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one could use linear approximation $g_{ab}=\eta_{ab}+h_{ab}$ to get linear ricci flow equation (2). How to do that? is any process ? I am studying general relativity , i just use the linear approximation : $g_{ab}=\eta_{ab}+\gamma_{ab}$, and calculate the christoffl symbol : $\Gamma_{a b}^{(1) c}=\frac{1}{2} \eta^{c d}\left(\partial_{a} \gamma_{b d}+\partial_{b} \gamma_{a d}-\partial_{d} \gamma_{a b}\right)$,finally, get the Ricci tensor:$R_{a b}^{(1)}=\partial^{c} \partial_{(a} \gamma_{b) c}-\frac{1}{2} \partial^{c} \partial_{c} \gamma_{a b}-\frac{1}{2} \partial_{a} \partial_{b} \gamma$ but there are two extra term :$\partial^{c} \partial_{(a} \gamma_{b) c}$ and $-\frac{1}{2} \partial_{a} \partial_{b} \gamma$, which not match the linearization ricci flow equation(2). what is wrong with me?

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  • $\begingroup$ You need to calculate the Gateaux derivative of the operator as shown in this Q&A: it is usually not difficult, even if perhaps someone may consider it a bit tedious. If I were more versed in differential geometry, I'd post an answer following the same lines of the one referred. $\endgroup$ Sep 30, 2019 at 6:48
  • $\begingroup$ i have modified my question, why am I wrong? $\endgroup$
    – explorer
    Oct 4, 2019 at 14:28
  • $\begingroup$ @Daniele Tampieri $\endgroup$
    – explorer
    Oct 10, 2019 at 1:07

2 Answers 2

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You are correct, the paper is mistaken. In fact it's well known that the linearization of the Ricci flow is not strongly parabolic, so it is impossible that the linearization should be the standard heat equation.

It looks like that article ("Modified Ricci flow and asymptotically non-flat spaces") also has some other errors or typos, such as its equation (3) which is an erroneous citation of the paper by Samuel and Roy Chowdhury.

See section 2 of Anderson and Chow "A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds" for a proper (brief) discussion of linearized Ricci flow.

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He used De Donder harmonic condition with respect to gamma (https://en.wikipedia.org/wiki/Harmonic_coordinate_condition) always possible: Einstein equation gives this freedom in the choice of coordinates (10 unknowns of the metric tensor - 6 independent equations). instead I would be curious to know what book this is

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