synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
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. But the main definition by far is the classical one.
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.
It is not necessary to make any assumption about whether there exists a countable base for the topology (second countable) or whether it has a countable dense subset (separable). This is because Tibor Rado (1923) proved that all Riemann surfaces, without such prior assumptions, must necessarily have a countable base. Thus for elegance this condition is customarily not assumed by specialists in Riemann surfaces.
There are generalizations, e.g., over local fields in rigid analytic geometry.
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 surface 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.
To see this, note that is compact, and an open subset by the open mapping theorem?, so the result follows by connectedness of .
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
and
To see this, we have tacitly observed that .
The Picard group of a Riemann surface is the group of holomorphic line bundles in it. Introductions include (Bobenko, section 8).
See also at Narasimhan–Seshadri theorem and at moduli space of connections – Flat connections over a torus.
In the theory of Riemann surfaces, there are several important theorems. Here are two:
The Riemann-Roch theorem, which analyzes the vector space of meromorphic functions satisfying certain conditions on zeros and poles;
The uniformization theorem?, which partially classifies Riemann surfaces.
A compact Riemann surface of genus is a homotopy 1-type. The fundamental groupoid is a Fuchsian group.
By the Riemann existence theorem, every connected compact Riemann surface admits the structure of a branched cover of the Riemann sphere. (MO discussion)
Historical references:
Hermann Weyl, Die Idee der Riemannschen Fläche, 1913 (The concept of a Riemann surface) (on the book, by Peter Schreiber, 2013: web)
Tibor Radó, Über den Begriff der Riemannschen Fläche, Acta Litt. Sci. Szeged, 2 (101-121), 10 (pdf, pdf)
(proving the triangulation theorem)
Monograph:
Lecture notes:
Alexander Bobenko, Compact Riemann Surfaces lecture notes (pdf)
Eberhard Freitag, Riemann surfaces – Sheaf theory, Riemann Surfaces, Automorphic forms, 2013 (pdf)
Joachim Wehler, Riemann surfaces 2020 (pdf)
Last revised on October 18, 2023 at 05:51:56. See the history of this page for a list of all contributions to it.