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European Journal of Applied Sciences – Vol. 12, No. 6

Publication Date: December 25, 2024

DOI:10.14738/aivp.126.17984.

Eltahir, A. E. M., & Abdel Rahman, A.-R. A.-R. A.-G. (2024). Complex Analytical Fiber Over Non-Compact Riemann Surfaces.

European Journal of Applied Sciences, Vol - 12(6). 728-741.

Services for Science and Education – United Kingdom

Complex Analytical Fiber Over Non-Compact Riemann Surfaces

Amna Eltahir Mokhtar Eltahir

Department of Mathematics, Faculty of Education,

Omdurman Islamic University, Omdurman, Sudan

Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman

Department of Mathematics, Faculty of Education,

Omdurman Islamic University, Omdurman, Sudan

ABSTRACT

In this paper we explained the Complex Analytical Fiber Over Non-Compact

Riemann Surface. We aimed to a relation between Complex analytic, Riemann

Surfaces and Riemann Hilbert problems. The Compact Complex Surface S to a

Riemann surface B such that the general fiber off is a Riemann surface of genus g.

and singular for Riemann Hilbert problems.

Keywords: Fiber Product, Complex, Non- Compact, Analytic, Riemann, Surfaces,

Diffeomorphism, Singular.

INTRODUCTION

A function f from a set A to a set B is a rule of correspondence that assigns to each element in A

one and only one element in B. A function f is thought of in this way, it is often referred to as a

mapping, or transformation. The image of a point z in the domain of definition S is the point. w

= f(z), and the set of images of all points in a set T that is contained in S is called the image of T.

[5], p73.The image of the entire domain of definition S is called the range of f. We often think of

a function as a rule or a machine that accepts inputs from the set A and returns outputs in the

set B. In elementary calculus we studied functions whose inputs and outputs were real

numbers. Such functions are called real-valued functions of a real variable. Here we began our

study of functions whose inputs and outputs are complex functions are simply generalizations

of well-known functions from calculus. A function f from a set A to a set B is a rule of

correspondence that assigns to each element in A one and only one element in B. [17], p3

COMPLEX ANALYTICAL OF FIBER PRODUCT

A complex function w = f (z) is said to be analytic at a point z if f is differentiable at z and at

every point in some neighborhood of Z. [13], p73.

We already defined analytic functions between Riemann surfaces and also from some analytic

structure and ask our- selves whether this data represents an integrable system. [18], p179

Even though the requirement of differentiability is a stringent demand, there is a class of

functions that is of great importance whose members satisfy even more severe requirements.

These functions are called analytic functions. [2], p145

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Eltahir, A. E. M., & Abdel Rahman, A.-R. A.-R. A.-G. (2024). Complex Analytical Fiber Over Non-Compact Riemann Surfaces. European Journal of

Applied Sciences, Vol - 12(6). 728-741.

URL: http://dx.doi.org/10.14738/aivp.126.17984

Definition (2.1)

An elliptic function with periods K and L two R -independent complex numbers are a

meromorphic function le ∶ C → C such that l(z) = l (z +mK + nL) for all m, n ∈ Z. We now

compute, by the fundamental theorem of calculus [1], P2.

FIBER PRODUCT OF RIEMANN SURFACES

Now, we introduce the concept of fiber product over Riemann surfaces. also, we explore the

irreducible components associated to the fiber product and we defined the normal fiber

product is true for all maps from non-compact Riemann surfaces into C, C∗

, the Riemann sphere

or complex tori. We devoted to study the fiber product at the level of connected Riemann

surfaces. Let S0, S1, S2 , be connected and not necessarily compact Riemann surfaces.

Definition (3.1)

Let us fix three Riemann surfaces,S0, S1 and S2, and two surjective holomorphic maps β1 : S0

→ S1 and β2: S2 → S0. The fiber product associated to the pairs (S1, β1) and (S2, β2) is defined

as

S1 × (β1, β2) S2:= {(z1, z2) ∈ S1 × S 2: β1 (z1) = β2( z2)} (1)

Endowed with the topology induced by the product topology of S1 × S2. There is associated a

natural continuous map β: S1 × (β1, β2) S2 → S0, such that:

(1) β = β1°π1 = β2 ◦ π2 (2)

Where πj

: S1 × (β1, β2) S2:= Sj

is the projection map πj

(z1, z2) = zj

, for j ∈ {1, 2}.

The fiber product of the pairs( S1, β1) and (S2, β2) enjoys the following universal property.

Definition (3.2)

The union of all the irreducible components is said the normal fiber product, which is denoted

by S1 × (β1, β2) S2 this is the normalization of the fiber product S S1 ×(β1, β2) S2 when it is

considered as a complex algebraic variety

Proposition (3.3)

If β1 and β2 both have finite degrees, then the number of irreducible components of the fiber

product of the two pairs ( S1, β1) and (S2, β2) is at most the greatest common divisor of the

degrees of β1 and β2. [9], p3 the concept of a conformal isomorphism. Note that any conformal

isomorphism has a conformal inverse. The Riemann surfaces with conformal structures

induced by (Q,z) and (n,z), respectively, do not have equivalent conformal structures but are

conformally isomorphic [8] ,p4.

A RIEMANN SURFACE

Is a two-dimensional, connected, Haus topological manifold M with a countable base for the

topology and with conformal transition maps between charts so complete analytic function"

had been introduced previously for the collection of all function elements obtained via analytic

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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024

continuation from however, from now on we shall use this term exclusively in the new sense.

Next, we turn to the special case where the Riemann surface is compact [3]P3.

COMPACT RIEMANN SURFACES

Corollary (5.1)

Let M be a compact Riemann surface and f: M ......+ N an analytic nonconstant map. Then f is

onto and N is compact.

Corollary (5.2)

Let M be a Riemann Surface. Then the following properties hold:

1. if M is Compact, then every holomorphic Function on M is constant.

2. Every nonconstant meromorphic function on a compact Riemann surface is onto C.

3. If f is a nonconstant holomorphic Function on a Riemann Surface M,

then If I attain neither a local maximum nor a positive local minimum on M. The analytical

ingredient in this proof consists of the uniqueness and open mapping theorems as well as the

removability theorem: the first two are reduced to the same properties in charts which then

require.

Let Σ be a bordered Riemann surface with genus g and analytic boundary components. [15],

p157

Definition (5.3)

Two atlases A and B are called equivalent, if A∪ B is a holomorphic atlas. An equivalence class

of atlases is called holomorphic structure. A Riemann Surface is a Haus’d or pace X together

with a holomorphic structure on X. Furthermore, a compact Riemann surface is a Riemann

surface X, such that any open cover of X has a finite subcover. [13], p158

Theorem (5.4)

There is a bijective correspondence between the set of conformal equivalence classes of

Compact Riemann surfaces and the set of birational equivalence classes of algebraic function

fields in one variable.

Example (5.5)

The developing map of the canonical projective structure is the inclusion map U→ p1 of the

universal cover of C. More generally, when the projective structure is induced by a quotient

map π : U → C = U/A like, then the developing map is the universal cover U →U of U and the

monodromic group is A, the open set U is not simply connected and the developing map is a

non trivial covering. Thus, the corresponding projective structure is not the canonical one.

Similarly, the developing map of a quasi-Fuchsian group is the uniformization map of the

corresponding quasi-disk and is not trivial; the projective structure is neither the canonical one,

nor of Schottky type.

The uniformization theorem for Compact Riemann Surfaces is