# Precise definition of conformal structure based on a Riemannian metric on a Riemann surface

As I read the literature, I keep having some doubt about what a " conformal structure on a Riemann surface " exactly means. ( You can assume all the Riemann surface in this literature have universal cover $\mathbb{D}$ ) .In some literature, it says a conformal structure is the same as a complex structure, which is okay with me. But sometime, after talking to people I get the impression that a putting conformal structure and putting a complex structure on a 2 dimensional smooth manifold $X$ are equivalent, but still not exactly the same. I get the idea : two 'conformally equivalent' Riemannnnian metrics determine the same angle, so a conformal structure should uniquely determine the angle between curves, which is done by a complex structure / Riemann surface structure. But then what exactly is / are the definition of a " conformal structure " ??

And, what is conformal metric then ? Is it an equivalence class of conformally equivalent metrics so that any metric in that class is called conformal metric ?

Also, according to the definition in your answers, what is / are the meaning of " conformal

structure based on a conformal metric ? "

If you want a reference to the literature I am taking this from, then please look at page 335 of Lipman Ber's paper " Quasiconformal Maps and Teichmüller's Theorems ", the last paragraph, where he says : " we define a new conformal structure based on the conformal metric $g = | dz + \mu(z) \; d\bar{z} |$. It is clear to me, however, that $g$ is conformal to the locally Euclidean metric $| dz|$ by the quasiconformal homeomorphism $w$ with the ( local ) Beltrami coefficient $\mu$

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Definition: Consider all Riemannian metrics on a topological surface $S$, which are classified by the conformal equivalence relation, {Riemmanian metrics on $S$}/~ , where each equivalence class is called a conformal structure .
Defintion: Suppose $g_1$, $g_2$ are two metrics on a manifold $M$, if $g_1$= $e^{2u}(g_2)$, u:$M \to \mathbb{R}$, then $g_1$ and $g_2$ are conformal equivalent. A conformal structure based on conformal metrics would be "made up" of such metrics.