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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]. www.indjst.org 110 4084 Indian Journal of Science and Technology Vol: 6 Issue: 2 February 2013 ISSN:0974-6846 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. 4085 www.indjst.org 111 Vol: 6 Issue: 2 February 2013 ISSN:0974-6846 Indian Journal of Science and Technology 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. www.indjst.org 112 4086 Indian Journal of Science and Technology 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: 4087 www.indjst.org 113 Vol: 6 Issue: 2 February 2013 ISSN:0974-6846 Indian Journal of Science and Technology ⊂ 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. www.indjst.org 114 4088 ⊂ ℝ are discussed from the following Indian Journal of Science and Technology 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 4089 www.indjst.org 115 Vol: 6 Issue: 2 February 2013 ISSN:0974-6846 ( × ) ( ) 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. www.indjst.org 116 | | , 4090 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 4091 www.indjst.org 117 Vol: 6 Issue: 2 February 2013 ISSN:0974-6846 ( ) → 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. 5. References 1. 2. 3. 4. A. E. El-Ahmady, The variation of the density on chaotic spheres in chaotic space-like Minkowski space time, Chaos, Solitons and Fractals, 31 (1272-1278) (2007). A. E. El-Ahmady, Folding of fuzzy hypertori and their retractions, Proc. Math. Phys. Soc. Egypt, Vol.85, No.1 (1-10) (2007). A. E. El-Ahmady, Limits of fuzzy retractions of fuzzy hyperspheres and their foldings, Tamkang Journal of Mathematics, Volume37, No. 1 (47-55) Spring, (2006). A. E. El-Ahmady, Fuzzy folding of fuzzy horocycle, CircoloMatematico di Palermo Serie II ,Tomo L III (443-450) (2004). www.indjst.org 118 4092 Indian Journal of Science and Technology 5. 6. 7. 8. Vol: 6 Issue: 2 February 2013 ISSN:0974-6846 A. E. El-Ahmady, Fuzzy Lobachevskian space and its folding, The Journal of Fuzzy Mathematics, Vol. 12 No. 2, (609-614) (2004). A. E. El-Ahmadyand H. Rafat, Retraction of chaotic Ricci space, Chaos, Solutions and Fractals, 41 (394-400) (2009). A. E. El-Ahmadyand H. Rafat, A calculation of geodesics in chaotic flat space and its folding, Chaos, Solutions and Fractals, 30 (836-844) (2006). A. E. El-Ahmadyand A. El-Araby, On fuzzy spheres in fuzzy Minkowski space, NuovoCimentoVol.125B(2010) . 9. A. E. El-Ahmady, Retraction of chaotic black hole, The Journal of Fuzzy Mathematics Vol.19, No. 3, (2011). 10. A. E. El-Ahmady, The geodesic deformation retract of Klein Bottle and its folding, The International Journal of Nonlinear Science, Vol. 12, No. 3, (2011). 11. E. Michael, Closed retract and perfect retracts, Topology and its Applications 12(451-468) (2003). 12. G. L. Naber, Topology, Geometry and Gauge fields: Foundations, Texts in Applied Mathematics 25, Edition, Springer-Verlage New York, Berlin, (2011). 13. Verlage New York, Berlin, (2011). 14. M. Baronti, E. Casini, C. Franchetti, The retraction constant in some Banach spaces, Journal of Approximation Theory, 120 (296-308) (2003). 15. M. Reid and BalazsSzendroi.Topology and Geometry, Cambridge University Press, Cambridge, New York, (2005). 16. P. DI-Francesco, Folding and coloring problem in Mathematics and Physics, Bulliten of the American Mathematics Society, Vol. 37, No. 3, (251-307) (2000). 17. P. Pellicer-Covarrubias, Retractions in hyperspaces, Topology and its Applications 135 (277-291) (2004). 18. Shick Pl. Topology: Point-Set and Geometry, New York, Wiley, (2007). 4093 www.indjst.org 119