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Let $U\subset X$ be an open subset of a connected Riemann surface $X$. Let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $P\in U$ be the unique point such that $z(P) =0$.

Suppose I take another point $Q\in U$. I want to use $z:U\longrightarrow B(0,1)$ to construct a coordinate around $Q$.

Question. Does the following work?

Consider an open set $V$ in $U$ whose image under $z$ is a small open disc around $z(Q)$. Then we define $w: V\longrightarrow B(0,1)$ by $$w(x) = z(x) - z(Q).$$ Is this is a coordinate around $Q$?

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Sure. You could also use a Möbius transformation of the unit disk to move $Q$ to $0$, that would let you get away with restricting to a smaller neighborhood $V$. – Gunnar Þór Magnússon Sep 29 '11 at 19:24
I don't quite understand. I'm moving $Q$ to $0$ by simply translating. This is not a Mobius transformation? If not, can I write down an explicit formula for this Mobius transformation? So my coordinate at Q would be w(x) = mobius(z(x)), right? What can one take for mobius? – shaye Sep 30 '11 at 15:09
The translation is a Möbius transformation, but you can also use a different one to map the entire unit disk holomorphically to itself in such a way that $Z(Q) \mapsto 0$. Your formula for the new coordinate is correct. Googling Möbius transformation or the automorphisms of the unit disk will turn up an explicit formula for this map if you want one. – Gunnar Þór Magnússon Oct 1 '11 at 9:30

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