Inextensible flows of S s surfaces of biharmonic

Transcription

1 Inxnsibl flows of s surfacs of biharmonic -curvs accordin o abban fram in Hisnbr Group His Tala Körpinar Es Turhan Fira Univrsiy, Dparmn of Mahmaics 9, Elazi, Turky alakorpinar@mailcom (Rcivd 0 April 0, accpd 9 Jun 0) Absrac In his papr, w sudy inxnsibl flows of s surfacs accordin o abban fram in h Hisnbr roup His W characriz h biharmonic curvs in rms of hir odsic curvaur w prov ha all of biharmonic curvs ar hlics in h Hisnbr roup His Finally, w find xplici paramric quaions of on paramr family of s surfacs accordin o abban Fram Kywords: Enry, Binry, Biharmonic curv, Hisnbr roup, s surfac Rsumn En s rabajo, sudiamos los flujos inxnsibls d s suprficis d acurdo al marco d abban n l rupo His d Hisnbr Caracrizamos las curvas biarmónicas n érminos d su curvaura odésica y dmosramos qu odas las curvas biarmónicas son hélics n l His rupo d Hisnbr Finalmn, nconramos cuacions paraméricas xplícias d familia d parámro uno d suprficis s qu concurdan con l marco d abban alabras clav: Enría, Binría, curva biarmónica, rupo d Hisnbr, suprfici s AC: 00-a, 065Fd IN I INTRODUCTION hysically, inxnsibl curv surfac flows iv ris o moions in which no srain nry is inducd Th swinin moion of a cord of fixd lnh, for xampl, or of a pic of papr carrid by h wind, can b dscribd by inxnsibl curv surfac flows uch moions aris qui naurally in a wid ran of physical applicaions Firsly, harmonic maps ar ivn as follows: Harmonic maps f :( M, ) ( N, h) bwn Rimannian manifolds ar h criical poins of h nry E( f ) = df v, M hy ar hrfor h soluions of h corrspondin Eulr-Laran quaion [,,, This quaion is ivn by h vanishin of h nsion fild ( f ) rac τ = df condly, biharmonic maps ar ivn as follows: Th binry of a map f by E ( f ) = τ ( f ) v, M say ha is biharmonic if i is a criical poin of h binry Jian drivd h firs h scond variaion formula for h binry in [5, showin ha h Eulr-Laran quaion associad o is E f N ( ) = J ( ( )) = Δ ( ) ( ( )) τ f τ f τ f rac R df, τ f df =0, f J is h Jacobi opraor of f Th quaion τ ( f ) whr = 0 f is calld h biharmonic quaion inc J is linar, any harmonic map is biharmonic [6, 7 In his papr, w sudy inxnsibl flows of s surfacs accordin o abban fram in h Hisnbr roup His W characriz h biharmonic curvs in rms of hir odsic curvaur w prov ha all of \ biharmonic curvs ar hlics in h Hisnbr roup His Finally, w find xplici paramric quaions of on paramr family of Fram s surfacs accordin o abban La Am J hys Educ ol 6, No, Jun 0 50 hp://wwwlajpor

2 Tala Körpinar Es Turhan II THE HEIENBERG GROU HEI Hisnbr roup His can b sn as h spac R ndowd wih h followin muliplicaion: ( x, yz, )( xyz,, ) = ( x+ xy, + yz, + z xy+ xy) (5) His is a hr-dimnsional, conncd, simply conncd -sp nilpon Li roup [8 Th Rimannian mric is ivn by = dx + dy + ( dz xdy) Th Li albra of His has an orhonormal basis =, = + x, =, (6) x y z z for which w hav h Li producs wih W obain [, =,[, = [, = 0 (, ) = (, ) = (, ) = = = = 0, = =, (7) = =, = = Th componns {R ijkl} of R rlaiv o {,, } ar dfind by ( ) R = R(, ), R = R(,,, ) = R(, ), ijk i j k ijkl i j k l i j l k Th non vanishin componns of h abov nsor filds ar R =, R =, R =, R =, R = R = III BIHARMONIC -CURE ACCORDING TO ABBAN FRAME IN THE HEIENBERG GROU HEI L γ : I His b a non odsic curv on h Hisnbr roup His paramrizd by arc lnh L { TNB,, } b h Frn fram filds ann o h Hisnbr roup His alon γ dfind as follows: T is h uni vcor fild γ ann o γ, N is h uni vcor fild in h dircion of T (normal o γ ), B T is chosn so ha { TNB,, } is a posiivly orind orhonormal basis [9, 0, Thn, w hav h followin Frn formulas: T T = N, N T = T + τ B, (8) B T = τ N, Whr k is h curvaur of γ τ is is orsion [, ( TT) ( NN) ( BB), =,, =,, =, ( TN) ( TB) ( NB), =, =, = 0 Now w iv a nw fram diffrn from Frn fram, [,,, 5, 6 L α : I b uni spd sphrical His curv W dno σ as h arc-lnh paramr of α L us dno ( σ) = ασ ( ), w call ( σ ) a uni ann vcor of α W now s a vcor s( σ ) = ασ ( ) ( σ) alon α This fram is calld h abban fram of α on h Hisnbr roup His Thn w hav h followin sphrical Frn-rr formula of α : α =, = α + s, (9) =, s whr is h odsic curvaur of h curv α on h His R =, R =, R =, ( ) ( αα) ( ss), =,, =,, =, (, ) (, s) (, s) α = = α = 0 La Am J hys Educ ol 6, No, Jun 0 5 hp://wwwlajpor

3 Inxnsibl flows of s surfacs of biharmonic -curvs accordin o abban fram in Hisnbr Group His Wih rspc o h orhonormal basis {,, }, w can Th s surfac of γ is a ruld surfac wri α = α + α + α, σ,u = ασ + us σ ( ) ( ) ( ) = + +, (0) s= s + s + s To spara a biharmonic curv accordin o abban fram from ha of Frn- rr fram, in h rs of h papr, w shall us noaion for h curv dfind abov as biharmonic -curv Thorm α : I is a biharmonic -curv if His only if = consan 0, + = [ s + [ αs, = αs + [ α roof U (5) abban formulas (9), w hav Lmma ([9) α : I curv if only if His is a biharmonic - = consan 0 + [ [ s α s = +, = αs [ α, Thn h followin rsul holds ( σ, u, ) Dfiniion A surfac voluion is flow ar said o b inxnsibl if is firs fundamnal form { EFG,, } saisfis E F G = = = 0 Dfiniion W can dfin h followin on-paramr family of dvlopabl ruld surfac ( σ, u, ) = ασ (, ) + us ( σ, ) Hnc, w hav h followin horm (5) Thorm L b on-paramr family of h s surfac of a uni spd non-odsic biharmonic -curv Thn is inxnsibl if only [ [ ( ) E M σ + M + ( ) σ+ + ( ) = [ E cos[ M M [ cos E 0, [ [E cos[ M σ + M E ( M + cos E ) cos[ M σ + M + M His Thorm ([9), All of biharmonic ar hlics -curvs in + [ [ E [ M σ + M ( M + cos E ) E + [ M I INEXTENIBLE FLOW OF s σ + M + M + [ [cose σ URFACE OF BIHARMONIC -CURE (6) ACCORDING TO ABBAN FRAME IN THE HEIENBERG GROU HEI ( ) σ + M( ) [ M( ) σ + M( ) E E To spara a s surfac accordin o abban fram from ha of Frn- rr fram, in h rs of h E( ) E( ) papr, w shall us noaion for his surfac as s [ [ M σ + M + M [ cos[ M σ + M surfac Th purpos of his scion is o sudy s surfacs of biharmonic -curv in h Hisnbr roup His La Am J hys Educ ol 6, No, Jun 0 5 hp://wwwlajpor ( )

4 Tala Körpinar Es Turhan M ( ) + + [ σ + + = M E M M M 0, ( ) = ( ( ) ) [ σ + ( ) σ E M M + ( ( ) ) E( ) cos[ M( ) σ+ M( ) + ( ( ) ) cose( ) whr M M, M, M, M + = ( cos E ) E ar smooh funcions of im = + E roof Assum ha ( σ, u, ) b a on-paramr family of h s surfac of a uni spd non-odsic biharmonic -curv From our assumpion, w h followin quaion ( ) ( ) σ ( ) + ( ) M σ + M + c E = E [ M + M E cos[ os whr MM, ar smooh funcions of im Obviously, w also obain (7) s = [ E cos[ M σ + M ( M + cos E ) E cos[ M σ + M + M + [ E [ M σ + M ( M + cos E ) E + [ M σ + M + M (8) σ + M [ Mσ + M + [coseσ E E E E [ [ Mσ + M + M[ cos[ Mσ + M + M M E M M + M, + [ σ + whr MM, ar smooh funcions of im + M = ( cos E ) ( ) = + E E { } F urhrmor, w hav h naural fram ( ),( ) σ ivn by u La Am J hys Educ ol 6, No, Jun 0 5 hp://wwwlajpor = [ E cos[ M σ + M ( M + cos E ) u ( ) E ( ) cos[ M σ + M + M + [ E [ M σ + M ( M + cos E ) E ( ) + si n[ M σ + M + M (9) ( ) σ + M ( ) + [cose σ E [ M σ + M E E( ) E( ) M σ M M cos[ M σ + M M ( ) σ [ [ + + [ + M + E [ M + M + M Th componns of h firs fundamnal form ar E = ( ( ),( ) ) = [( ) E σ σ [ M σ + M + [ ( ) E cos[ M σ + M + [ ( ) cos E, F = 0, G = (( ),( ) ) = [ [E u u E ( ) σ + M cos[ M ( M + cos E ) cos[ M σ + M + M + [ [ E [ M σ + M ( M + cos E )

5 Inxnsibl flows of s surfacs of biharmonic -curvs accordin o abban fram in Hisnbr Group His E ( ) cose( ) ( ) σ + M ( ) ( ) ( ) [ M σ + M + [ M σ + M + M (0) + σ E ( ) σ + M( ) [ M( ) σ + M( ) E + [ [cose σ E M E ( ) + E [ M σ + M + M + M, E( ) E( ) σ cos[ M σ + M [ [ M + M + M [ ( ) M + M + E [ M σ + M + M Hnc, is inxnsibl if only if Eq (6) is saisfid This concluds h proof of horm whr M, M, M, M M + = ( cos E ) E ar smooh funcions of im = + E roof By h abban formula, w hav h followin quaion Thorm L b on-paramr family of h s surfac of a uni spd non-odsic biharmonic - curv Thn, h paramric quaions of his family ar E( ) E( ) σ M cos[ M σ + M x ( σ, u, ) = cos[ M + + u[ E ( M + cos E ) cos[ Mσ + M + M + M, E M σ M M σ + M E y ( σ, u, ) = [ + + u[ La Am J hys Educ ol 6, No, Jun 0 5 ( ) [ E ( ) ( M + cos E ) + [ M σ + M + M +M ( ) σ + M ( ) z ( σ, u, ) = cose( ) σ E [ M σ + M E M ( ) + E [ M σ + M + u[cose + E E ( cos[ M( ) σ + M( ) + M( ) )cos[ M( ) σ + M( ) E + u[ ( M + cos E ) [ M σ + M, ( ) ( ) ( ) ( ) ( ) M + E = E [ M σ + M + E cos[ M σ + cos U () in (0), w obain ( ) ( ) σ ( ) ( ) ( ) σ ( ) = ( E [ M + M, E cos[ M + M, E ( ) + σ + + M ( ) ) cos[ M( ) σ + M ( ) ) cose E ( cos[ M M whr M, M ar smooh funcions of im Consqunly, h paramric quaions of can b found from, This concluds h proof of Thorm W can us Mahmaica in abov horm, yilds (a) hp://wwwlajpor

6 Tala Körpinar Es Turhan (b) FIGURE (a) (b) Th quaion is illusrad colour Rd, Blu, urpl, Oran, Mana, Cyan, Yllow, Grn a h im =, =, =, = 6, = 8, =, =, =, rspcivly REERENCE [ Caddo, R Monaldo,, Biharmonic submanifolds of, Inrna J Mah, (00) [ Chn, B Y, om opn problms conjcurs on submanifolds of fini yp, oochow J Mah 7, (99) m [ Dimiric, I, ubmanifolds of E wih harmonic man curvaur vcor, Bull Ins Mah Acad inica 0, 5-65 (99) [ Ells, J Lmair, L, A rpor on harmonic maps, Bull London Mah oc 0, -68 (978) [5 Jian, G Y, -harmonic maps hir firs scond variaional formulas, Chins Ann Mah r A 7, 89-0 (986) [6 Loubau, E Monaldo,, Biminimal immrsions in spac forms, prprin, 00, mahdg/0050 v [7 ruik, D J, Lcurs on Classical Diffrnial Gomry, (Dovr, Nw York, 998) [8 Rahmani,, Mriqus d Lornz sur ls roups d Li unimodulairs, d dimnsion rios Journal of Gomry hysics 9, 95-0 (99) [9 Körpinar, T Turhan, E, Biharmonic -Curvs Accordin o abban Fram in Hisnbr Group His, Bol oc aran Ma, 05- (0) [0 Turhan, E, Körpinar, T, On Characrizaion of Timlik Horizonal Biharmonic Curvs In Th Lornzian Hisnbr Group His, Zischrif für Naurforschun A- A Journal of hysical cincs 65a, 6-68 (00) [ Turhan, E, Körpinar, T, On Characrizaion Canal urfacs around Timlik Horizonal Biharmonic Curvs in Lornzian Hisnbr Group His, Zischrif für Naurforschun A- A Journal of hysical cincs 66a, -9 (0) [ Izumiya, Takuchi, N, pcial Curvs Ruld urfacs, Conribuions o Albra Gomry, 0- (00) [ Babaarslan, M Yayli, Y, Th characrizaions of consan slop surfacs Brr curvs, Inrnaional Journal of h hysical cincs 6, (0) [ Hlavay,, Diffrnial Lin Gomry, ( Nordhoff Ld, Groninn, 95) [5 O'Nill, B, mi-rimannian Gomry, (Acadmic rss, Nw York, 98) [6 Ravani, B, Wan, J W, Compur aidd d of lin consrucs, AME J Mch Ds, 6-7 (99) La Am J hys Educ ol 6, No, Jun 0 55 hp://wwwlajpor