Riemann surface

In complex analysis, a Riemann surface is a one-dimensional complex manifold. It can be thought of as a surface that locally looks like a patch of the complex plane C. Globally, a Riemann surface may appear very different from C however: it could look like a sphere or a torus or several sheets glued together.

The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially "multi-valued" ones such as the square root or the logarithm.

Every Riemann surface is a two-dimensional real analytic manifold, but it contains slightly more structure which is needed for the unambiguous definition of holomorphic functions. Some two-dimensional real manifolds are not Riemann surfaces; for instance, the Moebius strip is a two-manifold, but it cannot be viewed as a Riemann surface (because all Riemann surfaces are orientable.)

Formal definition

Let X be a topological space. A homeomorphism from an open subset of X to C in this context is called a chart. Two charts f and g whose domains intersect are said to be compatible if the maps h₀(x) = f(g^-1(x)) and h₁(x) = g(f^-1(x)) are holomorphic over their domains. If A is a collection of compatible charts and if any x in X is in the domain of some f in A, then we say that A is an atlas. When we endow X with an atlas A, we say that (X, A) is a Riemann surface. If the atlas is understood, we simply say that X is a Riemann surface.

Different atlases can give rise to essentially the same Riemann surface structure on X; to avoid this ambiguity, one sometimes demands that the given atlas on X be maximal, in the sense that it is not contained in any other atlas. Every atlas A is contained in a unique maximal one.

Examples

• The complex plane C is perhaps the most trivial Riemann surface. The map f(z) = z (the identity map) defines a chart for C, and {f} is an atlas for C. The map g(z) = z^* (the conjugate map) also defines a chart on C and {g} is an atlas for C. The charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its atlas A, the conjugate atlas B = {f^* ; f ∈ A} is never compatible with A, and endows X with a distinct, incompatible Riemann structure.

• In an analogous fashion, every open subset of the complex plane can be viewed as a Riemann surface in a natural way.

• Let S := C ∪ {∞} and let f(z) = z where z is in S \ {∞} and g(z) = 1 / z where z is in S \ {0} and 1/∞ is defined to be 0. Then f and g are charts, they are compatible, and { f, g } is an atlas for S, making S into a Riemann surface. This particular surface is called the Riemann sphere because it can be interpreted as wrapping the complex plane around the sphere. Unlike the complex plane, it is compact.

• Start with holomorphic functions a₀(z),...,a_n-1(z) and consider the set of all pairs (z, w) in C² such that

wⁿ + a_n-1(z)w^n-1 + a_n-2(z)w^n-2... + a₀(z) = 0.

This set can be turned into a Riemann surface in a natural fashion; Riemann surfaces of this type are called concrete.

• The main examples of Riemann surfaces are provided by analytic continuation (see below.)

Properties

We noted in the preamble that all Riemann surfaces are orientable. The details are beyond the scope of this article, but the basic idea is that if a Riemann surface weren't orientable, then there would be a point x and charts f and g whose domains include x, such that h = f(g^-1(z)) is locally a reflection. (Looking at h as a map from the plane to itself, its Jacobian would have a negative determinant.) No holomorphic map is allowed to behave this way, and by the compatibility requirement for charts in an atlas, h needs to be holomorphic.

Analytic continuation

Let

$f(z)=\sum _{k=0}^{\infty }\alpha _{k}(z-z_{0})^{k}$

be a power series converging in D_r(z₀) := {z in C : |z - z₀| < r} for r > 0. (Note, without loss of generality, here and in the sequel, we will always assume that a maximal such r was chosen, even if it is ∞.) Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector

g = (z₀, α₀, α₁, α₂, ...)

is a germ of f. The base g₀ of g is z₀, the stem of g is (α₀, α₁, α₂, ...) and the top g₁ of g is α₀. The top of g is the value of f at z₀, the bottom of g.

Any vector g = (z₀, α₀, α₁, ...) is a germ if it represents a power series of an analytic function around z₀ with some radius of convergence r > 0. Therefore, we can safely speak of the set of germs ${\mathcal {G}}$.

The topology of ${\mathcal {G}}$

If g and h are germs, if |h₀ - g₀| < r where r is the radius of convergence of g and if the power series that g and h represent define identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g, and we write g ≥ h. This compatibility condition is neither transitive, symmetric nor antisymmetric. If we extend the relation by transitivity, we obtain an equivalence relation on germs (not an ordering.) This extending by transitivity is sometimes called analytic continuation. The equivalence relation will be denoted $\cong$.

We can define a topology on ${\mathcal {G}}$. Let r > 0, and let

U_r(g) := {h ∈ ${\mathcal {G}}$ : g ≥ h , |g₀ - h₀| < r}

The sets U_r(g), for all r > 0 and g ∈ ${\mathcal {G}}$ define a basis of open sets for the topology on ${\mathcal {G}}$.

A connected component of ${\mathcal {G}}$ (i.e., an equivalence class) is called a sheaf. We also note that the map φ_g(h) = h₀ from U_r(g) to ${\mathbb {C}}$ where r is the radius of convergence of g, is a chart. The set of such charts forms an atlas for ${\mathcal {G}}$, hence ${\mathcal {G}}$ is a Riemann surface. ${\mathcal {G}}$ is sometimes called the universal analytic function.

Examples of analytic continuation

$L(z-1):=\sum _{k=1}^{\infty }(-1)^{k+1}(z-1)^{k}$

is a power series corresponding to the natural logarithm near z = 1. This power series can be turned into a germ

g = (1, 0, 1, -1, 1, -1, 1, -1, ...)

This germ has a radius of convergence of 1, and so there is a sheaf S corresponding to this germ. This is the sheaf of the logarithm function.

The uniqueness theorem for analytic functions also extends to sheaves of analytic function. If the sheaf of an analytic function contains the zero germ (i.e., the sheaf is uniformly zero in some neighborhood) then the entire sheaf is zero. Armed with this result, we can see that if we take any germ g of the sheaf S of the logarithm function, as described above, and turn it into a power series f(z) then this function will have the property that exp(f(z))=z. If we had decided to use a version of the inverse function theorem for analytic functions, we could construct a wide variety of inverses for the exponential map, but we would discover that they are all represented by some germ in S. In that sense, S is the "one true inverse" of the exponential map.

In older literature, sheaves of analytic functions were called multi-valued functions.

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See also: Monodromy theorem.