Trace of Ricci flow equation

Question

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.

Answer 1

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 $\partial_t(g^{ij} g_{ij}) = \partial_t n=0$.