Which textbook of differential geometry will introduce conformal transformation? [duplicate]

This question already has an answer here:

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.

Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers containing only a reference to a book or paper will be removed!

marked as duplicate by Qmechanic♦Aug 1 '15 at 4:46

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• Crossposted from math.stackexchange.com/q/761375/11127 – Qmechanic Apr 20 '14 at 6:23
• @Qmechanic Yes, there will be different results in these two forums – 346699 Apr 20 '14 at 6:26
• 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 7:26
• @Danu Yes, but I'm very curious why I can't find these formulas in a mathematical textbooks. – 346699 Apr 20 '14 at 8:37
• 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