A Geometrical Characterization of Spatially Curved Roberstion-Walker Space and Its Retractions

Our aim in the present article is to introduce and study new types of retractions of closed flat Robertson-Walker W model. Types of the deformation retract of closed flat Robertson-Walker W model are obtained. The relations between the retraction and the deformation retract of curves in W model are deduced. Types of minimal retractions of curves in W model are also presented. Also, the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been obtained. New types of homotopy maps are deduced. New types of conditional folding are presented. Some commutative diagrams are obtained.


Introduction
As is well known, the theory of retractions is always one of interesting topics in Euclidian and Non-Euclidian space and it has been investigated from the various viewpoints by many branches of topology and differential geometry El-Ahmady [1].
El-Ahmady [1][2][3][4][5][6][7][8][9][10][11][12][13] studied the variation of the density function on chaotic spheres in chaotic space-like Minkowski space time, folding of fuzzy hypertori and their retractions, limits of fuzzy retractions of fuzzy hyperspheres and their foldings, fuzzy folding of fuzzy horocycle, fuzzy Lobachevskian space and its folding, the deformation retract and topological folding of Buchdahi space, retraction of chaotic Ricci space, a calculation of geodesics in chaotic flat space and its folding, fuzzy deformation retract of fuzzy horospheres, on fuzzy spheres in fuzzy Minkowski space, retractions of spatially curved Robertson-Walker space, a calculation of geodesics in flat Robertson-Walker space and its folding, and retraction of chaotic black hole.
An n-dimensional topological manifold M is a Hausdorff topological space with a countable basis for the topology which is locally homeomorphic to .If is a homeomorphism of onto , then h is called a chart of M and U is the , the transformation chart between open sets of is defined, and if all of these charts transformation are n R c  -mappings, then the manifolds under consideration is a c  -manifolds.A differentiable structure on M is a differentiable atlas and a differentiable manifolds is a topological manifolds with a differentiable structure Arkowitz [14] Banchoff [15], Dubrovin [16], Kuhnel [17], Montiel [18].
Most folding problems are attractive from a pure mathematical standpoint, for the beauty of the problems themselves.The folding problems have close connections to important industrial applications Linkage folding has applications in robotics and hydraulic tube bending.Paper folding has application in sheet-metal bending, packaging, and air-bag folding Demainel [19].Following the great Soviet geometer Pogorelov [20], also, used folding to solve difficult problems related to shell structures in civil engineering and aero space design, namely buckling instability El Naschie [21].Isometric folding between two Riemannian manifold may be characterized as maps that send piecewise geodesic segments to a piecewise geodesic segments of the same length El-Ahmady [4].For a topological folding the maps do not preserves lengths El-Ahmady [5,6].
A subset of a topological space is called a retract of if there exists a continuous map such that where is closed and is open El-Ahmady [3,7].Also, let be a space and a subspace.A map such that r a for all a A,  is called a retraction of onto and is the called a retract of This can be re stated as follows.If i : A is the inclusion map, then is a map such that A If, in addition, X we call r a deformation retract and A a deformation retract of X Another simple-but extremely useful-idea is that of a retract.If then A is a retract of X if there is a commutative diagram.

Main Results
The flat Robertson-Walker Line element is one example of a homogeneous isotropic cosmological spacetime geometry, but not the only one.The general Robertson-Walker Line element for a homogeneous isotropic universe has the form where dl 2 is the line element of a homogeneous, isotropic threedimensional space.There are only three possibilities for this.Let's now look at the closed flat Robertson-Walker model.In the present work we give first some rigorous definitions of retractions, folding and deformation retraction as well as important theorems of closed flat Robertson-Walker model.In what follows, we would like to introduce the types of retraction, folding and deformation retraction of closed flat Robertson-Walker model El-Ahmady [11,12], Hartle [25], Straumann [26] with metric The coordinate of closed flat Robertson-Walker space are x = sin sin cos , x = sin cos , x = sin sin sin , x = cos , where the range of the three polar angles   , , To find a geodesic which is a subset of the closed flat Robertson-Walker space .Since Then the Lagrangian equations for closed flat Robertson-Walker space are.
d sin sin sin cos 0 ds  2 2   d sin sin 0 ds From Equation ( 5) we obtain x 0, x 0, x sin , x cos Which is the sphere , , it is a minimal geodesic and minimal retraction.Also, if x = 0,x = 0,x = sin ,x = cos This is the sphere , it is a minimal geodesic and minimal retraction.Also, if x = sin sin ,x = 0, x = sin cos ,x = c .os Which is the sphere , , it is a minimal geodesic and minimal retraction.Again, if x = 0,x = sin sin , x = sin cos ,x = cos

    
Which is the hypersphere , it is a minimal geodesic and minimal retraction.Also, if x = 0, x = 0, x = 0, x = 1 .
Which is the point of the hypersphere 4 , it is a minimal geodesic and minimal retraction .Also, if x sin cos , x = sin sin , cos ,x = 0 Which is the sphere , it is a minimal geodesic and minimal retraction .: sin sin cos , sin sin sin , sin cos , cos sin sin cos , sin sin sin ,sin cos , cos , 1 sin sin cos sin sin sin , sin cos , cos h 2h 2h 0, 0,sin , cos The deformation retract of the folded open flat Robertson-Walker space m, 1 h sin sin cos , sin sin sin , sin cos , cos sin πh 2 sin sin , 0,sin cos , cos Then, the following theorem has been proved.
Copyright © 2012 SciRes.AM it is a minimal geodesic., the inclusion map of is we get the coordinate of closed flat Robertson-Walker space which are given by 4 cos , x = sin sin , x = cos ,x = 0      Which is the sphere , it is a minimal geodesic and minimal retraction.If

Theorem 1 .
The retractions of closed flat Robertson-Walker space are minimal geodesics and geodesic spheres.In this position, we present some cases of deformation retract of open flat Robertson-Walker space .The deformation retract of open flat Robertson-Walker space is

I
h sin sin cos , sin sin sin , The deformation retract of the folded open flat Robertson-Walker space

Theorem 2 .
Under the defined folding and any folding homeomorphic to this type of folding, the deformation retract of the folded open flat Robertson-Walker space into the folded geodesics is the same as the deformation retract of open flat Robertson-Walker space into the geodesics.The deformation retract of the folded open flat Robertson-Walker space into the folded geodesic is: cos ,sin sin sin , sin cos , cos sin sin cos , sin sin sin ,sin cos , cos :

I 3 . 4 W 2 .
The deformation retract of the folded open flat Robertson-Walker space into the folded geodesic is: sin cos sin sin sin , sin cos , cos πh sin sin sin , 0,sin cos , 0, cos 2 following theorem has been proved.Theorem Under the defined folding and any folding homeomorphic to this type of folding, the deformation retract of the folded open flat Robertson-Walker space geodesics is different from the deformation retract of open flat Robertson-Walker space into the geodesics.Lemma 1.The relations between the retractions and the limits of the folding of open flat Robertson-Walker space discussed from the following commutative diagrams Lemma The end of limits of the folding of closed flat Robertson-Walker space is a 0-dimensional space.

Lemma 3 . 4 W 4 .
The relation between the retraction and the deformation retract of open flat Robertson-Walker space discussed from the following commutative diagram Theorem Any

5 .
Then there are induces retractions such that the following diagram is commutative.j : S S 