= Levi-Civita connection under conformal rescaling
{c}
{title2=$\widetilde\nabla_XY=\nabla_XY+d\sigma(X)Y+d\sigma(Y)X-g(X,Y)\operatorname{grad}_g\sigma$}
For $\widetilde g=e^{2\sigma}g$, the displayed correction is a symmetric tensor, so adding it to $\nabla$ gives a <torsion-free connection>. The two extra metric-compatibility pairings sum to $2d\sigma(X)g(Y,Z)$; this is exactly the derivative of the conformal factor. Uniqueness of the <Levi-Civita connection> proves the formula. It also yields the <critical-point criterion for equal conformal connections>.
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