Riemann surface

In complex analysis, a Riemann surface is a one-dimensional complex manifold. It can be thought of as a "deformed version" of the complex plane: locally near every point it looks like a patch of the complex plane, but the overall global geometry of it can be quite different from the complex plane: it can look like a sphere or a torus or a couple of sheets glued together or like any ordinary (real) two-dimensional orientable surface. The Möbius strip is not orientable, and hence cannot be turned into a Riemann surface.

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 more structure which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface (usually in several different ways) if and only if it is orientable.

Table of contents 1 Formal definition 2 Examples 3 Properties and further definitions 4 Analytic continuation 5 The topology of <math>\mathcal G</math> 6 Examples of analytic continuation

Formal definition

Let X be a Hausdorff 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. More generally, every open subset of a Riemann surface is a Riemann surface.

• 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.

• The theory of compact Riemann surfaces can be shown to be equivalent to that of algebraic curves that are defined over the complex numbers and non-singular. Important examples of non-compact Riemann surfaces are provided by analytic continuation (see below.)

Properties and further definitions

A function f : M → N between two Riemann surfaces M and N is called holomorphic if for every chart g in the atlas of M and every chart h in the atlas of N, the map h o f o g^-1 is holomorphic (as a function from C to C) wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces M and N are called conformally equivalent if there exists a bijective holomorphic function from M to N whose inverse is also holomorphic (it turns out that the latter condition is automatic and can therefore be omitted). Two conformally equivalent Riemann surfaces are for all practical purposes identical.

Every simply connected Riemann surface is conformally equivalent to C or to the Riemann sphere C ∪ {∞} or to the open disk {z ∈ C : |z| < 1}. This statement is known as the uniformization theorem[?].

Every connected Riemann surface can be turned into a complete 2-dimensional real Riemannian manifold with constant curvature -1, 0 or 1. This Riemann structure is unique up to scalings of the metric. The Riemann surfaces with curvature -1 are called hyperbolic; the open disk is the canonical example. The Riemann surfaces with curvature 0 are called parabolic; C is a typical parabolic Riemann surface. Finally, the surfaces with curvature +1 are known as elliptic; the Riemann sphere C ∪ {∞} is an example.

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

<math>f(z)=\sum_{k=0}^\infin \alpha_k (z-z_0)^k</math>

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 <math>\mathcal G</math>.

The topology of <math>\mathcal G</math>

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 <math>\cong</math>.

We can define a topology on <math>\mathcal G</math>. Let r > 0, and let

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

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

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

Examples of analytic continuation

<math>L(z-1):=\sum _{k=1}^\infin (-1)^{k+1}(z-1)^k</math>

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. See sheaf for the general concept.

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See also: Monodromy theorem[?].