Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
Examples of Riemann surfaces include graphs of multivalued functions such as √z or log, e.g. the subset of pairs with.
Every Riemann surface is a surface: a two-dimensional real manifold, but it contains more structure. Conversely, a two-dimensional real manifold can be turned into a Riemann surface if and only if it is orientable and metrizable. Given this, the sphere and torus admit complex structures but the Möbius strip, Klein bottle and real projective plane do not. Every compact Riemann surface is a complex algebraic curve by Chow's theorem and the Riemann–Roch theorem.
Definitions
There are several equivalent definitions of a Riemann surface.- A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point there is a neighbourhood of x that is homeomorphic to the open unit disk of the complex plane, and the transition maps between two overlapping charts are required to be holomorphic.
- A Riemann surface is a oriented manifold of dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the space is homeomorphic to a subset of the real plane. The supplement "Riemann" signifies that X is endowed with an additional structure that allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same. Choosing an equivalence class of metrics on X is the additional datum of the conformal structure.
Examples
Algebraic curves
Further definitions and properties
As with any map between complex manifolds, a function 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 is holomorphic wherever it is defined. The composition of two holomorphic maps is holomorphic. The two Riemann surfaces M and N are called biholomorphic if there exists a bijective holomorphic function from M to N whose inverse is also holomorphic. Two conformally equivalent Riemann surfaces are for all practical purposes identical.Orientability
Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts f and g with transition function, h can be considered as a map from an open set of R2 to R2 whose Jacobian in a point z is just the real linear map given by multiplication by the complex number h′. However, the real determinant of multiplication by a complex number α equals 2, so the Jacobian of h has positive determinant. Consequently, the complex atlas is an oriented atlas.Functions
Every non-compact Riemann surface admits non-constant holomorphic functions. In fact, every non-compact Riemann surface is a Stein manifold.In contrast, on a compact Riemann surface X every holomorphic function with values in C is constant due to the maximum principle. However, there always exist non-constant meromorphic functions. More precisely, the function field of X is a finite extension of C, the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see. Meromorphic functions can be given fairly explicitly, in terms of Riemann theta functions and the Abel–Jacobi map of the surface.
Algebraicity
All compact Riemann surfaces are algebraic curves since they can be embedded into some CPn. This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve.Analytic vs. algebraic
The existence of non-constant meromorphic functions can be used to show that any compact Riemann surface is a projective variety, i.e. can be given by polynomial equations inside a projective space. Actually, it can be shown that every compact Riemann surface can be embedded into complex projective 3-space. This is a surprising theorem: Riemann surfaces are given by locally patching charts. If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of analytic or algebraic geometry. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic. On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem.As an example, consider the torus T :=. The Weierstrass function ℘τ belonging to the lattice is a meromorphic function on T. This function and its derivative ℘τ′ generate the function field of T. There is an equation
where the coefficients g2 and g3 depend on τ, thus giving an elliptic curve Eτ in the sense of algebraic geometry. Reversing this is accomplished by the j-invariant j, which can be used to determine τ and hence a torus.
Classification of Riemann surfaces
The set of all Riemann surfaces can be divided into three subsets: hyperbolic, parabolic and elliptic Riemann surfaces. Geometrically, these correspond to surfaces with negative, vanishing or positive constant sectional curvature. That is, every connected Riemann surface X admits a unique complete 2-dimensional real Riemann metric with constant curvature equal to −1, 0 or 1 that belongs to the conformal class of Riemannian metrics determined by its structure as a Riemann surface. This can be seen as a consequence of the existence of isothermal coordinates.In complex analytic terms, the Poincaré–Koebe uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of the following:
- The Riemann sphere, which is isomorphic to P1;
- The complex plane C;
- The open disk, which is isomorphic to the upper half-plane.
Elliptic Riemann surfaces
The Riemann sphere P1 is the only example, as there is no group acting on it by biholomorphic transformations freely and properly discontinuously and so any Riemann surface whose universal cover is isomorphic to P1 must itself be isomorphic to it.Parabolic Riemann surfaces
If X is a Riemann surface whose universal cover is isomorphic to the complex plane C then it is isomorphic to one of the following surfaces:- C itself;
- The quotient ;
- A quotient, where with.
Hyperbolic Riemann surfaces
In the remaining cases, X is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a Fuchsian group. The topological type of X can be any orientable surface save the torus and sphere.A case of particular interest is when X is compact. Then its topological type is described by its genus. Its Teichmüller space and moduli space are -dimensional. A similar classification of Riemann surfaces of finite type can be given. However, in general the moduli space of Riemann surfaces of infinite topological type is too large to admit such a description.