Trace of Ricci flow equation The Ricci flow equation as is known is given as:
$\partial_t g_{ij} = -2R_{ij}$.
If I take the trace/contract the indices of both sides, does this imply that:
$\partial_t g = -2R$,
where $g$ is the trace of the metric tensor, and $R$ is the Ricci curvature scalar?
Thank you.

To take the trace of the Ricci tensor you are using the metric, so you need to be a little bit more careful: the correct expression is 
$$ g^{ij} \partial_t g_{ij} = -2R.$$
Since $g^{ij}$ is varying in time you cannot replace the LHS by $\partial_t(g^{ij} g_{ij}) = \partial_t n=0$.
