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.
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.
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.
Theorem 1 Let be a regular map. If is compact and is nonconstant, then is surjective and compact.
To see this, note that is compact, and by the open mapping theorem, open, so the result follows by connectedness of .
Since a Riemann surface is a smooth 2-dimensional manifold, 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 I will in the future just abbreviate ; this will not be confusing since I will only do this when I talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.
I think the natural setting of all this leads to the theory of almost complex manifolds, which sometime soon I will try to learn and/or blog about.
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.
I 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 .
Admittedly this isn’t the best place to stop. Next time, I’ll discuss more about how one can manipulate 1- and 2-forms. Pretty soon we can then get to some big theorems.