Download Foldings and Deformation Retractions of Hypercylinder

Vol: 6 Issue: 2 February 2013 ISSN:0974-6846
Indian Journal of Science and Technology
Foldings and Deformation Retractions of Hypercylinder
1,2
2
A. E. El-Ahmady, 2 N.Al-Hazmi
Mathematics Department, Faculty of Science, Taibah University, Madinah, Saudi Arabia
1
Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt.
[email protected]
Abstract
Our aim in the present work is to introduce and study new types of retractions of hypercylinder. Types of the deformation retracts
of hypercylinder are presented. The relations between the folding and the deformation retract of hypercylinder are deduced.Types
of minimal retractions of hypercylinder are introduced. Also, the isometric and topological folding in each case and the relation
between the deformation retracts after and before folding are obtained. New types of homotopy maps are deduced. Theorems
governing this connection are achieved.
Keywords: Hypercylinder, Retraction, Folding.
1. Introduction and definition:
As is well known, the theory of retraction 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
[1,2,3,6,7,8,9,10,11,12,13,14,15,17,18] .
The aim of this paper is to describe and study new types ofretraction,deformation retract andfolding of hypercylinder from
view point of the variation of the density function on chaotic spheres in chaotic space-like Minkowski space time, folding of
fuzzyhypertori and their retractions , limits of fuzzy retractions of fuzzy hyperspheres and their foldings, fuzzy folding of fuzzy
horocycle, fuzzy Lobachevskian space and its folding, retraction of chaotic Ricci space, a calculation of geodesics in chaotic flat
space and its folding, on fuzzy spheres in fuzzy Minkowskispace,retraction of chaotic black hole and the geodesic deformation
retract of Klein Bottle and its folding as presented by El-Ahmady [1,2,3,4,5,6,7,8,9,10]. A subset A of a topological space X is
called a retract of X if there exists a continuous map
r: X 
 A such that :
(i) X is open,
(ii)
r (a)  a
 a  A [3,6,9,11,12,13, 14,17].
A subset A of a topological space X is a deformation retracts of X if there exists a retraction r : X  A and a
homotopy : X  I 
 X such that:
( x,0)  x

 x X
 ( x ,1)  r ( x ) 
 ( a , t )  a , a  A, t [ 0 ,1] [10,12,13,15,18] .
A map : M 
 N , where M and N are c  –Riemannian manifolds of dimension m , n respectively is said to be an
isometric folding of M into N , iff for any piecewise geodesic path  : J 
 M , the induced path    : J 
 N is a
piecewise geodesic and of the same length as  [2,4,5,7]. If does not preserve length, then  is a topological folding
[1,5,6,10,16].
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2. Main results
To achieve our aims, we will introduce the following:
is a subspace of ℝ and
Definition 1. IF
is a subspace of ℝ , then the Cartesian product
can be identified in a natural
is a subspace ofℝ and (−1,1) is a subspace of
and thereby acquires a subspace topology. e. g.
way with a subset ofℝ
×
×(−1,1) can be viewed as a subspace of ℝ , usually called a hypercylinder, see Fig. (1).
so
ℝ are the cylinders, sphere, great circles and generator .
Theorem 1. The retraction geodesics of the hypercylinder
Proof: Now, we discuss the retraction of the hypercylinder used cylindrical coordinates ,
and
with
The coordinates of the hypercylinder are given by
( ,
,
)= (
,
Now, we use lagrangian equations
,
+
,
,
,
,
,
(
(
,
,
+
are
(2)
)
,
)
(3)
(4)
)
,
From equation (2) we have
(1)
(1)
,i=1,2,3, to find a geodesic which is a subset of the hypercylinder . Since
,
Then the lagrangian equations for the hypercylinder
(
)
(
from (1)
, 0,
(
,
,
if
, then from (1)
if
, especially if
Constant say
),it is clear that
, 0,
(0,0,0,
if
,
, then
is the cylinder, when
), it follows immediately that
, we have
and we obtain from
, which is a geodesic retraction. If
is a cylinder, it is retraction and geodesic when
), it is a geodesic and retraction. Moreover, from equation (4) we have
, then from (1)
(
, which is geodesic and retraction. Again if
,
then
,
,then from (1)
, 0). It is clear that
(
,
,
0. Also,
constant say
is the generator, when
, 0,0). It is clear that
is the
great circle, which is a retraction geodesic.
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ℝ . Let
Now, we are going to discuss the deformation retract of the hypercylinder
− { })
(
((
,
,
,
∶ ((
,
,
) { }) be the open hypercylinder, the homotopy map is defined as
,
)-{ })× → ((
,
,
)- { })
,
Where = [0,1] then we have the following cases of deformation retracts.
− { })
The deformation retract of (
( , ℎ)
Where
((
,
,
( , 0) ((
and ( , 1) (
, 0,
,
( , ℎ) =
− { })
The deformation retract of (
, 0,
,
)
)-{ })
,
ℝ into a geodesic
,
,
− { })
{((
(
ℝ is given by:
)
,
(1 − ℎ)((
The deformation retract of (
− { })
(
)-{ })
,
,
The deformation retract of (
( , ℎ) =
ℝ into a geodesic
,
− { })
( , ℎ) = (1 − ℎ){((
,
) − { })
,
{0,0,0,
− { })
(
,
) − { })
,
),
, 0,
− { }) ⊂ ℝ is defined by:
(
ℝ into a geodesic
,
ℝ is defined as:
) − { }) +ℎ(
,
ℝ into a geodesic
,
− { })
(
.
ℝ is given by:
{(
,
,
, 0) .
Now, we are going to discuss the folding ℱof the hypercylinder . let
⊂ℝ
( ,
⊂ ℝ , where
,
,
,
,
,
)
ℝ into itself may by defined as
An isometric folding of the hypercylinder
ℱ ∶ ((
) ( , | |,
)-{ })
,
,|
((
|,
)-{ })
,
The deformation retract of the folded hypercylinderℱ( ) ⊂ ℝ into the folded geodesic ℱ( ) ⊂ ℝ is:
ℱ
,|
∶ ((
|,
,
,|
)-{ })× → ((
|,
)- { })
,
with
ℱ(
((
,|
|,
,
)-{ })
, 0) = ((
,|
|,
,
)-{ }) and
, ℎ)
where
ℱ(
(
ℱ(
, 1)
, 0,
(
,
, 0,
)
,
)
Then, the following theorem is proved.
Theorem2. Under the defined folding, the deformation retract of the folded hypercylinder into the folded geodesic is the same as
the deformation retract of the hypercylinder into the retraction geodesic.
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If the folding
( ,
defined as
Vol: 6 Issue: 2 February 2013 ISSN:0974-6846
,
) (| |,
,
|,
)
,
ℝ is
The isometric folded of the hypercylinder
G =((|
,
,
,
)- { })
The deformation retract of the folded hypercylinder ( ) ⊂ ℝ into the folded retraction ( ) ⊂ ℝ is:
ℱ(
, ℎ) =
(1 − ℎ)((|
|,
,
)- { })
,
|,
ℎ(2ℎ + ℎ)(|
, 0,
),
Then, the following theorem is proved.
Theorem3. Under the defined folding, the deformation retract of the folded hypercylinder into the folded geodesic is different
from the deformation retract of the hyper cylinder into the retraction geodesic.
Theorem4. Let M be a compact smooth submanifold of the hypercylinder
thehypercylinder ⊂ ℝ .
Proof: Let the parametric equations of the hypercylinder
( , ,
)={(
,a
× [0,1]
following continuous map
( , )= ( ,
⊂ ℝ be defined as
, )- { }}, and
,
⊂ ℝ . Then M is a strong deformation retract of
⊂
⊂ ℝ be a retraction of
×
. Now consider the
such that:
), then it is easy to see that
( , 0)= ( , 0 ) = 0
( , 1)= ( ,
)=
⊂
( , )=
)=
⊂
( ,
⊂ ℝ lim
→
⊂
ℝ ,
[0,1].
Corollary 1.Let be a deformation retract of thehypercylinder
singularity. Then the following diagram is commutative
ℝ
If the retraction of the hypercylinder
: ((
,
,
⊂ℝ →
⊂ℝ
folding without
isCorollary 2.
) − { }) →
,
⊂ ℝ and Ϝ:
∗
ℝ and the folding of
ℝ into
itself is
,
∶ ((
((|
|, |
,
|, |
,
)-{ })
|, | |)-{ })
Then there are commutative diagram between retraction and folding such that:
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⊂ ℝ and the limit of folding discussed from the
Corollary 3. The relation between the retraction of the hyper cylinder
following commutative diagram.
Theorem 5.The retractions of the hypercylinder are three types of retractions.
Proof: Let the parametric equations of the hypercylinder be defined as
( , ,
×
)={(
is called a retraction of
( )= ∀ ∈
×
,a
,
×
,
×
,
)−{ }}, since
and
if there exists a continuous map
is an open manifold and
be two topological manifolds. A subset
×
such that
is a closed subset of
. Since the retractions of the hypercylinder defined only for open manifold, then if
is closed. Then (
×
(
)
)
of
×
is open, where
is open and
, Since ( ) is not defined, See Fig.(2).
Fig.(2)
where r (
−
)=
. If
×
is open, where
defined where r ( ) = r ( ) = 0-dimensional space. If
is closed and is open .Then
×
is open , where
and
(
×
)
are open. Then
( ), since (
(
×
)
) is not
(
)
( ).
Corollary 4. The relation between the retraction and the limit of foldings of the sphere
commutative diagram.
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⊂ ℝ are discussed from the following
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Vol: 6 Issue: 2 February 2013 ISSN:0974-6846
Theorem 6.The retractions of the solid hypercylinderare the base and sides.
Proof. Let ={ ∈
:‖ ‖ ≤ 1} be an 3-dimensional disc in . The boundary of
is defined by
={ ∈
:‖ ‖ = 1} is
just the sphere . Now consider the solid hypercylinder
× [0,1] and the subset
× {0} ∪ × [0,1] consisting of its base and
sides. We retract
× [0,1] onto × {0} ∪ × [0,1] by the projection from the point (0, 0, 0, 0, 2) in , this means that, let
r:{(
× [0,1])- }
{
× {0} ∪
× [0,1]} be defined as
is a retraction map.
Corollary 5.The deformation retract of the solid hypercylinder are
× {0}or
Corollary 6.The strong deformation retract of the solid hypercylinder is the
× [0,1].
={-1,1} which is the 0-sphere in
.
Theorem7. The foldings of the hypercylinder are three types of foldings.
×
Proof: Let
(
( (
×
, ,
)
be the hypercylinder, then the folding of the hyper cylinder can be defined as follows
(
)
))={
,where
,a
,|
|, }, ℑ
thenℑ is an isometric folding and the set of singularities of ℑ , the set of points of
circle (
,
where ℑ fails to be differentials, is the great
, 0) , see Fig. (3).
Also, if
( (
, ,
|, |a
))={|
|, |
|, }, ℑ
thenℑ is an isometric folding and the set of singularities of ℑ is a graph consisting of the intersection of the three coordinate
planes
,
,
, with six vertices ( a,0,0),(0,±a,0),(0,0, a) and the image is the octant, see Fig. (4).
Fig .4
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(
×
) (
)
minimum folding of
Indian Journal of Science and Technology
( ) where the folding of
. Again ℎ(
( ) is a subspace of
) = ℎ(
×
) × ℎ( ). Moreover, the
can be defined as following, see Fig. (5).
Corollary 7. The folding of the Cartesian product (
×
) is either geodesics or a minimal space .
Corollary 8. The limit of the folding of the Cartesian product (
×
) is not equal to the Cartesian product of their limits.
Corollary 9. The retraction and folding of the hypercylinder are not commutative, i.e (ℑ(
×
)) ℑ (
×
) .
Corollary 10.The retraction which preserves the dimension of the
hypercylinder is a type of folding .
Corollary 11. The retraction which decreases the dimension of the
hypercylinder is a limit of folding of the hyper cylinder .
In this position let ∏
⊂ℝ →
|
∏
|
⊂ ℝ be given by
(∗)
Then, the isometric chain folding of hypercylinder
⨅ {
,
⨅ :{
→{
⨅ :{
Then we get lim
,
,
,
, | |}
⊂ ℝ into itself may be defined by :
→{
,
,
→∞ ⨅
,
{
}→{
,
0}which is geodesic retraction sphere
From the above discussion we will arrive to the following theorem.
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| |
,
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in hypercylinder .
Indian Journal of Science and Technology
Vol: 6 Issue: 2 February 2013 ISSN:0974-6846
⊂ ℝ into itself, under condition (∗), is the same as the geodesic retraction
Theorem 8. The limit folding of the hypercylinder
sphere
in hypercylinder.
⊂ℝ →
If we let ∏
⊂ ℝ be given by
|
∏
|
|
|
(∗∗)
Then, the isometric chain folding of hypercylinder
⨅ {
,
,
|
⨅ :{
,
|
→{
|,
| |
, | |}
|,
→{
⊂ ℝ into itself may be defined by :
,
⨅ :{
,
Then we get lim
,
→∞ ⨅
which is hypersurface
}→{
{
,
0}
in hypercylinder .
From the above discussion we will arrive to the following theorem.
Theorem 9.The limit folding of the hypercylinder
hypercylinder ⊂ ℝ .
:
If the folding is defined by
|
| |
| |
⊂ℝ
| |
|
⊂ ℝ into itself, under condition(∗∗), is different from the retraction of the
⊂ ℝ such that
, (∗∗∗)
Then, the isometric chain folding of the hypercylinder
:{
|
,
,
||
: {|
,
||
||
| | |}
||
| | |}
}
}
:
Then we get lim
⊂ ℝ into itself may be defined by :
→∞
}
, which a zero- dimensional hypersphere
in hypercylinder
.
Thus the following theorem is obtained.
Theorem 10. The limit folding of the hypercylinder
dimensional space.
⊂ ℝ into itself, under condition (∗∗∗), is equivalent to the zero -
Theorem11. The end of the limits of the foldings of the n-dimensional hypercylinder
⊂ℝ
is a 0- dimensional space .
Proof : Let
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( ) →
Indian Journal of Science and Technology
( ) ,…,
…
…
lim
…
→∞
.
Let
(
) →
(
) ,…,
…
…
lim
…
→∞
Let ℎ
ℎ
ℎ
.,…,
ℎ
ℎ ℎ ℎ ( ) → ℎ ℎ ( ) ,…,
ℎ ℎ
ℎ
ℎ
ℎ
lim
→∞ ℎ
… ℎ
… ℎ
ℎ
ℎ
ℎ
ℎ
ℎ
ℎ
… ℎ
Consequently ,lim lim lim ℎ …
→∞ →∞ →∞
ℎ
ℎ
o-dimensional hypercylinder, it is a minimal retraction.
3. Conclusion
In this paper we achieved the approval of the important of the geodesic retractions of the hypercylinder .The relations between
folding, retractions, deformation retract and limits of folding of the hypercylinder are discussed. Theorems which govern these
relations are presented.
4. Acknowledgments
The author is deeply indebted to the team work at the deanship of the scientific research, Taibah University for their valuable help
and critical guidance and for facilitating many administrative procedures. This research work was financed supported by Grant no.
3066/1434 from the deanship of the scientific research at Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia.
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