A Riemann surface is a -dimensional algebro-geometric object with good properties. The name ‘surface’ comes from the classical case, which is -dimensional over the complex numbers and therefore -dimensional over the real numbers.
There are several distinct meaning of what is a Riemann surface, and it can be considered in several generalities.
Classically, a Riemann surface is a connected complex--dimensional complex manifold, in the strictest sense of ‘manifold’. In other words, it’s a Hausdorff second countable space which is locally homeomorphic to the complex plane via charts (i.e., homeomorphisms) for open and such that is holomorphic.
Evidently an open subspace 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 we 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.
Indeed, we must first put a topology on . If with centered at , then let a neighborhood of be given by all function elements for ; these form a basis for a suitable topology on . Then the coordinate projections form appropriate local coordinates. In fact, there is a globally defined map , whose image in general will be a proper subset of .
Since we have local coordinates, we can define a map of Riemann surfaces to be holomorphic or regular if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map ; for meromorphicity, this becomes .
Many of the usual theorems of elementary complex analysis (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a Laurent expansion, etc.
Let be a regular map. If is compact and is nonconstant, then is surjective and compact.
Since a Riemann surface is a -dimensional smooth manifold in the usual (real) sense, it is possible to do the usual exterior calculus. We could consider a 1-form to be a section of the (usual) cotangent bundle , but it is more natural to take the complexified cotangent bundle , which we will in the future just abbreviate ; this should not be confusing since we will only do this when we talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.
If is a local coordinate on , defined say on , define the (complex) differentials
These form a basis for the complexified cotangent space at each point of . There is also a dual basis
for the complexified tangent space.
We now claim that we can split the tangent space , where the former consists of multiples of and the latter of multiples of ; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given for open and (just for convenience), we can write
where , which we will often abbreviate as . If is holomorphic and conformal sending , we have
in particular, preserves the decomposition of .
Given smooth, we can consider the projections of the 1-form onto and , respectively; these will be called . Similarly, we define the corresponding operators on 1-forms: to define , first project onto (the reversal is intentional!) and then apply , and vice versa for .
In particular, if we write in local coordinates , then
To see this, we have tacitly observed that .
In the theory of Riemann surfaces, there are several important theorems. Here are two: