Retractions of One Dimensional Manifold

Our aim in the present article is to introduce and study types of retraction of one dimensional manifold. New types of geodesics in one dimensional manifold are presented. The deformation retracts of one dimensional manifold into itself and onto geodesics is deduced. Also, the isometric and topological folding in each case and the relation between the deformations retracts after and before folding has been obtained. New types of conditional folding are described.


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][2][3].
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 El-Ahmady [2,3].Following the great Soviet geometer El-Ahmady [4], also used folding to solve difficult problems related to shell structures in civil engineering and aero space design, namely buckling instability El-Ahmady [4].
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 [2][3][4].For a topological folding the maps do not preserves lengths El-Ahmady [5,6], i.e.

A map
, where M and N are -Riemannian manifolds of dimension m, n respectively is said to be an isometric folding of M into N, iff for any piecewise geodesic path  is a piecewise geodesic and of the same length as  , If does not preserve length, then is a topological folding El-Ahmady [7,8].

I I
A subset A of a topological space X is called a retract of X if there exists a continuous map such that where A is closed and X is open El-Ahmady [9][10][11].Also, let X be a space and A a subspace.A map such that : r X A    , r a a  for all , a A  is called a retraction of onto and is called a retract of X A A X .This can be restated as follows.

If
is the inclusion map, then is a map such that If, in addition, we call r a deformation retract and , X ri id  A a deformation retract of X Another simple-but extremely useful-idea is that of a retract.If A, X M  then A is a retract of X if there is a commutative diagram.

On a Closed Interval
In what follows, we discuss the retractions, let the closed interval be

 
, , I a b  since the closed interval I is closed then firstly take a point of it to make a re- be the open interval, and then the homotopy map is defined as , where   0,1 , I  then we present the following cases of deformation retracts From the above discussion, we obtain the following theorem.
The deformation retracts of the folded closed interval The deformation retract of the folded closed interval The deformation retract of the folded closed interval Then, the following theorem has been proved.Then we have the retraction of the zero dimensional manifold is the identity map, i.e.

On a Cartesian Product of Closed Interval
In this position, we introduce the retraction of Cartesian product of closed interval.Consider two closed intervals . The Cartesian product is defined as The retraction is defined as.Consider the square 2 I with vertices removing only one vertex then the retraction is given by, ; 0,1,2,3 Also, removing two adjacent vertices is equivalent to removing an edge of the square, and then the retraction is defined as follows, , 0 and 1 closed interva.In what follows, we discuss the deformation retract of the square 2 I as follows.The deformation retract of the square where I is the closed interval   0,1 .Then we have the following cases of deformation retract.The deformation retract of the square 2 ,0 ; 0,1, 2,3 , The deformation retract of the square 2 From all the above discussion, we arrive to the following theorem.
Theorem 5.The limit of retractions sequence of the square is the 0-dimensional manifold.Also, the deformation retract of the square is either subsquare or zero-dimensional manifold.
Proposition 3. If the retraction of the square 2 2 and the folding of 2,3 ; 0,1 Then there are commutative diagrams between retractions and foldings such that : ; 0,1, 2,3 ; 0,1, 2,3 , and the retraction of 2 2 point and the folding of I   into itself is Then there are induces deformation retractions, and folding such that the following diagram is commutative.
the folding of I I  , and are defined by 1 : acent edges and the retractions of , and Hence, the following diagram is commutative Also, the end of limits of the folding and the end of limits of retractions of 2 2 I   induces the 0-dimensional space which is a point and in this case the retraction and folding of 0-dimensional space coincide.Proposition 6.The limit of the folding of 0-dimensional space M is 0-dimensional space.
Proof.Let M be an n-dimensional space, consider the limit of the folding lim : x y x y x y x y i I Then,  2 , but in this case dimension not invariant.Then is the same dimension of I 2 .But, the limit of the foldings of 2-dimensional manifold into itself is a manifold of dimension n − 1.Then, the limit of foldings is a one-manifold.
, the retractions of and 0 S   0 1 i S are defined by and is the inclusion map of   The purpose of this position is to introduce the relation between the deformation retract and folding of the circle, the parametric equation of the open circle in the plane is given by Now consider some types of retractions of the circle x t e t a t a t p a a t t t Now, we are going to discuss the folding g of the circle .let , where x x x  x g .An isometric folding of the circle into itself may by defined by The deformation retracts of the folded circle Then, the following theorem has been proved.Theorem 9.Under the defined folding and any folding homeomorphic to this type of folding, the deformation retracts of the folded circle into the folded retractions is the same as the deformation retracts of the circle into the retractions.
If the folding be defined by     The isometric folded of the circle is defined as The deformation retract of the folded circle   The deformation retract of the folded circle   Then, the following theorem has been proved.Theorem 10.Under the defined folding and any folding homeomorphic to this type of folding, the deformation retract of the folded circle into the folded retractions is different from the deformation retract of the circle into the retractions.
Proposition 10.If the retraction of the circle is the circle with center and radius l .The intersection of all circles is denoted by 0. Let G   : : , , Hence, we can introduce the following theorems: , is the inclusion map, the retractions of and   C n  are defined by and , the inclusion map of is , and the retractions of , and are given by , and , where , is a retract of i Theorem

 
C n 

Conclusion
In this paper we achieved the approval of the important of the curves in the Euclidean space by using some geometrical transformations.The relations between folding, retractions, deformation retracts, limits of folding and limits of retractions of the curves in the Euclidean space are discussed.New types of minimial retractions on curves in the Euclidean space are deduced.

1 . 2 .
are in a position to formulate the following two theorems.TheoremAll types of retraction of a closed interval are semi-open set or open set or zero-space.Theorem The limit of retraction of closed interval is a zero-manifold.Now, we are going to discuss the deformation retract of the closed interval.Let  

Theorem 3 .g
The deformation retracts of a closed interval gives semi-open set, open set and zero-dimensional space.Now, we are going to discuss the folding g of closed interval I.Let :An isometric folding of closed interval I into itself may by

Theorem 4 . 1 .
The deformation retract of the folded closed interval into the folded retractions is the same as or different from the deformation retract of the closed interval into the retractions Proposition If the retraction of the closed interval are commutative diagram between retraction and folding such that Proof.Let a retraction

Proposition 2 .
The relation between the retraction of the closed interval and the limit of folding discussed from the following commutative diagram.
two non-adjacent vertices gives a retraction, which is directly the zero-dimensional manifold,

Proposition 4 . 2 I 2 I
The relation between the retraction of the square 2   and the limit of folding discussed from the following commutative diagrams.Also Again where the limit of the folding of the Cartesian product of 2   is not equal to the Cartesian product of their limits.Proposition 5.If the deformation retract of 2 2

2
I .To be specific, the k-homotopy D is assumed:

Proposition 7 .
The retraction of is a two-dimensional manifold and the limit of foldings is a one-manifold.

2 I
  has the fixed point property.Observe that certainly does not have the fixed point property since, for example, the antipodal map-1 S id is continuous, but has no fixed points.Then can therefore not be a retract of 1

Proposition 9 .
The relation between the retraction, the limit of the folding and the inclusion map of circle discussed from the following commutative diagram 1