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Riemann Surfaces

Field: Complex Analysis

One-dimensional complex manifolds.

Sequence of Expressions

Definition

Definitions

There are several equivalent definitions of a Riemann surface. - A Riemann surface X is a connectedcomplex 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 x ∈ X 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 (connected) oriented manifold of (real) 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. A complex structure gives rise to a conformal structure by choosing the standard Euclidean metric given on the complex plane and transporting it to X by means of the charts. Showing that a conformal structure determines a complex structure is more difficult. - ^Farkas & Kra 1980, Miranda 1995 - ^See (Jost 2006, Ch. 3.11) for the construction of a corresponding complex structure.
As with any map between complex manifolds, a functionf : 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 ∘ f ∘ 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 biholomorphic (or conformally equivalent to emphasize the conformal point of view) 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. Each Riemann surface, being a complex manifold, is orientable as a real manifold. For complex charts f and g with transition function h = f(g^{−1}(z)), h can be considered as a map from an open set of R^{2} to R^{2} whose Jacobian in a point z is just the real linear map given by multiplication by the complex numberh′(z). However, the real determinant of multiplication by a complex number α equals |α|^{2}, so the Jacobian of h has positive determinant. Consequently, the complex atlas is an oriented atlas. Every non-compact Riemann surface admits non-constant holomorphic functions (with values in C). 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 (holomorphic functions with values in the Riemann sphereC ∪ {∞}). More precisely, the function field of X is a finite extension of C(t), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent. This statement generalizes to higher dimensions, see Siegel (1955). Meromorphic functions can be given fairly explicitly, in terms of Riemann theta functions and the Abel–Jacobi map of the surface. All compact Riemann surfaces are algebraic curves since they can be embedded into some CP^{n}. This follows from the Kodaira embedding theorem and the fact there exists a positive line bundle on any complex curve. - ^Nollet, Scott. "KODAIRA'S THEOREM AND COMPACTIFICATION OF MUMFORD'S MODULI SPACE Mg"(PDF).
- Branching theorem - Hurwitz's automorphisms theorem - Identity theorem for Riemann surfaces - Riemann–Roch theorem - Riemann–Hurwitz formula
Let RR be a Hausdorff topological space. A Riemann surface is a structure (R,A)(R, \mathcal{A}) such that A={(Ui,ϕi)}i∈I\mathcal{A} = \{(U_i, \phi_i)\}_{i \in I} is an atlas satisfying the following conditions:\n\n1. **Covering:** The collection of open sets {Ui}i∈I\{U_i\}_{i \in I} covers RR (i.e., R=⋃i∈IUiR = \bigcup_{i \in I} U_i).\n2. **Local Homeomorphism:** For each i∈Ii \in I, ϕi:Ui→Vi⊂C\phi_i: U_i \to V_i \subset \mathbb{C} is a homeomorphism, where ViV_i is an open subset of C\mathbb{C}.\n3. **Holomorphic Compatibility:** For any pair of indices i,j∈Ii, j \in I such that Ui∩Uj≠∅U_i \cap U_j \neq \emptyset, the transition map ψji=ϕj∘ϕi−1\psi_{ji} = \phi_j \circ \phi_i^{-1} restricted to ϕi(Ui∩Uj)\phi_i(U_i \cap U_j) must be a biholomorphic map from ϕi(Ui∩Uj)\phi_i(U_i \cap U_j) to ϕj(Ui∩Uj)\phi_j(U_i \cap U_j).\n\nEquivalently, RR is a one-dimensional complex manifold whose transition functions are holomorphic.