In the late 1700’s and early 1800’s, the French mathematician Augustin-Louis Cauchy (1789 – 1857) and German mathematician Bernhard Riemann (1826 – 1866) worked independently to create a theory of differentiating complex-valued functions . In order to define such functions, they realized they would need a way to define their domains of definition. In this lecture, we review some of the ideas surrounding Riemann Surfaces.
Riemann Surfaces
A compact, connected Riemann Surface is a triple satisfying the following five properties:
- Atlas: For each from a finite indexing set , the pair is a chart. That is, is a “cover” of , and is an injective map.
I like to think of an atlas as a book, where the are the page numbers, the table contents is the collection of injective maps, and each page is a set . Indeed, if you have an atlas of the Earth, you can lay out various photos of the globe as flat sheets of paper, but when you combine all of the photos in the atlas you recover a spherical planet.
- Locally Euclidean: Each is a connected, open subset of . Moreover, the transition maps defined as the compositions are analytic functions, that is, complex-valued functions where the Cauchy-Riemann equations hold: and where while .
The following diagram may be helpful to keep track of the transition maps:
- Hausdorff: Points in can be separated by neighborhoods. That is, for distinct and there exist open subsets such that
and in while in . - Compact: Say that is an open cover of , that is, while are open sets. Then we require that it has a finite subcover .
- Connected: The only continuous functions are the constant functions.
Here, we endow with the discrete topology (that is, every subset is considered open), and say that a map is continuous if the inverse image is open for every subset and .
We often abuse notation and say “ is a Riemann surface” instead of “ is a compact, connected Riemann surface.” We say that a function is analytic on if each composition is analytic as a function on the subset for each index .
Instead of defining as a Riemann surface, we can define it as an -dimensional real manifold if we replace with and “analytic function” with “smooth function.” For this reason, we may think of a Riemann surface as a 1-dimensional complex manifold, that is, a complex curve.
The Riemann Sphere
Recall that the sphere of radius is the 2-dimensional surface . We explain why is a Riemann surface.
We will use the following charts:
Since is just the sphere minus the “north pole” while is the sphere minus the “south pole”, we have the union . It is easy to see that and are well-defined, injective maps because , for example, has inverse
.
This map is usually called stereographic projection; you should compare this formula with those given in Lecture 2. You’ll note that and are open sets regardless of whether you use the Euclidean topology on or the Zariski topology on .
Finally, it is easy to check that we have the transition maps on :
Clearly and are both examples of analytic functions because they are rational functions. We leave it as an exercise to show that points on can be separated by neighborhoods, that is compact, and that it is connected.
Endomorphisms of the Sphere
We end today’s lecture with a discussion about how rational functions are related to rotations of the sphere.
Let me begin with an observation and a definition which will be useful later. Fix two relatively prime polynomials and , that is, polynomials such that the determinant
of the Sylvester Matrix is nonzero. The rational function defines a mapping which sends , so we can use the ideas above to extend to a mapping which sends . I’ll explain.
For any radius , we define a map using stereographic projection: in terms of the complex number . (Recall that is the complex conjugate of .) Amazingly, even there are issues defining when the denominator , the function is always well-defined. We may consider this to be an endomorphism of the sphere because it maps the sphere to itself.
Let me give a simple example. Say that in terms of a complex number with . Since for some angle , we see that is a rotation through angle .
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