Riemann surface: Difference between revisions

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 ${\mathbb {C}}$. Globally, a Riemann surface may appear very different from ${\mathbb {C}}$ 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 surface; for instance, the Moebius strip is a two-manifold, but it can not be viewed as a Riemann surface (because all Riemann surfaces are orientable.)

Technical discussion

Let X be a topological space. A homeomorphism from an open subset of X to ${\mathbb {C}}$ in this context is called a chart. Two charts f and g whose domains intersect are said to be compatible if the maps $h_{0}(x)=f(g^{-1}(x))$ and $h_{1}(x)=g(f^{-1}(x))$ are analytic over their domains. If ${\mathcal {A}}$ is a collection of compatible charts and if any $x\in X$ is in the domain of some $f\in {\mathcal {A}}$ then we say that ${\mathcal {A}}$ is an atlas. When we endow X with an atlas ${\mathcal {A}}$, we say that $(X,{\mathcal {A}})$ is a Riemann surface. If the atlas is understood, we simply say that X is a Riemann surface.

If ${\mathcal {A}}$ and ${\mathcal {B}}$ are atlases on X and if ${\mathcal {A}}\cup {\mathcal {B}}$ is also an atlas on X, then we say that ${\mathcal {A}}$ and ${\mathcal {B}}$ are compatible and we write ${\mathcal {A}}\cong B$. Given an atlas ${\mathcal {A}}$ then we can look at ${\hat {\mathcal {A}}}:=\cup _{{\mathcal {B}}\cong {\mathcal {A}}}{\mathcal {B}}$. Since compatibility of atlases is an equivalence relation, ${\hat {\mathcal {A}}}$ is also an atlas, and it contains all atlases compatible with itself. In that sense ${\hat {\mathcal {A}}}$ is said to be maximal. Some demand that an atlas be maximal to begin with.

If we are not particularly concerned with the precise choice of atlas within a family of compatible atlases, and if the family of compatible atlases is well understood, we can speak of the Riemann surface structure on X.

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 analytic map is allowed to behave this way, and by the compatibility requirement for charts in an atlas, h needs to be analytic.

Analytic continuation

We could begin with an analytic function defined on some open subset of ${\mathbb {C}}$, but it is simpler to instead use power series.

Let $f(z)=\sum _{k=0}^{\infty }\alpha _{k}(z-z_{0})^{k}$ be a power series converging in $D_{r}(z_{0}):=\{z\in {\mathbb {C}}|\left|z-z_{0}\right|<r\}$ 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 $\infty$.) Then we say that the vector $g=(z_{0},\alpha _{0},\alpha _{1},\alpha _{2},...)$ is a germ of $f$. The base $g_{0}$ of g is $z_{0}$, the stem of g is $(\alpha _{0},\alpha _{1},\alpha _{2},...)$ and the top $g_{1}$ of g is $\alpha _{0}$. The top of g is the value of f at $z_{0}$, the bottom of g.

Any vector $g=(z_{0},\alpha _{0},\alpha _{1},...)$ is a germ if it represents a power series of an analytic function around $z_{0}$ with some radius of convergence $r>0$. Therefore, we can safely speak of the set of germs ${\mathcal {G}}$.

If g and h are germs, if $|h_{0}-g_{0}|<r$ 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 $g\geq h$. This compatibility condition is not transitive nor reflexive, however, we can extend it by transitivity, and we obtain an equivalence relation on germs. 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 $U_{r}(g):=\{h\in {\mathcal {G}}:g\geq h,|g_{0}-h_{0}|<r\}$. The sets $U_{r}(g)$, for all $r>0$ and $g\in {\mathcal {G}}$ 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 $\varphi _{g}(h)=h_{0}$ 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.

As an example, the power series

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

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 corresponding to this germ. This is the sheaf of the logarithm function.

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

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See also: Monodromy theorem.