Holomorphic Maps Between Riemann Surfaces

Holomorphic Maps Between Riemann Surfaces

Recall from the Riemann Surfaces page that a Riemann surface $R$ is a second countable, connected, and Hausdorff topological space with a collection of pairs $\{ (U_{\alpha}, \phi_{\alpha}) : \alpha \in \Gamma \}$ called an atlas of $R$ which satisfy the following properties:

  • 1. For each $\alpha \in \Gamma$, $U_{\alpha}$ is open in $R$.
  • 2. For each $\alpha \in \Gamma$, $\phi_{\alpha}$ is a homeomorphism of $U_{\alpha}$ onto an open subset $V_{\alpha}$ of $\mathbb{C}$.
  • 3. $\displaystyle{R = \bigcup_{\alpha \in \Gamma} U_{\alpha}}$.
  • 4. For all $\alpha, \beta \in \Gamma$, $\phi_{\beta} \circ \phi_{\alpha}^{-1}$ is complex analytic from $\phi_{\alpha}(U_{\alpha} \cap U_{\beta})$ to $\phi_{\beta} (U_{\alpha} \cap U_{\beta})$.

If $R_1$ and $R_2$ are Riemann surfaces then we can define what it means for a map $f : R_1 \to R_2$ to be holomorphic.

Definition: Let $R_1$ and $R_2$ be Riemann surfaces with atlases $\{ (U_{\alpha}, \phi_{\alpha}) \}$ and $\{ (V_{\beta}, \psi_{\beta}) \}$ respectively. A map $f : R_1 \to R_2$ is said to be Holomorphic if for all charts $(U_{\alpha}, \phi_{\alpha})$ of $R_1$ and all charts $(V_{\beta}, \psi_{\beta})$ of $R_2$ we have that $\psi_{\beta} \circ f \circ \phi_{\alpha}^{-1}$ is holomorphic.

Note that $\phi_{\alpha}^{-1}$ maps an open subset of $\mathbb{C}$ onto $U_{\alpha} \subseteq R_1$. Then $f$ maps $U_{\alpha}$ into some subset of $R_2$. If $V_{\beta}$ intersects this subset of $R_2$, then $\psi_{\beta}$ maps that portion to some open subset of $\mathbb{C}$ (contained in the open subset for which $\psi_{\beta}$ maps $V_{\beta}$ onto).

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