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 which is locally homeomorphic to via charts (i.e., homeomorphisms) for open and such that is holomorphic.

**Examples **

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of is a Riemann surface in a natural manner.

The Riemann sphere or is a Riemann sphere with the open sets and the charts

The transition map is and thus holomorphic on .

An important example comes from analytic continuation, which I will briefly sketch below. A **function element** is a pair where is holomorphic and is an open disk. Two function elements are said to be **direct analytic continuations** of each other if and on . 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 , we can consider the totality of all equivalence classes of function elements that can be obtained by continuing along curves in . Then is actually a Riemann surface. (more…)