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We know by definition a conformal Killing vector X satisfies the equation $$L_X g = \kappa g$$ with the conformal factor $\kappa$ satisfying the equation $$(n-2)\partial_\mu\partial_\nu \kappa + g_{\mu\nu} \Delta_g \kappa = 0$$ for flat space. It was claimed the conformal factor satisfies the same equation with the derivatives replaced by covariant derivatives in generic curved space in page 15 of "A mathematical introduction to conformal field theory" by Prof. Schottenloher. It seems the derivation of the equation in flat space depends on the commutativity of partial derivatives, does any one know a derivation of this equation in curved space where covariant derivatives do not commute?

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