Finally, it’s time to get to the definition of a Riemann surface. A Riemann surface is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space ${M}$ which is locally homeomorphic to ${\mathbb{C}}$ via charts (i.e., homeomorphisms) ${\phi_i:U_i \rightarrow V_i}$ for ${U_i \subset M, V_i \subset \mathbb{C}}$ open and such that ${\phi_j \circ \phi_i^{-1}: V_i \cap V_j \rightarrow V_i \cap V_j}$ is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of ${\mathbb{C}}$ is a Riemann surface in a natural manner.

The Riemann sphere ${P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}}$ or ${S^2}$ is a Riemann sphere with the open sets ${U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}}$ and the charts $\displaystyle \phi_1 =z, \ \phi_2 = \frac{1}{z}.$

The transition map is ${\frac{1}{z}}$ and thus holomorphic on ${U_1 \cap U_2 = \mathbb{C}^*}$.

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair ${(f,V)}$ where ${f: V \rightarrow \mathbb{C}}$ is holomorphic and ${V \subset \mathbb{C}}$ is an open disk. Two function elements ${(f,V), (g,W)}$ are said to be direct analytic continuations of each other if ${V \cap W \neq \emptyset}$ and ${f \equiv g }$ on ${V \cap W}$. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element ${\gamma = (f,V)}$, we can consider the totality ${X}$ of all equivalence classes of function elements that can be obtained by continuing ${\gamma}$ along curves in ${\mathbb{C}}$. Then ${X}$ is actually a Riemann surface. (more…)