1 Broad Sense
In its broadest meaning, the term Riemann surface means a one-dimensional
complex manifold. Spelled out explicitly, this means that a Riemann surface is a Hausdorff
topological space together with a set of homeomorphisms between certain open subsets
of that space and open subsets of the complex plane which satisfy the following two
conditions:
- Every point of the Riemann surface lies in the domain of at least one of the
homeomorphisms.
- If the domains of two homeomorphisms overlap, the composition of one
homeomorphism with the inverse of the restriction of the other homeomorphism to the
overlap region is a complex analytic function.
The simplest example of a Riemann surface which is not a subset of the complex plane is the
Riemann sphere.
The main reason Riemann surfaces are interesting is that one can speak of analytic functions on
Riemann surfaces. A complex-valued function on a Riemann surface is said to be analytic if the
composition of this function with the inverse of any of the homeomorphisms mentioned in the
definition is a complex analytic function.
2 Narrow Sense
The term “Riemann surface” is often used in a narrower sense. A Riemann surface in the narrower
sense is a branched covering of the complex plane. That means that we have a one-dimensional
complex manifold together with a projection map from a dense open subset of that manifold to the
complex plane. (Note that the image of this projection map need not be the whole complex plane
— in fact, in the case of functions with natural boundary, it may not even be a dense subset
thereof.)
A particularly important motivation for this definition is the result that, given a (possibly
multiply-valued) analytic function defined on an open subset of the complex plane, there exists a
(single-valued) analytic function defined on a Riemann surface (in the narrow sense)
such that the pullback of the restriction of this function to a suitable open subset of
the Riemann surface under the projection map corresponds to the original function
defined on a subset of the complex plane. To show this result, one needs to exhibit a
method whereby, given a function defined on an open subset of the complex plane,
one can construct a suitable Riemann surface. Over the years, mathematicians have
devised several means for accomplishing these ends, and the remainder of this entry is
devoted to an exposition of some of these methods, starting with Riemann’s concrete
geometric approach and ending with more abstract approaches typical of contemporary
mathematics.
3 Riemann’s Construction
To begin our discussion, we may go back to Riemann’s original motivation for introducing his
surfaces. He was trying to make sense of “many-valued functions” such as the square root and the
logarithm. One way of making sense of such entities is by making “branch cuts” — i.e. removing
certain curves from the complex plane such that one is left with a dense open set on which the
function is well-defined (single-valued). For instance, to study the square root or the logarithm, one
typically removes the negative real axis.
Of course, if the original function is multiple-valued, there is more than one way of defining it in
the dense open set. Each of these possible definitions is called a branch of the function. For
instance, there is the negative branch and the positive branch of the square root. In the
case of the logarithm, the different branches differ by an additive factor of the form
2πin.
Riemann’s clever idea was to combine these different branches by means of a geometric device. He
imagined taking as many copies of the open set as there are branches of the function and joining
them together along the branch cuts. To understand how this works, imagine cutting out
sheets along the branch curves and stacking them on top of the complex plane. On
each sheet, we define one branch of the function. We glue the different sheets to each
other in such a way that the branch of the function on one sheet joins continuously
at the seam with the branch defined on the other sheet. For instance, in the case of
the square root, we join each end of the sheet corresponding to the positive branch
with the opposite end of the sheet corresponding to the negative branch. In the case
of the logarithm, we join one end of the sheet corresponding to the 2πn branch with
an end of the (2n + 1)πn sheet to obtain a spiral structure which looks like a parking
garage.
The advantage of Riemann’s construction is that one has now constructed a geometric space on
which the function is well-defined and single valued. The multi-valuedness and the branches are
easily understood form this viewpoint —the function appears to have many values at a single point
because we did not distinguish between different points on the Riemann surface which
project to the same point on the complex plane and instead tried to think (somewhat
illogically) of the function as having more than one value at a single point on the complex
plane rather than as having different values at several points which correspond to this
point.
The modern reader will recognize in Riemann’s cut-and-paste procedure the same idea of
combining open sets to create a space which underlies the modern definition of manifold cited at
the beginning of this article. In the nineteeenth century, even though such topological concepts as
manifolds were known, they were not rigorously defined. Indeed, even the definition of open and
closed sets would not be introduced until the end of the nineteenth century and the
beginning of the twentieth century, some 50 years after Riemann. In the meanwhile,
Riemann, Betti, Maxwell, Tait, Moebius and others had to rely on intuitive topological
ideas.
This old description of Riemann surfaces is worth knowing about because it explains terminology
such as “sheet of the Riemann surface” which is still in use today. Moreover, the lead author
believes that there is no better way of coming to terms with Riemann surfaces than by taking
scissors, paper, and tape and constructing models of Riemann surfaces. (he has done this himself
several times) It is easiest to start with the Riemann surfaces for the square root and the
logarithm. After one gains some experience cutting and gluing together Riemann surfaces, one can
try some more complicated examples as the Riemann surface of the function f(z) = 
.
When one has constructed this surface and convinced oneself that it has the topology of a
torus, one is well on one’s way to developing an intuitive understanding of Riemann
surfaces.
4 Weyl’s Approach
A half-century after Riemann, H. Weyl provided a rigorous construction of Riemann surfaces using
techniques of topology. Aside from its historical interest, this construction, which is based upon
analytic continuation by power series, is rather interesting in its own right, so we shall discuss it
here.
The basic strategy of this construction is to construct the surface as a subspace of a larger
topological space Ω. The elements of the underlying set of Ω are pairs consisting of a point
of the complex plane and a convergent power series about that point. The topology
may be described via a subbasis as follows. Given an element (z,{ak}k=0∞) ∈ Ω and
real number 𝜖 between zero and the radius of convergence of {ak}k=0∞, we define a
subbaisis element B ⊂ Ω as follows: A pair (w,{bk}k=0∞) belongs to B if |w − z| < 𝜖 and
∑
k=0∞a
k(ζ − z)k = ∑
k=0∞b
k(ζ − w)k for all ζ ∈ ℂ such that both series converge.
(Because |w − z| is less than the radius of convergence of the former series, there exists an
infinity of such points ζ and, in fact, the bk’s are uniquely determined in terms of the
αk’s.)
A Riemann surface is then defined to be a closed, connected subspace of Ω. The projection map
from the Riemann surface to the complex plane simply is the map which sends each pair to its first
component. Using this map, one can show that what has been constructed is in fact a complex
manifold. This is essentially a routine verification; for every subbasis element this map is an
isomorphism to an open subset of the complex plane, so the totality of subbasis elements form an
atlas.
An fascinating feature of this construction is that it provides all possible Riemann surfaces at once.
The topological space Ω has many connected components, each of which is a different Riemann
surface.
5 Space of Paths
Another approach is based upon analytic continuation along paths and resembles the construction
of the universal cover of a manifold as a set of paths. In this approach, we start with an
analytic function defined in the neighborhood of some point. We consider the totality of all
paths along which this function may be analytically continued. Next, we identify paths
which have the same endpoints and can be deformed homotopically into each other by a
one-parameter family of paths along which the function can be analytically continued. (By
the monodromy theorem, analytically continuing the function along paths which are
to be identified will lead to the same result.) It turns out that this set of identified
paths has the structure of a complex manifold. Since the proof of this latter assertion is
somewhat technical and would interrupt the flow of exposition, it has been relegated to an
attachment.
As it turns out, this construction does not produce all Riemann surfaces — it only produces the
simply connected ones. To produce the remaining surfaces, one needs to perform yet
another identification. One must identify all closed loops such that analytic continuation
along these loops leads to the same function and identify paths which differ by such a
loop. A beautiful example of this identification is provided by an example considered
earlier, the Riemann surface of f(z) = 
. If one does not make identifications,
one obtains a plane on which f lifts to a doubly periodic elliptic function. Identifying
by the two periods forms a torus which, as we saw earlier, is the Riemann surface of
f.
6 Sheaves
Just as the two foregoing approaches may be seen as being based upon the analytic continuation
by power series and along paths, respectively, so the approach to be described now may be seen as
based upon analytic continuation along chains of open sets. Define a function element to be a pair
(D,f) where D is an open subset of the complex plane and f : D → ℂ is analytic. The set of
all function elements form a sheaf over the complex plane, which we shall denote as
𝒮.
Next, we consider the germs of our sheaf 𝒮. One may show that the germs of 𝒮 correspond to
subsets of 𝒮 which satisfy the following properties:
- If (D,f) ∈ G and (E,g) ∈ G, then D ∩ E is not empty.
- If (D,f) ∈ G and (E,g) ∈ G, then f(x) = g(x) for all x ∈ D ∩ E.
- No proper superset of G satisfies both properties 1 and 2 above.
Given a germ G, there will exist exactly one point p such that p ∈ D for all (D,f) ∈ G. We call
this point the basepoint of the germ and have a projection map which sends germs to their
basepoints.
Analogously to what Weyl did, we now introduce a topology on the set of germs. To every function
element (D,f), we will assign an set B(D,f) consisting of all germs X ∈𝒢 such that (D,f) ∈ X.
The collection of such sets satisfies the defining conditions for a basis of a topology, hence defines a
topology on 𝒢. Just as above, we define a Riemann surface is then defined to be a closed,
connected subspace of 𝒢 under this topology; in fact this topology is hoemomorphic to the space Ω
in Weyl’s approach.
7 Comparison of Approaches
8 Branch Points
Bibliography
H. Cohn, Conformal Mapping on Riemann Surfaces, Dover Publishing, 1967
H. Weyl, The Concept of a Riemann Surface
