[On page 16](https://arxiv.org/abs/1412.7559) of this lecture notes and [in this lecture](https://www.youtube.com/watch?v=fTlRYcyISfA), 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](https://www.math.auckland.ac.nz/mathwiki/images/c/cf/StaffordMSc.pdf), 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.


NB:I feel like this problem is too basic for this site. If you find an answer on MathStackExchange, please transfer this post to MathStackExchange.



Definition of the conformal metric
Let $(M,[g])$ be a conformal manifold;
i.e. $(M,g)$ is a Riemannian manifold and $[g] = \{ u^2g \mathrel{}:\mathrel{} u \in C^\infty(M), u>0 \}$ is the set of Riemannian metrics conformal to $g$.
It is clear that $[g] = [u^2g]$ for all positive $u \in C^\infty(M)$.
Let me now change notation and write $c:=[g]$, to emphasize that there is no canonical choice of representative of the given conformal class.


A conformal manifold $(M,c)$ determines a $\mathbb{R}\_+$-subbundle $\mathcal{Q} \subset S^2T^\ast M$ by
$$ \mathcal{Q} = \{ (x,g\_x) \mathrel{}:\mathrel{} x \in M, g \in c \} . $$
Define the action of $\mathbb{R}\_+$ on $\mathcal{Q}$ by $\delta\_s(x,g\_x) := (x,s^2g\_x)$ for $s>0$.
Let $\pi \colon \mathcal{Q} \to M$ be the projection $\pi(x,g\_x)=x$ inherited from $S^2T^\ast M$.
Then $\mathcal{Q}$ is a principal $\mathbb{R}\_+$-bundle.


The **canonical metric** is the degenerate metric $\mathbf{g}$ on $\mathcal{Q}$ defined as follows:
Let $X,Y \in T\_{(x,g\_x)}\mathcal{Q}$ be tangent vectors at a point $(x,g\_x)$ in $\mathcal{Q}$.
Then $\pi\_\ast X, \pi\_\ast Y \in T\_xM$, and so we may define
$$ \mathbf{g}(X,Y) := g\_x(\pi\_\ast X, \pi\_\ast Y) . $$
Since this metric depends only on $\mathcal{Q}$, it is an invariant of $(M,c)$.


Next observe that if $X \in T\_{(x,g\_x)}\mathcal{Q}$ and $s>0$, then $(\delta\_s)\_\ast X \in T\_{(x,s^2g\_x)}\mathcal{Q}$ and
$$ \pi\_\ast (\delta\_s)\_\ast X = (\pi \circ \delta\_s)\_\ast X = \pi\_\ast X , $$
where the last equality uses the observation that $\pi \circ \delta\_s = \pi$.
It readily follows that if $X,Y \in T\_{(x,g\_x)}\mathcal{Q}$, then
$$ (\delta\_s^\ast \mathbf{g})(X,Y) = \mathbf{g}((\delta\_s)\_\ast X, (\delta\_s)\_\ast Y) = s^2g\_x(\pi\_\ast X,\pi\_\ast Y) = s^2\mathbf{g}(X,Y) . $$
That is, $\mathbf{g}$ is homogeneous of degree $2$ with respect to the dilations $\delta\_s$.


Finally, a positive conformal density of weight $1$ may be regarded a positive function $\sigma \in C^\infty(\mathcal{Q})$ such that $\delta\_s\sigma = s\sigma$ for all $s \in \mathbb{R}\_+$.
It immediately follows that $\delta\_s^\ast (\sigma^{-2}\mathbf{g}) = \sigma^{-2}\mathbf{g}$.
Conversely, if $h \in c$ is a choice of conformal metric, then
$$ \underline{h}(X,Y) := h(\pi\_\ast X, \pi\_\ast Y), $$
$X,Y \in T\_{(x,g\_x)}\mathcal{Q}$, is such that $\delta\_s^\ast\underline{h}=\underline{h}$.
It follows that there is some positive $\sigma \in \Gamma(\mathcal{E}[1])$ such that $\underline{h} = \sigma^{-2}\mathbf{g}$.
This is the correspondence between choices of metric in $c$ and positive sections of $\mathcal{E}[1]$ indicated by Curry and Gover.


