Conformal gravity

Remarks

The action is invariant under the Weyl rescaling $g \mapsto \Omega^2 g$ for every smooth positive function $\Omega$ on $M$. Indeed $C^a{}_{bcd}$ is unchanged, $C_{abcd}C^{abcd}$ is multiplied by $\Omega^{-4}$, and $\sqrt{-\det g}$ is multiplied by $\Omega^4$ in four dimensions. Example: every Einstein metric, with $R_{ab} = \Lambda g_{ab}$, solves $B_{ab} = 0$, and so does every metric conformal to an Einstein metric. Non-example: in dimension $n \ne 4$ the same integral is not Weyl invariant, since $\sqrt{-\det g}$ then scales by $\Omega^n$, which does not cancel $\Omega^{-4}$. The theory belongs to conformal geometry: it depends only on the conformal class of $g$.