133r7/ as a possible reference. Any book on Ricci Flow should have a good description of this.