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…)