Curvature-dependent conformal transformations such as g-tilde = F(R[g])g are not local changes of metric variables: their inverse requires solving a differential equation, so no finite-jet metric-only inverse exists generically.
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Differential Obstructions to Curvature-Dependent Conformal Transformations
Curvature-dependent conformal transformations such as g-tilde = F(R[g])g are not local changes of metric variables: their inverse requires solving a differential equation, so no finite-jet metric-only inverse exists generically.