On page 16 of this lecture notes and in this lecture, at some point (somewhere at 1:37:45), Rod Gover defined the $\text{conformal metric}$ $\mathbb{g}$ on a conformal manifold $(M, [g])$. He mentioned that this metric is canonical. Does this mean it is an invariant of the conformal manifold $(M, [g])$? Also, he mentioned that $\text{conformal metric}$ $\mathbb{g}$ gives a $1-1$ correspondence between the positive section of conformal $1$ densities and the metrics of $[g].$ Since I don't understand the definition, I cannot see this correspondence.

Could you please elaborate on this definition of $\text{conformal metric}$ $\mathbb{g}$ on a conformal manifold $(M, [g])$? Note that on page $12$ of this, one can find the definition, but to me, the definition is not well stated.

I would appreciate any detailed answer from the scratch. Note that I am comfortable with the associated bundle associated with a given principal bundle and conformal densities of weight $w.$ We can visualize/interpret a conformal manifold $(M, [g])$ as a ray bundle, an $\mathbb{R_{>0}}-$ principal bundle and its sections correspond to the metrics $g \in [g]$. I would appreciate any help you could provide. Thanks so much.

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