Folding of Hyperbolic Manifolds

Transcription

1 It. J. Cotmp. Math. Scics, Vol. 7, 0, o. 6, Foldig of Hyprbolic Maifolds H. I. Attiya Basic Scic Dpartmt, Collg of Idustrial Educatio BANE - SUEF Uivrsity, Egypt hala_attiya005@yahoo.com Abstract I this papr, w itroduc th dfiitio of hyprbolic maifold. Th foldig of hyprbolic maifold ito itslf is dfid ad discussd. Typs of ths foldigs ar dducd. Thorms govrig ths typs ar achivd. Mathmatics Subjct Classificatio, 5H0, 57N0 Kywords: Foldig, hyprbolic maifolds Itroductio Th word topology is drivd from th Grk words tops ad logos ad mas th scic of plac. Adoptd from th Frch topologic, th word cam ito us i Eglish i th 600s. Its origial maig Th brach of botay that dals with th localitis of particular plats has fall ito discuss, alog with a host of latr os. (I th 860s, th word primarily rfrrd to th art of assistig th mmory by associatig th thig to b rmmbrd with som wll kow plac). Topology was itroducd as a mathmatical trm by th Grma mathmaticia Joha Bdict listig i 847. Th first o who itroducd th foldig of Rimaia maifolds is S.A- Robrtso 977[9].Mor studis o th foldig of ral maifolds ar studid by

2 79 H. I. Attiya E.El- kholy[8].ad M.EL-Ghoul[,,,4,5,6,7].I this articl w will itroduc th foldig of hyprbolic maifold. Dfiitios ad backgroud W will giv som dfiitios which w will d thm i this papr: () Map f : M N,whr M, N ar C - Rimaia maifolds of dimsios m,,rspctivly is said to b a isomtric foldig of M ito N,if ad oly if for ay picwis godsic path γ : J N,th iducd path f o γ : J M is picwis γ, γ = 0, [9]. godsic ad of th sam lgth as [ ] If f ot prsrvs lgths th f is a topological foldig. () Thr xist two modls of hyprbolic pla th first modl is th gomtry i th uppr pla s Fig. ().ad th scod modl is th gomtry itrior th circl s Fig.(). Fig. () Fig. () Th mai rsults Aimig to our study w will itroduc som dfiitio:

3 Foldig of hyprbolic maifolds 79 [] I Fig.() : Fig.(4) Fig.() AB = Log Au Bu I Fig.(4) : AB = Log A u B u B v A v [] Foldig of hyprbolic pla is a map f : H H such that H H. Thr xist thr typs of foldigs of hyprbolic pla w will discuss ths typs: () Foldig which fold th hyprbolic pla ito athr hyprbolic pla ad prsrvig th basic proprtis of hyprbolic pla s Fig.(5)ad Fig.(6). Fig. (5)

4 794 H. I. Attiya Fig. (6) () foldig which fold th hyprbolic pla to a subst of hyprbolic pla s Fig.(7)ad Fig.(8). Fig.(7)

5 Foldig of hyprbolic maifolds 795 Fig.(8) () Lt f : H A which A is a subst of H with boudary s Fig.(9)ad Fig.(0). Fig.(9)

6 796 H. I. Attiya Thorm (): Proof: Fig.(0) Th limit of foldigs of th hyprbolic pla ito itslf of th first typ is a poit. This foldig which fold th hyprbolic pla ito itslf is dcliatio of th hyprbolic pla ad dcliatio of lis ad ca fold th hyprbolic pla ay mor as follows, H f f f f f H H H po it s Fig. () modl ()

7 Foldig of hyprbolic maifolds 797 Thorm(): modl () Fig.() Th imag of th scod ad th third typ of foldigs of th hyprbolic pla is ot hyprbolic pla. Proof: Lt f : H N is a foldig of th hyprbolic pla H to subst of itslf N of th scod typ th imag of this foldig ot satisfis th proprtis of th hyprbolic pla. Ad lt f : H A which is a foldig of th third typ which fold th hyprbolic pla H to a subst A with boudary th th imag of this foldig A is a disk ot hyprbolic pla. Foldig of fuzzy hyprbolic pla: is A map ~ ~ ~ ~ ~ f : H H such that H H ad f µ = µ as fallows: ( i) j

8 798 H. I. Attiya Thorm (): Th isomtric foldig of Rimaia maifolds icrasig th dimsios but th foldig of hyprbolic pla prsrv th dimsios. Proof: Lt f : M N b a isomtric foldig, M, N ar two Rimaia maifolds, th dimsios M dimsios N. If g: H H is isomtric foldig from hyprbolic pla H to aothr o H, th dimsios H = dimsios, from th dfiitio of lgth i hyprbolic pla. H AB AB Au = Log if A = B Bu = Log= 0, Ad th aothr th dfiitio of lgth i hyprbolic pla AB A u = Log B u B v A v AB = Log= 0 No rprstatio of a li to b a hyprbolic spac. Thorm(4): Proof: Th foldig of th hyprbolic pla cosidr as th rtractio of it. Lt r : H H, H H r : H H... r : H H + Th r f, thr ar homomorphisms h i i i such that:

9 Foldig of hyprbolic maifolds 799 r r r r H H H H L r H h h h h4 h + f f f f H H H H L f + H such that h + o r = f o h Rfrcs [] M. El-Ghoul : Foldig of maifolds, ph. D, Thsis, Tata Uiv., Egypt, (985). [] M. El-Ghoul : Ufoldig of Rimaia maifold, Commu, Fac. Sic. Uivrsity of Akara. 7, -4, (988). [] M. El-Ghoul : Foldig of fuzzy graphs ad fuzzy sphrs, Fuzzy Sts ad Systms, Grmay, 58, 55-6, (99). [4] M. El-Ghoul : Foldig of fuzzy torus ad fuzzy graphs, Fuzzy Sts ad Systms, Grmay, 80, 89-96, (995). [5] M. El-Ghoul : Th limit of foldig of a maifold ad its dformatio rtract, Joural of Egyptia, Mathmatic Socity, 5, -40, (997) [6]M. El-Ghoul: Th most gral st chaos graph: Chaos, solitos ad UK,,8-88, (00). [7]M. El-Ghoul, H.I. Attiya: Th dyamical fuzzy topological spac:th fuzzy mathmatics Vol. No., Los Agls, U.S.A,685-69,(004). [8] E.M. El-Kholy: Isomtric ad topological foldig of maifold, ph. D. Thsis uivrsity of South Thampto, UK (98). [9] S. A. Robrtso, Isomtric Foldig of Rimaia Maifolds, Proc. Roy. Soc. Ediburgh, 77, 75-89, (977). Fractals, Joural of Rcivd: April, 0