A Geometrical Characterization of Spatially Curved Roberstion-Walker Space and Its Retractions ()
1. 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 [1-13] 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 associated chart domain. A collection (
) is said to be an atlas for M if
. Given two charts 
such that
, the transformation chart 
between open sets of 
is defined, and if all of these charts transformation are 
-mappings, then the manifolds under consideration is a 
-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 ElAhmady [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 
for all 
is called a retraction of 
onto 
and 
is the called a retract of 
This can be re stated as follows. If 
is the inclusion map, then 
is a map such that 
If, in addition, 
we call r a deformation retract and 
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.
If 
and 
then 
is a retract of g if there is a commutative diagram Arkowitz [14], Naber [22], Shick [23] and Strom [ 24].
2. 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 RobertsonWalker 
Line element for a homogeneous isotropic universe has the form 
where dl2 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 RobertsonWalker 
model El-Ahmady [11,12], Hartle [25], Straumann [26] with metric
(1)
The coordinate of closed flat Robertson-Walker space 
are
(2)
where the range of the three polar angles 
is given by 
and 
Now, we use Lagrangian equations
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.
(3)
(4)
(5)
From Equation (5) we obtain 
constant say
, if
, we obtain the following cases:
If 
hence we get the coordinates of closed flat Robertson-Walker space 
which are given by
.
Which is the sphere
, 
, it is a minimal geodesic and minimal retraction. Also, if
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, 
, it is a minimal geodesic and minimal retraction. Again, if
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, 
it is a minimal geodesic and minimal retraction. Also, if
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, 
, it is a minimal geodesic and minimal retraction. If 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, 
, it is a minimal geodesic and minimal retraction. Again, if 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
.
This is the sphere
, 
it is a minimal geodesic and minimal retraction. Also, if 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, 
, it is a minimal geodesic and minimal retraction. If 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the sphere
, 
, it is a minimal geodesic and minimal retraction. Again, if
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, it is a minimal geodesic and minimal retraction. Also, if
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, it is a minimal geodesic and minimal retraction. If 
hence we get the coordinate of closed flat RobertsonWalker space 
which are given by
Which is the hypersphere
, it is a minimal geodesic and minimal retraction. Again, if
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the hypersphere
, it is a minimal geodesic and minimal retraction. Also, if 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the sphere
, it is a minimal geodesic and minimal retraction. If 
hence we get the coordinate of closed flat RobertsonWalker space 
which are given by
This is the sphere
, it is a minimal geodesic and minimal retraction. Again, if 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
.
Which is the point of the hypersphere 
, it is a minimal geodesic and minimal retraction. Also, if 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the sphere
, it is a minimal geodesic and minimal retraction. If 
hence we get the coordinate of closed flat RobertsonWalker space 
which are given by
.
Which is the point of the hypersphere 
, it is a minimal geodesic and minimal retraction . Also, if 
hence we get the coordinate of closed flat Robertson-Walker space 
which are given by
Which is the sphere
, it is a minimal geodesic and minimal retraction .
Theorem 1. The retractions of closed flat RobertsonWalker 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
where 
be the open flat Robertson-Walker space 
and is the closed interval [0, 1], be present as
The deformation retract of the open flat RobertsonWalker space 
into the sphere 
is
where
and
The deformation retract of the open flat RobertsonWalker space 
into the sphere 
is
The deformation retract of the open flat RobertsonWalker space 
into the sphere 
is
Now, we are going to discuss the folding 
of the open flat Robertson-Walker 
space Let
, where
(6)
An isometric folding of the open flat RobertsonWalker 
space into itself may be defined by
The deformation retract of the folded open flat Robertson-Walker space 
into the folded geodesic 
is:
with
The deformation retract of the folded open flat Robertson-Walker space 
into the folded geodesic 
is:
The deformation retract of the folded open flat Robertson-Walker space 
into the folded geodesic 
is:
Then, the following theorem has been proved.
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.
Now, let the folding be defined by:
where
(7)
The isometric folded open flat Robertson-Walker space 
is:
The deformation retract of the folded open flat Robertson-Walker space 
into the folded geodesic 
is:
with
The deformation retract of the folded open flat Robertson-Walker space 
into the folded geodesic 
is:
The deformation retract of the folded open flat Robertson-Walker space 
into the folded geodesic 
is:
Then, the following theorem has been proved.
Theorem 3. 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 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 2. The end of limits of the folding of closed flat Robertson-Walker space 
is a 0-dimensional space.
Proof. Let
Let
.
Consequently, 
-dimensional sphere, it is a minimal geodesic.
Lemma 3. The relation between the retraction and the deformation retract of open flat Robertson-Walker space 
discussed from the following commutative diagram
Theorem 4. Any folding of 
into 
induces folding of B
into 
from
Proof. Let 
, then there is an induced folding
such that
and
such that the following diagram is commutative
i.e. 
Theorem 5. Any retraction of 
into
induces retraction of 
into
.
Proof. Let r be a retraction map,
where 
and 
are the open sphere in
. Also, let
and
such that
.
Then we have the retraction 
such that
Theorem 6. Any retraction
then the map 
induced by the exponential map.
Proof. Let a retraction
, be a retraction of 
into
. Also, Let 
and
.
Then we have the retraction 
such that 
Theorem 7. Any retraction
, then the map 
induced by the inverse exponential map.
Proof. Let a retraction
, be a retraction of 
int
. Also, Let
and
.
Then we have the retraction
such that
Theorem 8. If the retraction of the sphere 
is
, the inclusion map of 
is
, and inclusion map of 
is
. Then there are induces retractions such that the following diagram is commutative.
Proof. Let the retraction map of the hypersphere 
is
, the inclusion map of 
is 
, the retraction map of 
is
, the retraction map of
is given by
, and
. Hence, the following diagram is commutative.
Theorem 9. If the retraction of the sphere 
is
, 
and
.
Then there are induces exponential inverse map such that the following diagram is commutative.
Proof. Let the retraction map of the hypersphere 
is
,
,
, 
, and
. Hence, the following diagram is commutative.
3. Conclusion
The present article deals what we consider to be closed flat Robertson-Walker 
model. The retractions of closed flat Robertson-Walker 
model are presented. The deformation retract of closed flat Robertson-Walker 
model will be deduced. The connection between folding and deformation retract is achieved. New types of conditional folding are presented. Also, the relations between the limits of folding and retractions are discussed. Some commutative diagrams are presented.