# Vanishing Ricci flow on a curved manifold

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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 Im 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?

asked Dec 11, 2011
retagged Mar 25, 2014
Ive just seen that at theoretical physics SE a related question is asked: theoreticalphysics.stackexchange.com/questions/675/… So I`ll observe both places for answers.

This post imported from StackExchange Physics at 2014-03-17 03:35 (UCT), posted by SE-user Dilaton

## 1 Answer

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I get the impression that OP is referring to Normalized Ricci Flow (NRF):

$$\frac{1}{2} \partial_t g_{\mu\nu} ~=~ -R_{\mu\nu} + \frac{\langle R \rangle}{n} g_{\mu\nu}~.$$

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.

This post imported from StackExchange Physics at 2014-03-17 03:35 (UCT), posted by SE-user Qmechanic
answered Dec 12, 2011 by (3,110 points)
Thanks @Qmechanics You are right ...

This post imported from StackExchange Physics at 2014-03-17 03:35 (UCT), posted by SE-user Dilaton

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