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Let $(M, \tilde{g})$ be a Riemannian manifold and let $g:=e^{2u}\tilde{g}$ be a conformal change. I'm trying to find a resource (ideally a book, or a paper where it's mentioned or derived) for the formula for the sectional curvature under this conformal change.

For any $\tilde{g}$-orthonormal pair $e_1,e_2$ spanning a two-plane $\Pi \subset T_y \widetilde{M}$, $$K_g(\Pi) = e^{-2u} \Big( K_{\tilde{g}}(\Pi) - \mathrm{Hess}_{\tilde{g}}u(e_1,e_1) - \mathrm{Hess}_{\tilde{g}}u(e_2,e_2) + (du(e_1))^2 + (du(e_2))^2 - \|\nabla_{\tilde{g}}u\|^2 \Big).$$

I'd appreciate if you could please mention a book/paper that derives this formula so I can put in the bibliographic references.

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    $\begingroup$ It's not stated in exactly this form, but you this formula follows easily from formula (7.44) in my Introduction to Riemannian Manifolds (2nd ed.), page 217. $\endgroup$ Commented Nov 18 at 23:33
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    $\begingroup$ Another source, which I found from math.stackexchange.com/questions/98113/… is Theorem 1.159 on page 58 of Besse's Einstein Manifolds. $\endgroup$ Commented Nov 18 at 23:58
  • $\begingroup$ Many thanks, Prof. Lee and Yang! I'll put both of them as references! $\endgroup$ Commented Nov 19 at 10:39

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