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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.

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 [1-3].

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 [

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 [2-4]. 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, the induced 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].

A subset of a topological space is called a retract of if there exists a continuous map such that where A is closed and is open El-Ahmady [9-11]. Also, let be a space and a subspace. A map such that for all is called a retraction of onto and is called a retract of. 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 a deformation retract of Another simple-but extremely useful-idea is that of a retract. If A, then is a retract of if there is a commutative diagram.

If and then f is a retract of g if there is a commutative diagram El-Ahmady [3,7], Arkowitz [

In what follows, we discuss the retractions, let the closed interval be since the closed interval I is closed then firstly take a point of it to make a retraction, is open. Consider some types of retractions of.

If

then we can get

.

Now, we are in a position to formulate the following two theorems.

Theorem 1. All types of retraction of a closed interval are semi-open set or open set or zero-space.

Theorem 2. 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 be the open interval, and then the homotopy map is defined as

where

then we present the following cases of deformation retracts

where

and

From the above discussion, we obtain the following theorem.

Theorem 3. 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 of closed interval I.

Let, where

An isometric folding of closed interval I into itself may by

The deformation retracts of the folded closed interval into the folded retraction iswith,where and

The deformation retract of the folded closed interval into the folded retraction is

The deformation retract of the folded closed interval into the folded retraction is

Then, the following theorem has been proved.

Theorem 4. 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 1. If the retraction of the closed interval is and the folding of () into itself is, then there are commutative diagram between retraction and folding such that

Proof. Let a retraction, be a retraction of into. Also, let the folding is and the folding be. Then we have the retraction such that

Proposition 2. The relation between the retraction of the closed interval and the limit of folding discussed from the following commutative diagram.

Proof. Let a retraction, be a retraction of into. Also, let the limits of folding are given by

and.

Then we have the retraction of the zero dimensional manifold is the identity map, i.e. such that

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 with vertices removing only one vertex then the retraction is given by,

.

Also, removing two adjacent vertices is equivalent to removing an edge of the square, and then the retraction is defined as follows,

Moreover, removing two non-adjacent vertices gives a retraction, which is directly the zero-dimensional manifold,

In what follows, we discuss the deformation retract of the square as follows. The deformation retract of the square is defined by

where is the closed interval. Then we have the following cases of deformation retract. The deformation retract of the square onto a is given by

where

and

The deformation retract of the square onto closed interval will be

The deformation retract of the square onto zero-points is

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 is

and the folding of into itself is

Then there are commutative diagrams between retractions and foldings such that

Also

And also

Proposition 4. The relation between the retraction of the square and the limit of folding discussed from the following commutative diagrams.

Also

Again

where the limit of the folding of the Cartesian product of is not equal to the Cartesian product of their limits.

Proposition 5. If the deformation retract of

where and the retraction of is ,and the folding of into itself is Then there are induces deformation retractions, and folding such that the following diagram is commutative.

Proof. Let the deformation retract of is

the folding of, and

are defined by also

, the deformation retract of isand the retractions of, and

are given by,. Hence, the following diagram is commutative

Also, the end of limits of the folding and the end of limits of retractions of 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 be an n-dimensional space, consider the limit of the folding,, thenbut if M has 0-dimension. Then. Since , then .

Theorem 6. is a strong deformation retract of.

Proof. Let, where, is a strong deformation retracting of, into. To be specific, the k-homotopy D is assumed:

, , and, and.

Let, be defined as

Also.

Then, is a strong deformation retract of.

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

Proof. If A is a retraction of, then either dimension A = dimension or dimension A ≠ dimension, 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.

Theorem 7. If has the fixed point property, then is not a retract of.

Proof. Let has the fixed point property. Observe that certainly does not have the fixed point property since, for example, the antipodal map- is continuous, but has no fixed points. Then can therefore not be a retract of.

Proposition 8. If has the fixed point property, then is not a retract of, where

.

Proof. Since, and does not have the fixed point property. Then does not have the fixed point property,. Then is not a retract of.

Theorem 8. If is a 0-manifold, then is a retract of, such that,, andwhere and.

Proof. Now, let, ,be the retraction map of defined as. Let the inclusion map of, where, is, the retractions of and are defined by and, where,is the inclusion map of, the retraction of is, also the retractions of and are given by and.

Hence the following diagram is commutative:

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

where

and

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

, if

then we can get

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,

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,

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,

,

,

,

.

Now, we are going to discuss the deformation retract of the circle. Let be the open circle, then the homotopy map is defined as

where,

Then we have the following cases of deformation retracts.

where

and,

,

,

Now, we are going to discuss the folding of the circle. let, where.

An isometric folding of the circle into itself may by defined by

The deformation retracts of the folded circle into the folded retraction is

with

where

and

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 into the folded retraction is

with

The deformation retract of the folded circle into the folded retraction is

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:, and the folding ofinto itself is

).

Then there are commutative diagrams between retraction and folding such that:

Proof. Let a retraction, be a retraction of into. Also, let the folding is given by

and, also,such that

Let where

is the circle with center and radius. The intersection of all circles is denoted by 0. Let

, be the retraction map of such that

,

,

,

,

,

,

,

,

Hence, we can introduce the following theorems:

Theorem 11. Any circle with center and radius, where, is a retract of

.

Proof. Let and is a retract of, where. Also, If , is the inclusion map, the retractions of and are defined byand, the inclusion map of is, and the retractions of, and are given by, and

. Hence, the following diagram is commutative:

Proposition 11. Any circle with center and radius, where, is a retract of

.

Theorem 12. Any circle with center and radius, where, is a retract of

.

Proof. Since is a retract of and any circle in with center and radius, where, is a retract of, then any circle with center and radius, where, is a retract of. Also, Since, is a retract of and any circle with center and radius, where, is a retract of, then any circle with center and radius, were, is a retract of.

Theorem 13. Any retract of circle, in with center and radius, where

, is a retract of.

Proof. Let is a retract of, then there is a continuous map, , where. Then the circle in with center and radius, where, is a retract of, then there is a continuous map,, where.Then, is a continuous map. Also,. Then any retract of circle in with center and radiuswhere, is a retract of.

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.