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Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ \delta^k_i\partial_j\varphi + \delta^k_j\partial_i\varphi-g_{ij}\nabla^k\varphi $$ $$\tilde R_{ijkl} = e^{2\varphi}\left( R_{ijkl} - \left[ g {~\wedge\!\!\!\!\!\!\bigcirc~} \left( \nabla\partial\varphi - \partial\varphi\partial\varphi + \frac{1}{2}\|\nabla\varphi\|^2g \right)\right]_{ijkl} \right)$$ $$\tilde R = e^{-2\varphi}\left[R + \frac{4(n-1)}{(n-2)}e^{-(n-2)\varphi/2}\triangle\left( e^{(n-2)\varphi/2} \right) \right] $$

I've read many textbooks about differential geometry, such as Do Carmo, Kobayshi, Novikov and so on. But I never found these formulas. Who can give me a reference about these formulas.

Qmechanic
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346699
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    Crossposted from http://math.stackexchange.com/q/761375/11127 – Qmechanic Apr 20 '14 at 06:23
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    @Qmechanic Yes, there will be different results in these two forums – 346699 Apr 20 '14 at 06:26
  • I don't know if it's exactly what you're looking for, but Carroll's textbook on general relativity has some pretty similar formulae in appendix G (on conformal transformations) – Danu Apr 20 '14 at 07:26
  • @Danu Yes, but I'm very curious why I can't find these formulas in a mathematical textbooks. – 346699 Apr 20 '14 at 08:37
  • Have a look the appendices of Wald's book on general relativity where some advanced issues on differential geometry, like the use of paracompactness, are discussed. Notice that therein $e^\varphi$ is denoted by $\Omega$. – Valter Moretti Apr 20 '14 at 12:34
  • This question was marked as the duplicate of a completely different one. – Daniel Teixeira Aug 28 '20 at 12:35

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