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.

  • 1
    $\begingroup$ Crossposted from math.stackexchange.com/q/761375/11127 $\endgroup$
    – Qmechanic
    Commented Apr 20, 2014 at 6:23
  • 2
    $\begingroup$ @Qmechanic Yes, there will be different results in these two forums $\endgroup$
    – 346699
    Commented Apr 20, 2014 at 6:26
  • $\begingroup$ 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) $\endgroup$
    – Danu
    Commented Apr 20, 2014 at 7:26
  • $\begingroup$ @Danu Yes, but I'm very curious why I can't find these formulas in a mathematical textbooks. $\endgroup$
    – 346699
    Commented Apr 20, 2014 at 8:37
  • $\begingroup$ 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 $\Omega$. $\endgroup$ Commented Apr 20, 2014 at 12:34