Riemann surface

A Riemann surface is a one-dimensional complex manifold in the mathematical sub-area of function theory (complex analysis) . Riemann surfaces are the simplest geometric objects that have the structure of complex numbers locally . They are named after the mathematician Bernhard Riemann . The investigation of Riemann surfaces falls into the mathematical field of function theory and depends largely on methods of algebraic topology and algebraic geometry .

Riemann area of ​​the complex logarithm . The leaves arise due to the ambiguity, i. H. because the complex exponential function is not injective .

Historically, the Riemann surface is the answer to the fact that holomorphic functions do not always have unambiguous continuations. For example, the main branch of the complex logarithm (which is defined in a neighborhood of ) receives the additional argument when continuing along a positively oriented circle around 0 . ${\displaystyle z=1}$ ${\displaystyle 2\pi \mathrm {i} }$

history

The theory of Riemann surfaces arose from the fact that different function values ​​can arise in the analytical continuation of holomorphic functions along different paths, as is the case, for example, with the complex logarithm. In order to get unambiguous continuations again, the domain of definition was replaced by a multi-leaf area that had as many leaves as there were possibilities to continue the function. The analytical continuation is clear again on such an overlapping surface. Bernhard Riemann initially explained the areas named after him as follows: Several (possibly infinitely many) complex number levels are superimposed, provided with certain (for example straight) cuts and then glued together along these cuts. This vivid idea was very fruitful at first, although it was criticized as inexact. Today's definition comes from Hermann Weyl . In his book The Idea of ​​the Riemann Surface (1913) he defined what is now the fundamental concept of (real or complex) manifold . If Riemann was concerned with the analytical continuation of a concretely given function, then with the abstract definition of a Riemann surface by Weyl the question arises whether complex functions exist at all on such a manifold. The Riemann mapping theorem and the Riemann-Roch theorem provide an answer.

definition

A Riemann surface is a complex manifold of dimension one. ${\displaystyle X}$

This means that it is a Hausdorff room that is equipped with a complex structure . (The second axiom of countability , which is otherwise required in the definition of complex manifolds, does not need to be presupposed in the definition of Riemann surfaces, because there it follows from the other properties according to Radó's theorem.) ${\displaystyle X}$

Many authors also demand that Riemann surfaces should be contiguous .

Complex curve

Every compact Riemann surface is biholomorphic into a smooth, complex projective variety of dimension one. In algebraic geometry , a Riemann surface is called a smooth complex curve .

Examples

The Riemann number ball

• The complex plane is the simplest Riemannian surface. The identical figure defines a map for whole , therefore the set is an atlas for . ${\displaystyle \mathbb {C} }$ ${\displaystyle f(z)=z}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \{f\}}$ ${\displaystyle \mathbb {C} }$

A torus

• Each area is also a Riemannian area. Here, too, is the identical illustration, a map for the entire area. More generally, every open subset of a Riemann surface is again a Riemann surface. ${\displaystyle G\subset \mathbb {C} }$

• The Riemann number ball is a compact Riemann surface. It is sometimes also referred to as a complex projective straight line or short . ${\displaystyle \mathbb {C} \cup \{\infty \}}$ ${\displaystyle \mathbb {C} P^{1}}$ ${\displaystyle \mathbb {P} _{1}}$

• The torus surface for a lattice on which the elliptic functions are explained is a compact Riemann surface. ${\displaystyle \mathbb {C} /\Gamma }$ ${\displaystyle \Gamma }$

Theory of Riemann surfaces

Due to the complex structure on the Riemann surface, it is possible to define holomorphic and meromorphic images on and between Riemann surfaces. Many of the theorems from function theory on the complex level about holomorphic and meromorphic functions can be generalized for Riemann surfaces. In this way, Riemann's theorem of liftability , the principle of identity and the principle of maximum can be transferred to Riemannian surfaces. However, it must be noted that the holomorphic functions are not particularly rich, especially on compact Riemann surfaces. Precisely this means that a holomorphic function must always be constant on a coherent, compact surface . A coherent, compact Riemann surface is therefore not holomorphically separable , only the constant holomorphic functions exist on it. (For disjointed, compact Riemann surfaces, these statements apply if constant is replaced by locally constant .) Cauchy's integral theorem and Cauchy's integral formula , two central theorems of the function theory of the complex plane, cannot be proven analogously on Riemann surfaces. On differentiable manifolds in general and on Riemannian surfaces in particular, the integration must be explained with the help of differential forms so that it is independent of the choice of map. However, Stokes' theorem, which is central to integration theory, does exist . With its help one can prove the residual theorem , which follows from Cauchy's integral formula in the complex plane, also for Riemann surfaces. ${\displaystyle f\colon X\to \mathbb {C} }$ ${\displaystyle X}$

In addition to continuation sentences, statements about zero and pole positions are of particular interest in the theory of Riemann surfaces . A simple proof for the fundamental theorem of algebra could already be found in the function theory of the complex plane with the help of Liouville's theorem . In the theory of Riemann surfaces, for example, one obtains the following relatively simple theorem. Let be and Riemannian surfaces and an actual , non-constant holomorphic mapping. Then there is a natural number , so that every value is calculated with a multiplicity of- times. Since meromorphic functions can be understood as holomorphic mappings , where the Riemann number sphere denotes, it follows that on a compact Riemann surface every non-constant meromorphic function has as many zeros as there are poles. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle f}$ ${\displaystyle c\in Y}$ ${\displaystyle n}$ ${\displaystyle f\colon X\to \mathbb {C} }$ ${\displaystyle f\colon X\to \mathbb {P} _{1}}$ ${\displaystyle \mathbb {P} _{1}}$ ${\displaystyle f\colon X\to \mathbb {C} }$

literature

• Otto Forster : Riemann surfaces. (= Heidelberg Pocket Books 184). Springer, Berlin et al. 1977, ISBN 3-540-08034-1 .
  • English: Lectures on Riemann Surfaces. (= Graduate Texts in Mathematics 81). Corrected 2nd printing. Springer, Berlin et al. 1991, ISBN 3-540-90617-7 .
• Athanase Papadopoulos (Ed.): Handbook of Teichmüller theory. Vol. I, European Mathematical Society (EMS), Zurich 2007, ISBN 978-3-03719-029-6 , doi : 10.4171 / 029 . (IRMA Lectures in Mathematics and Theoretical Physics 11)
• Athanase Papadopoulos (Ed.): Handbook of Teichmüller theory. Vol. II, European Mathematical Society (EMS), Zurich 2009, ISBN 978-3-03719-055-5 , doi : 10.4171 / 055 . (IRMA Lectures in Mathematics and Theoretical Physics 13)
• Athanase Papadopoulos (Ed.): Handbook of Teichmüller theory. Vol. III, European Mathematical Society (EMS), Zurich 2012, ISBN 978-3-03719-103-3 , doi : 10.4171 / 103. (IRMA Lectures in Mathematics and Theoretical Physics 19)