Applied Mathematics
Vol.3 No.10(2012), Article ID:23372,9 pages DOI:10.4236/am.2012.310167
Retractions of One Dimensional Manifold
1Mathematics Department, Faculty of Science, Taibah University, Madinah, KSA
2Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt
Email: hk.mishra@juet.ac.in
Received July 23, 2012; revised September 10, 2012; accepted September 17, 2012
Keywords: Retractions; Deformation Retracts; Folding; One Dimensional Manifold
ABSTRACT
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.
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 [1-3].
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 [2-4]. 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 [12], Naber [13,14], Reid [15], Shick [16] and Strom [17].
2. Main Result
2.1. On a Closed Interval
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
is
with
,
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
2.2. 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 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
is
and 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
,
, then
but 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 I2. 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.
2.3. On a Circle
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
,
, and
where
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
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
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 of
into 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 by
and
, 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 radius
where
, is a retract of
.
3. 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.
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