Surface of Riemann

In differential Geometry, a surface of Riemann is a differential Variété analytical complex of dimension 1. By Lapse of memory of structure, a surface of Riemann is presented in the form of a real differential variety of dimension 2, from where the name surface . They were named in homage to the German mathematician Bernhard Riemann. Any directional real surface can be provided with a complex structure, in other words to be looked like a surface of Riemann. That is specified by the Théorème of standardization.

The study of surfaces of Riemann to is crossed many other mathematical fields of which, except the differential geometry, the Théorie of the numbers, the algebraic Topologie, the algebraic Géométrie, the partial derivative equations…

Elementary theory

Definition

A surface of Riemann is a topological Espace separate and countable ad infinitum X , admitting an atlas modelled on the plan complexes C whose applications of board swapping are biholomorphes applications. In other words X admits a covering by open U_i homeomorphic to the open ones of C ; these charts known as holomorphic $f_i: U_i \ to V_i$ are such as the functions of board swapping $f_i \ circ f_j^ {- 1}$ is holomorphic functions between open of C .

One can add new charts as long as they are compatible with the preceding ones with the direction where the applications of board swapping remain holomorphic. In fact, there exists thus a maximum atlas for the surface of Riemann. One will identify two superficial structures of Riemann on the same topological space when they are compatible, i.e. lead to the same maximum atlas.

If X and Y is two surfaces of Riemann, an application of X in Y is known as holomorphic when, read in the holomorphic charts, it is holomorphic.

The plan complexes C is identified naturally with R ². As holomorphic implies differentiable, any surface of Riemann inherits a structure of differential variety of dimension 2. As any holomorphic application preserves the orientation of C , any surface of Riemann inherits an orientation as a real variety. In fact:

Any surface of Riemann is presented in the form of a directed real surface.

These considerations spread for all the holomorphic varieties.

On the other hand, any directed real differential variety of even size does not admit a complex structure inevitably. It is a remarkable fact in dimension 2, that any directed real surface admits indeed a superficial structure of Riemann. But this structure is not inevitably single.

Examples

the Plan complex C is presented very simply like surfaces of Riemann. The identity makes it possible to define an atlas reduced to a single chart.

the plane complex combined $\ overline {C}$ is topologically C , but one provides it like single chart with the complex conjugation.

simplest of compact surfaces of Riemann is the Sphère of Riemann . Topologically, it is defined like the Compactifié d' Alexandroff of the complex plan, namely S ² = C ∪ {∞}. It is covered with two holomorphic charts, defined respectively on C and C ^*∪ {∞}: the identity Z $\ mapsto$ Z and the inversion Z $\ mapsto$ 1/ Z .

the plane projective complex P ¹ C is another representation of the sphere of Riemann. It seems the quotient of C ²- {(0; 0)} by the natural action by multiplication of the group $\ mathbb C^*$ (passage to the quotient for the analytical varieties).

the hyperbolic plane $\ mathbb H^2$ is a fundamental example of surface of Riemann corresponding to the open disc of C , or to the higher plane half, or, by the theorem of standardization, with all open simply related $\ mathbb C$ (and different from $\ mathbb C$ ).

To judge importance of these examples: Any the universal surface coating of related Riemann is a surface of Riemann simply related isomorph with $\ mathbb C$ , or $\ mathbb S^2$ or $\ mathbb H^2$ .

For example: $\ mathbb C^*$ is the quotient of the plan complexes $\ mathbb C$ by the group of the translations $2i \ pi \ mathbb Z$ . More precisely, the coating $\ mathbb C \ rightarrow \ mathbb C^*$ is given by the exponential complex one.

Hyperbolic surfaces

Geometry of Riemann for surfaces

It is advisable a priori to distinguish surfaces from Riemann, complex analytical varieties of dimension 1 and the varieties riemanniennes which are surfaces, i.e. varieties of dimension two provided with a metric Tenseur. However the two concepts are very close.

If Σ is a surface directed provided with a structure of riemannienne variety, it is possible to define a almost complex Structure associated J on Σ, which is always integrable, i.e. Σ can be naturally seen like a surface of Riemann. The application J is defined on each tangent space by requiring that J (v) be of the same standard than v and than (v, J (v)) is orthogonal direct.

Reciprocally, if Σ is a surface of Riemann, it is possible to define several metric riemanniennes compatible with its complex structure. Among them, there exists about it such as the riemannienne variety obtained either complete and of constant Courbure -1,0 or 1. Such metric is single except for a factor.

See too

References

Hershel Mr. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4
Jürgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X
Eric Reyssat, Some aspects of surfaces of Riemann (1989), Birkhaüser, Boston. ISBN 0-8176-3743-5

Curve pseudoholomorphe

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