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When conformal gravity is written in terms of connections of the conformal group it is a common trick to impose an additional constraint, namely torsion is equal to zero, to observe that it is equivalent to the standard Weyl theory. (See for example Kaku, M. Townsend, P. K. & van Nieuwenhuizen, P., Phys. Lett., 1977, B69, 304-308 [1]). If one doesn't do this and simply integrates out the auxiliary special conformal connection the remaining action still remains a functional of both solder form and Lorentz connection which is not right for Weyl gravity. But this constrain doesn't follow as a consequence of equations of motion for Lorentz-connection or at least I don't know how it can be shown. So why is it even legit?

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As I managed to understand from James T. Wheeler "Weyl gravity as general relativity" vanishing of the torsion does not follow from the equations of motion but imposing this additional condition provides that all so found solutions of the theory are conformally equivalent to the solutions of the vacuum Einstein equaiton which is not the case for Weyl gravity in general.

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