# Vanishing Ricci flow on a curved manifold

If I understand this right the Ricci flow on a compact manifold given by

$\partial g_{\mu \nu} = - 2R_{\mu \nu} + \frac{2}{n}\!R_{\alpha}^{\alpha} \,g_{\mu \nu}$

tends to expand negatively curved regions and to shrink positively curved regions.

Looking at the above definition I`m wondering if the parameter n can be used to achieve $\partial g_{\mu \nu} = 0$ even if the Ricci tensor is not zero such that the validity of physics, that depends on the metric to be constant (as a precondition), could be extrapolated to curved manifolds to describe an expanding universe with a positive cosmological constant?

$$\frac{1}{2} \partial_t g_{\mu\nu} ~=~ -R_{\mu\nu} + \frac{\langle R \rangle}{n} g_{\mu\nu}~.$$
Here $\langle R \rangle$ is the average scalar curvature over the full space-time $M$. The average procedure is often weighted with an Einstein-Hilbert Boltzmann factor. It is just a number (as opposed to a space-time dependent scalar quantity).
Also $n$ is the space-time dimension, which is fixed, and hence cannot be easily varied as OP suggests.
• Apparently here $[g] = [R][t]$, $[g_{\mu\nu}] [x]^2= [g]$, $[R_{\mu\nu}] [x]^2=[R]=1$. – Qmechanic Nov 22 '19 at 22:29