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In Jost's "Compact Riemann surfaces" he defines (Definition 2.3.1) a conformal Riemannian metric on a Riemann surface $\Sigma$ to be given in local coordinates by $$\lambda^2(z) dz d \overline{z}. $$

I find the triplet conformal-Riemannian-metric a bit confusing:

I can think of a Riemann surface $\Sigma$ as a $2$-dimensional smooth manifold $M$ + a conformal structure $C$ on $M$, so that $\Sigma=(M, C)$.

(I am using the fact that specifying a conformal structure on $M$ is equivalent to specifying a complex structure on $M$).

A conformal structure on $M$ is an equivalence class of Riemannian metrics $g$ on $M$.

What exactly does it mean for a metric on $\Sigma$ to be a conformal Riemannian metric?

Is it a particular choice of Riemannian metric $g \in C$?

user7090
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    I think this is just to say that the metric belongs in the conformal class – Didier Apr 11 '21 at 19:36
  • So just some $g \in C$? – user7090 Apr 11 '21 at 19:36
  • Yes. The author insists on the fact $\lambda(z)^2 \mathrm{d}z\mathrm{d}\overline{z}$ belongs to the conformal class induced by the complex structure – Didier Apr 11 '21 at 19:47
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    Since $dz,d\bar{z}$ is a flat metric, it appears that he is talking about locally conformally flat metrics. Presumably he also shows that every Riemannian metric on a Riemann surface is conformally flat. – Deane Apr 11 '21 at 22:25
  • I think, Jost is very clear: A Riemannian metric $g$ on a Riemann surface $X$ is conformal iff $g\in C$, i.e. the conformal atlas defined by $g$ (via orientation-preserving isothermal coordinates) is equivalent to the holomorphic atlas of $X$. – Moishe Kohan Apr 14 '21 at 00:59

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