Trying to understand the conformal transformation law of Ricci curvature

Asked Mar 05 '19 at 00:58

Active Mar 05 '19 at 00:58

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I'm trying to understand the conformal transformation law of Ricci curvature. It states that if the metric $g$ now changes to $e^{2\phi}g$, then $$Ric(e^{2\phi}g)=Ric^g-(n-2)Hess^g(\phi)-\Delta_g(\phi)g-(n-2)|grad^g(\phi)|^2g+(n-2)d\phi\otimes d\phi$$

I am trying to understand why that is. It seems to me that $Ric(X,Y)=g\big(R(X\wedge e_i),(Y\wedge e_i)\big)$. Hence, if the metric has now changed to $e^{2\phi}g$, then $$Ric^{e^{2\phi}g}(X,Y)=e^{2\phi}g(R(X\wedge e_i),(Y\wedge e_i))=e^{2\phi}Ric(X,Y)$$ Why is this not true?

asked Mar 05 '19 at 00:58

1

Because the conformal transformation affects the Christoffel symbols. Here https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry is, I hope, useful information. – FormerMath Mar 05 '19 at 03:54
You may also check my answer here. – Yuri Vyatkin Mar 05 '19 at 05:11

Trying to understand the conformal transformation law of Ricci curvature

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