On Tiling for Some Types of Manifolds. and their Folding

Transcription

1 Appled Mathematcal Scences, Vol. 3, 009, no. 6, On Tlng for Some Types of Manfolds and ther Foldng H. Rafat Mathematcs Department, Faculty of Scence Tanta Unversty, Tanta Egypt Abstract In ths paper we ntroduce new types of tlng for some manfolds whch have coverng spaces. The relatons between tlngs of manfolds and ther coverng space are dscussed. Theorems governng these relatons are acheved. Some applcaton n medcne, chemstry and envronment are presented. Mathematcs Subjected Classfcaton: 5H0, 57N0 Keywords: Manfolds, foldng, tlng Introducton In Grunbaum and shephard s book Tllngs and Patterns [7], a tllng s defned as a countable famly of closed sets (tles) that cover the plane wthout gaps or overlaps. Tlngs are known as tesselatons or pavngs, they have appeared n human actvtes snce prehstorc tmes. Ther mathematcal theory s mostly elementary, but nevertheless t contans a rch supply of ntrestng problems at varous levels. The foldng of a manfold was, frstly ntroduced by S.Robertson n 977 [8]. Whle the unfoldng of a manfold appeared n [3]. Snce then many authors have studed some types of condtonal foldngs of manfolds such as n [,,4-6]. Defntons () A tlng of the plane s a collecton of closed topologcal dscs (tles) whch covers the Eucldean plane Ε and s such that the nterors of the tles are dsjont [7].

2 76 H. Rafat () Let M and M be arcwse connected, locally arcwse connected space, and let P:M M be contnuous. The par ( M,P ) s called a coverng space of M f: () P s surjectve, and () For each x M, there exst open sets U n M contanng x such that p - ( U) s a dsjont unon of open sets, each of whch s mapped homeomorphcally onto U by P. Such an open set U wll be called admssble [9]. (3) let M and N be two smooth manfolds of dmenson m and n respectvely. A map f:m N s sad to be sometrc foldng of M nto N f for every pecewse geodesc path υ :I M the nduced path f υ :I N, Is a closed nterval [0,], s pecewse geodesc and of the same length as υ [ 8] f does not preserve length t s called topologcal foldng [4]. (4) let M and N be two smooth manfolds of the same dmenson. A map g:m N s sad to be unfoldng of M nto N f every pecewse geodesc path υ :I M, the nduced path go υ :I Ns pecewse geodesc wth length grater than υ [ 3]. o. If Man results To obtan the man results n ths artcle, we wll ntroduce the followng defntons: () Any tlng of any fundamental regon F of a space S correspondng to a tlng for the space S, as n Fg.() Fundamental regon Fg. () From ths fgure, f any tlng t for F. Then, there are nduced tlngs t, t for the space S whch s smlar to t. Ths type of tlng s called the fundamental tlng. () There are two types of tlngs for the torus, the frst s a tlng by patch and the second s tlng by a fnte cylnder, whch s called the cylndrcal tlng, see Fg.().

3 On tlng for some types of manfolds 77 Second type Fg. () Theorem (): Any tlng of a manfold homeomorphc to S nduce an n- fundamental tlng for t coverng space, where n. Proof π r Let P:R S be a coverng projecton where P( r) e R,P s a coverng space of S, as n Fg.(3). =, then ( ) Fg. (3) If we consder any tlng of S ts nterors are open ntervals, then we get an nduced tlng to the fundamental regon for the space R. Also, for any tlng of R ts nterors are open ntervals for an nfnte number of closed bounded ntervals, whch nduced a tllng for S. Thus the followng dagram s commutatve..e., top Pot. Corollary (): The -fundamental tlng of a coverng space of a manfold M, whch homeomorphc to S s not correspondng to a tlng for M. Proof: Consder R s coverng S and tlng of R by ts -fundamental regon then S s not a tlng, as n Fg.(4).

4 78 H. Rafat Fg.(4) Theorem (): The foldng of a manfold M, whch s homeomorphc to S n (foldng of t tles) correspondng to foldng wthout sngular pont for the fundamental regon of ts coverng space (smlar to t). Proof: Let f be defned by f ( a cosθ, a snθ) ( b cosθ, b snθ) where a>b, whch nduce a type of condtonal foldng of S to S as n Fg. (5). Fg. (5). Then we have any fundamental tlng of the folded crcle S. There s an nduced tllng for ts coverng space, for all =,...,n. But n the lmt case lm S n = a pont, whch s not a coverng space for n dagram. R. Thus, we get the followng.e., P o f fop +, =,,...,n. Whch means that the dagram s commutatve for all =,...,n. But n the lmt case the dagram s not commutatve. Theorem (3): The unfoldng of a manfold M, whch homeomorphc to S n (unfoldng of ts tles) correspondng to unfoldng for the fundamental regon of ts coverng space up to. unf : a cosθ, a snθ b cosθ, b snθ where Proof: Let unf be gven by ( ) ( ) a<b, whch nduced unfoldng of S to S, see Fg.(6).

5 On tlng for some types of manfolds 79 Fg.(6). Then we have any fundamental tlng of the unfolded crcle S nduced a tlng of ts coverng space, for all =,...,n. Bur n the lmt case lms = R, whch s a n coverng space for R. Thus the followng dagram s obtaned. n.e., P ounf u n fop +, =,,...,n. Whch means that the dagram s commutatve for all. Theorem (4): The foldng whch preserves the tlng but does not preserve the curvature of the coverng space s the same tlng foldng of M whch s homeomorphc to S. n Proof: If we consder the foldng whch preserves the curvature of R, gven by F:R R, where the curvature of R s not equal to zero. Then any tlng for R s correspondng to tlng of S n, as n Fg. (7) below. Fg. (7).

6 80 H. Rafat Thus the followng dagram s obtaned. P o F FoP +, =,,...,n. Corollary (): The convers of the above theorem s true. Theorem (5): The foldng of the coverng space wth sngular ponts correspondng to foldng of M whch s homeomorphc to S wth sngular ponts. n Proof: Let f :M M, M s homeomorphc to S s a foldng from S nto tself, does not homeomorphc to S, lke f:a ( c o s θ, a s n θ) ( a c o s θ, a s n θ ), see Fg.(8). Fg. (8) Then, the tlng of foldng of S nto a subspace of S, whch s not homeomorphc to S, not nduces a tlng for ts coverng space. Theorem (6): The two types of fundamental tllng for a tours of ts coverng space. Proof: If Χ = T= S = S, Χ = R, P:Χ Χ be gven by ( ) ( nr nr = ), then (,P) P r,r e,e Χ s a coverng space of Χ, as n Fg. (9). Fg. (9).

7 On tlng for some types of manfolds 8 If we consder any tlng of T, then we get an nduced tlng to the fundamental regon of the space R. Also, for any tlng of the fundamental regon of R nduced tlng to T. Thus the followng dagram s commutatve..e., t o P Pot. Theorem (7): The foldng of the fundamental regon of the torus by cylndrcal tllng does not nduce foldng wth sngular ponts to ts coverng space. Proof: From the followng fgures we can see that the foldng s not a coverng space, but the sum f () t s a coverng space, see Fg.(0).. Fg. (0). Theorem (8): The frst type of followng a tours wthout sngular ponts s equvalent to foldng of the fundamental regon of ts coverng space. Proof: Let f be gven by f: ( acosψ+ b ) cosψ, ( a cosψ+ b ) snψ, a snψ = ( acosψ + b ) cosψ, ( acosψ+ b ) snψ, asnψ, o < ψ < x and o<ψ x where a < a x, then ths type of foldng nduced a foldng of the fundamental regon of ts coverng space, as n Fg.().

8 8 H. Rafat Fg. (). Theorem (9): The unfoldng of a torus s equvalent to unfoldng of the fundamental regon of ts coverng space. Proof: Let u n f be defned by unf { ( a cosψ+ b) cosψ, ( a cosψ+ b) snψ, a snψ } = { ( acosψ + b ) cosψ, a cosψ + b snψ, a snψ ( ) } where a >a, b >b, o<ψ < x and o<ψ < x. Then we get the unfoldng of the fundamental regon of ts coverng space, see Fg.().

9 On tlng for some types of manfolds 83 Thus the followng dagram s obtaned Fg. (). Corollary (3): The sectonal curvature of the fundamental regon s an nvarent. Applcatons: The medcal study of any secton of human body as blood, tssues, bones etc., we can take a part of these or any cells of lver. Take a secton f these, whch are homogenous then t wll gve the medcal general study on all parts of body.. In chemcal reacton to determne the propertes of some substance, the labor takes an sotropc secton from the medum, from homogenety; any character covers the whole medum. In the envronment the polluton dscussed from a homogeneous part ar, sea,, etc. any homogeneous medum. References [] P.DI-Francesco: Foldng and colorng problem n mathematcs and physcs, Bulletn of the Amercan Mathematcal Socety, Vol. 37, No. 3, (000),

10 84 H. Rafat [] A. E. El-Ahmady: Fuzzy foldng of fuzzy hoxocycle, Crcolo Mathematca Palerme, seres II, Tomol III, (004), ( [3] M. El-Ghoul: Unfoldng of Remannan manfolds, commum, Fac. Sc. Unv. Ankara. Seres A37, (998), -4,. [4] M. El-Ghoul: the deformaton retract of the complex projectve space and ts topologcal foldng, Journal of materal scence (30), England, (995), [5] M. El-Ghoul: Fractonal foldng of a manfold, Chaos, soltons and Fracton, UK, Vol., (00), 09-03,. [6] M. El-Ghout, A. E. El-Ahmady H. Rafat: Foldng retracton of chaotc dynamcal manfold and the VAK of vacuum fluctaton, Choos, soltons, and Froctals, UK, Vol. 0, (004), [7] B. Grunbaum and G.C. Shephard: Tllngs and patterns, New York, WH Freeman, 987. [8] S. A Robertson: Isometrc foldng of Remannan manfolds, Proc. Roy S.C. Ednburgh 77, (977), [9] J. M. Snger and J. A. Thorp: Lecture notes on elementary topology and geometry, Sprnger-Verlag, New York, 967. Receved: November, 008