TITLE: the left hand side of the  Ricci flow equation at the initial value
QUESTION [6 upvotes]: I just started to learn about the Ricci flow and try to understand the Ricci flow evolution equation. It states that a one-parameter family $g_t$, $t\in[0,T)$ of Riemannian metrics on a smooth closed manifold $M$ is a solution of the equation
$$ \frac{\partial g_t}{\partial t}=-2 Ric_{g_t}. $$
But, in order to be able to write down the expression
$$ \frac{\partial g_t}{\partial t} $$ 
I need a little bit more than the fact that $g_t$ is a one-parameter family of Riemannian metrics. 
There are several ways to understand this expression, which probably are equivalent to each other. I think the most naive idea is to consider $g_t$ as a smooth curve in the infinite-dimensional vector space of all Riemannian metrics on $M$ and $\frac{\partial g_t}{\partial t}$ as tangent vectors to this curve.
Now, my question is: How do I have to understand the left hand side of the Ricci flow equation at the initial value $t=0$?
Does one consider $g_t$ as a smooth family at the open interval $(0,T)$ and requires continuity at the point $t=0$ or has one to deal with differentiability from the right? 
Thank you very much!

REPLY [3 votes]: Pardon, this is not an answer, but it was too long to put in as a comment, and I wasn't sure how to include this random thought otherwise. Let $g$ be a Riemannian metric on a closed manifold $M$. We can define its temperature $\tau(g)\in\lbrack0,\infty]$ to be the supremum of
all $T$ such that there exists a solution to the Ricci flow on $M\times
\lbrack0,T]$ with $g(T)=g$. By Brett Kotschwar's backward uniqueness result,
there exists a unique solution to the Ricci flow on $M\times(0,\tau(g)]$ with
$g\left(  \tau\left(  g\right)  \right)  =g$. This solution may or may not be
extendable to $t=0$. The temperature of an Einstein metric with nonnegative
scalar curvature is infinity (when $R>0$, more generally, a shrinking gradient
Ricci soliton). The temperature of an Einstein metric with negative scalar
represents its scale. More generally, $\tau (cg) =c\tau (g)$. By Shigetoshi Bando's result, if $\tau(g)>0$, then $g$ is real analytic (as Otis Chodosh wrote above). I'm not sure what this invariant, nontrivial only for real analytic metrics, is good for, though.