Conformal gravity is a generic name for gravity theories which are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations where is the metric tensor and is a function on spacetime.
Read more about Conformal Gravity: Weyl-squared Theories, Four Derivative Theories, Conformal Unification To The Standard Model
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“Grown beyond nature now, soft food for worms,
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—Derek Mahon (b. 1941)