# conformal coordinates on a Riemann surface

Let $$\Sigma$$ be a Riemann surface with complex structure $$j$$ and a volume form $$dvol_\Sigma$$. I read somewhere that one can take the so-called 'conformal coordinates' $$z=s+it$$ so that $$j\partial_s = \partial_t$$ and $$dvol_\Sigma = f^2(ds\wedge dt)$$ for some function $$f$$.

Question: When can we find a conformal coordinate so that locally $$dvol_\Sigma =ds\wedge dt$$, equivalently, the induced metric is locally $$ds^2+dt^2$$? For example, can we simply use a new coordinate $$s'=f(s,t) s$$ and $$t' =f(s,t) t$$?

I guess it would be sufficient if $$\Sigma$$ is flat but I am not sure. Can someone tell me the whole picture of this question? Thanks!