Riemann surface

A Riemann surface is in the mathematical branch of function theory (german complex analysis) one-dimensional complex manifold. Riemann surfaces are the simplest geometric objects that own the local structure of the complex numbers. They are named after the mathematician Bernhard Riemann. The study of Riemann surfaces falls in the mathematical field of complex analysis and essentially depends on methods of algebraic topology and algebraic geometry from.

The Riemann surface is - historically - the answer that holomorphic functions do not always have unique continuations. For example, receives the main branch of the complex logarithm ( which is actually defined in a neighborhood of ) in continuation along a positively oriented circle around 0 the additional argument.

History

The theory of Riemann surfaces arose from the fact that in the analytic continuation of holomorphic functions along different paths different function values ​​may arise, as for example in the complex logarithm is the case. In order to obtain unique sequels again, were replaced by the domain of a multi-sheet surface that had so many leaves as there were ways to continue the function. In such an overlay surface, the analytic continuation is clearly again. Abstracted to this concept further, we get to today's concept of Riemannian surface. Bernhard Riemann explained the areas named after him initially as follows: several (possibly infinitely many ) complex number planes are superimposed, provided with certain sections (eg straight ) and then glued together along these cuts. This vivid idea was initially very fruitful, 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 the basic concept of today (real or complex ) manifold.

Definition

A Riemann surface is a complex manifold of dimension one. This means that Hausdorff space is equipped with a complex structure.

Examples

• The complex plane is the simplest Riemannian surface. The identity map defines a map for the whole, therefore, the amount is an atlas for.

• Each area is also a Riemannian surface. Here again is also the identity map a map for the whole area. General even every open subset of a Riemannian surface is a Riemann surface again.

• The Riemann number sphere is a compact Riemann surface. It is sometimes referred to as complex - projective line or short.

• The toroidal surface for a grid on which the elliptic functions are explained, is a compact Riemann surface.

Theory of Riemann surfaces

Because of the complex structure on the Riemann surface, it is possible holomorphic and meromorphic pictures on and define between Riemann surfaces. Many of the phrases used in the theory of functions on the complex plane over holomorphic and meromorphic functions can be generalized for Riemannian surfaces. This allows the Riemannian Hebbarkeitssatz, the identity theorem and the maximum principle applied to Riemann surfaces. However, one must realize that the holomorphic functions are not particularly rich in particular on compact Riemann surfaces. Precisely, this means that a holomorphic function on the compact surface must always be constant. A compact Riemann surface is thus not separable holomorphic on her exist only constant holomorphic functions. The Cauchy integral theorem and Cauchy's integral formula, two key sets of function theory of complex plane, can not be proved by analogy to Riemannian surfaces. On differentiable manifolds in general or on Riemann surfaces, in particular, the integration with the help of differential forms must be explained so that it is independent of the choice of the map. However, the key for the integration theory Stokes' theorem, by which one can prove also for Riemann surfaces the residue theorem, which follows from the Cauchy integral formula in the complex plane exists.

In addition to continuation rates are in the theory of Riemann surfaces statements about zeros and poles of particular interest. Thus was already in the theory of functions of complex plane with the help of the set of Liouville a simple proof of the fundamental theorem of algebra are found. In the theory of Riemann surfaces is obtained, for example, the following relatively simple sentence. Be and Riemann surfaces and an actual, non-constant holomorphic map. Then there exists a natural number, so that each value with multiplicity assumes expected times. As meromorphic functions can be regarded as holomorphic maps, where the Riemann number sphere referred to, it follows that on a compact Riemannian surface every non- constant meromorphic function has as many zeros as poles.